Potential and Flux Field Landscape Theory of Spatially
Inhomogeneous Non-Equilibrium Systems
A Dissertation presented
by
Wei Wu
to
The Graduate School
in Partial Fulfillment of the
Requirements
for the Degree of
Doctor of Philosophy
in
Physics
Stony Brook University
December 2014
Stony Brook University
The Graduate School
Wei Wu
We, the dissertation committee for the above candidate for the
Doctor of Philosophy degree, hereby recommend
acceptance of this dissertation
Jin Wang - Dissertation Advisor
Associate Professor, Departments of Chemistry and Physics
Philip B. Allen - Chairperson of Defense
Professor, Department of Physics and Astronomy
Thomas K. Allison
Assistant Professor, Departments of Chemistry and Physics
Huilin Li
Professor, Department of Biochemistry and Cell Biology
Stony Brook University
This dissertation is accepted by the Graduate School
Charles Taber
Dean of the Graduate School
ii
Abstract of the Dissertation
Potential and Flux Field Landscape Theory of Spatially
Inhomogeneous Non-Equilibrium Systems
by
Wei Wu
Doctor of Philosophy
in
Physics
Stony Brook University
2014
In this dissertation we establish a potential and flux field landscape theory
for studying the global stability and dynamics as well as the non-equilibrium
thermodynamics of spatially inhomogeneous non-equilibrium dynamical systems.
The potential and flux landscape theory developed previously for spatially homogeneous non-equilibrium stochastic systems described by Langevin and FokkerPlanck equations is refined and further extended to spatially inhomogeneous nonequilibrium stochastic systems described by functional Langevin and Fokker-Planck
equations. The probability flux field is found to be crucial in breaking detailed
balance and characterizing non-equilibrium effects of spatially inhomogeneous
systems. It also plays a pivotal role in governing the global dynamics and formulating a set of non-equilibrium thermodynamic equations for a generic class of
spatially inhomogeneous stochastic systems. The general formalism is illustrated
by studying more specific systems and processes, such as the reaction diffusion
system, the Ornstein-Uhlenbeck process, the Brusselator reaction diffusion model,
and the spatial stochastic neuronal model. The theory can be applied to a variety
of physical, chemical and biological spatially inhomogeneous non-equilibrium
systems abundant in nature.
iii
Contents
List of Figures
vii
List of Tables
viii
Acknowledgements
ix
1
Introduction
1.1 Global Stability and Dynamics . . . . . . . . . . . . . . . . . . .
1.2 Non-Equilibrium Thermodynamics . . . . . . . . . . . . . . . . .
1.3 Scope and Structure . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
5
6
2
Global Stability and Dynamics of Non-Equilibrium Systems
2.1 Global Stability and Dynamics of Spatially Homogeneous NonEquilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Deterministic and Stochastic Dynamics of Spatially Homogeneous Systems . . . . . . . . . . . . . . . . . . . .
2.1.2 Potential and Flux Landscape Theory for Spatially Homogeneous Non-Equilibrium Systems . . . . . . . . . . . .
2.1.3 Lyapunov Function Quantifying the Global Stability of Spatially Homogeneous Non-Equilibrium Systems . . . . .
2.1.4 An Illustrative Example . . . . . . . . . . . . . . . . . .
2.2 Global Stability and Dynamics of Spatially Inhomogeneous NonEquilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Method of Formal Extension . . . . . . . . . . . . .
2.2.2 Deterministic and Stochastic Dynamics of Spatially Inhomogeneous Systems . . . . . . . . . . . . . . . . . . . .
2.2.3 Potential and Flux Field Landscape Theory for Spatially
Inhomogeneous Non-Equilibrium Systems . . . . . . . .
8
iv
8
9
12
16
24
33
34
35
40
2.2.4
2.3
2.4
3
Lyapunov Functional Quantifying the Global Stability of
Spatially Inhomogeneous Non-Equilibrium Systems . . .
Reaction Diffusion Systems . . . . . . . . . . . . . . . . . . . . .
2.3.1 Dynamics of Reaction Diffusion Systems . . . . . . . . .
2.3.2 Potential and Flux Field Landscape for Reaction Diffusion Systems . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Lyapunov Functional Quantifying the Global Stability of
Reaction Diffusion Systems . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Non-Equilibrium Thermodynamics of Stochastic Systems
3.1 Non-Equilibrium Thermodynamics for Spatially Homogeneous Stochastic Systems with One State Transition Mechanism . . . . .
3.1.1 Stochastic Dynamics . . . . . . . . . . . . . . . . . . . .
3.1.2 Generalized Potential-Flux Landscape Framework . . . .
3.1.3 Non-Equilibrium Thermodynamic Context . . . . . . . .
3.1.4 State Functions of Non-Equilibrium Isothermal Processes
3.1.5 Thermodynamic Laws of Non-Equilibrium Isothermal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.6 Summary and Discussion . . . . . . . . . . . . . . . . . .
3.1.7 Extension to Systems with One General State Transition
Mechanism . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Non-Equilibrium Thermodynamics for Spatially Homogeneous Stochastic Systems with Multiple State Transition Mechanisms . .
3.2.1 Stochastic Dynamics for Multiple State Transition Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Potential-Flux Landscape Framework for Multiple State
Transition Mechanisms . . . . . . . . . . . . . . . . . . .
3.2.3 Non-Equilibrium Thermodynamics for Multiple State Transition Mechanisms . . . . . . . . . . . . . . . . . . . . .
3.2.4 Necessary and Sufficient Condition for the Collective Definition Property . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Ornstein-Uhlenbeck Processes of Spatially Homogeneous
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Non-Equilibrium Thermodynamics for Spatially Inhomogeneous
Stochastic Dynamical Systems . . . . . . . . . . . . . . . . . . .
3.3.1 Description of Spatially Inhomogeneous Systems . . . . .
3.3.2 Stochastic Dynamics of Spatially Inhomogeneous Systems
v
45
49
49
55
56
59
61
61
62
64
69
71
80
91
95
99
100
102
104
109
112
121
122
125
3.3.3
3.4
4
Potential-Flux Field Landscape Framework for Spatially
Inhomogeneous Systems . . . . . . . . . . . . . . . . . .
3.3.4 Non-Equilibrium Thermodynamics of Spatially Inhomogeneous Systems . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Ornstein-Uhlenbeck Processes of Spatially Inhomogeneous
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.6 Spatial Stochastic Neuronal Model . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion
128
131
140
149
164
166
Bibliography
168
A Proof of Several Equations
179
B Proof of the Necessary and Sufficient Condition for the Collective Definition Property
181
C Ornstein-Uhlenbeck Processes for Spatially Homogeneous Systems
183
D Abstract Representation and Representation Transformation
189
D.1 Abstract Representation . . . . . . . . . . . . . . . . . . . . . . . 189
D.2 Representation Transformation . . . . . . . . . . . . . . . . . . . 194
E Ornstein-Uhlenbeck Processes for Spatially Inhomogeneous Systems 202
E.1 Dynamical Equations in the Abstract Representation . . . . . . . 202
E.2 Thermodynamic Expressions in the Space Configuration Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
F Spatial Stochastic Neuronal Model
vi
208
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
Streamlines of the deterministic driving force . . . . . . . .
Steady state probability distribution for D = 1 . . . . . . . .
Potential landscape for D = 1 . . . . . . . . . . . . . . . .
Streamlines of the negative gradient of potential landscape
D=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Streamlines of the flux velocity . . . . . . . . . . . . . . . .
The intrinsic potential landscape as the Lyapunov function .
Probability transport dynamics . . . . . . . . . . . . . . . .
. . .
. . .
. . .
for
. . .
. . .
. . .
. . .
25
27
28
29
30
31
40
Temporal profile of the system’s (renormalized) transient entropy
S, cross entropy U and relative entropy A in the process of relaxing to the equilibrium state in the spatial stochastic neuronal
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
vii
List of Tables
1.1
Comparison of different approaches . . . . . . . . . . . . . . . .
2.1
2.2
2.3
2.4
Types of (non-)equilibrium dynamics . . . . . . . . . . . . . . .
Lyapunov function and stability of a fixed point . . . . . . . . .
(non-)equilibrium dynamics . . . . . . . . . . . . . . . . . . .
Characterization of chemical reactions in the Brusselator model
viii
.
.
.
.
4
14
17
44
52
Acknowledgements
I am grateful for my advisor Prof. Jin Wang’s assistance, support and advice
during my academic endeavor as a Ph.D student. I am very glad to have been given
the opportunity to work on a series of related research projects in a systematic way.
What I have learned in these experiences has no doubt become a treasure for me
and will continue to be cherished in my future scientific pursuit.
I am thankful to the members (present and former) in our research group, who
have provided an inviting and stimulating research environment. They include
Nathan Borggren, Yuan Yao, Zhedong Zhang, Chunhe Li, Cong Chen, Zaizhi Lai,
Haidong Feng, Zhiqiang Yan, Bo Han, Xiaosheng Luo, Ruonan Lin, Fang Liu,
Weixin Xu, Ronaldo Olviera, Qiang Lu, and Jeremy Adler.
I am also deeply indebted to my family for their unconditional support.
The text of this dissertation in part is a reprint of the materials as it appears in
the following publications.
W. Wu and J. Wang, Potential and flux field landscape theory. I. Global stability and dynamics of spatially dependent non-equilibrium systems, The Journal
of Chemical Physics, 139, 121920 (2013). Copyright 2013, AIP Publishing LLC.
W. Wu and J. Wang, Potential and flux field landscape theory. II. Non-equilibrium
thermodynamics of spatially inhomogeneous stochastic dynamical systems, The
Journal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIP Publishing LLC.
Permission to use these publications in this dissertation is granted by AIP Publishing LLC. The co-author listed in these publications directed and supervised the
research that forms the basis for this dissertation.
ix
Chapter 1
Introduction
We live in a non-equilibrium world.1 Non-equilibrium dynamical systems
constantly exchanging matter, energy and information with the environment are
ubiquitous in nature. Our earth is a non-equilibrium system constantly receiving
energy from the Sun. Our human body is a non-equilibrium system constantly
consuming energy for survival. Uncovering the principles and physical mechanisms underlying these activated processes is vital for understanding the physical,
chemical and biological non-equilibrium systems. In this dissertation we focus on
two aspects of non-equilibrium systems, namely the global stability and dynamics
as well as the non-equilibrium thermodynamics.
1.1
Global Stability and Dynamics
Significant efforts and progresses have been made on the study of the global stability and dynamics of non-equilibrium systems [1–14]. Many studies have
been focused on spatially homogeneous systems, where spatial dependence due
to spatial inhomogeneity is ignored. The deterministic dynamics of these systems
can be described by an ordinary differential equation of the system’s finite degrees
of freedom. When extrinsic or intrinsic stochastic fluctuations have to be taken
into account, the dynamics governing these non-equilibrium systems becomes s1
Most of the material in this chapter was originally co-authored with Jin Wang in two journal
articles. Reprinted with permission from W. Wu and J. Wang, The Journal of Chemical Physics,
139, 121920 (2013). Copyright 2013, AIP Publishing LLC. Reprinted with permission from W.
Wu and J. Wang, The Journal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIP
Publishing LLC.
1
tochastic. Stochastic Markovian dynamics, governed by Langevin, Fokker-Planck
and (discrete-state) master equations [15, 16], has found extensive applications in
studying such systems. For example, Langevin equations have been used to study
critical dynamics [17], chemical reactions [18–20], soft colloidal systems [21] and
cell migration [22]. Fokker-Planck equations have been employed in the study of
anomalous transport [23], plasma physics [24], chemical reaction processes [25]
and polymer dynamics [26]. Master equations have been utilized to model, for
instance, systems weakly coupled to heat baths [27], gas phase chemical kinetics [28] and complex reaction networks [29]. Besides Markovian dynamics, nonMarkovian dynamics with memory effects [30, 31] has also been used in some studies, among which is the generalized Langevin equation [32] employed to study
non-equilibrium molecular dynamics [33] and its further generalization utilized to
investigate chemical reactions taking place in changing environments [34].
It is a well-known and readily observable fact that spatial inhomogeneity is
characteristic of non-equilibrium systems in the physical world. Some typical examples of non-equilibrium spatially inhomogeneous systems with spatial-temporal
dynamics of pattern formation and self-organization [1,35–37] are Rayleigh-Bénard
convection in fluids [38], Turing pattern in chemical morphogenesis [39], Drosophila embryo differentiation in developmental biology [40], and plant distributions in
the population dynamics of ecological systems [41]. These systems have been
studied using the reaction diffusion equation, where the spatial dependence of the
system is accounted for on the mean field level, ignoring intrinsic and extrinsic
stochastic fluctuations. The mean field description can give local dynamics and
local stability analysis for these systems. However, it cannot address the global
nature of the system, such as global stability, dynamics and robustness. Besides,
it is well known that in some cases fluctuations can play an important role where
the mean field description breaks down. A typical example is when the system
is approaching the critical point of a continuous phase transition where the correlation length diverges. Therefore a stochastic description is needed to account
for the effect of fluctuations. Large classes of spatially inhomogeneous systems with stochastic fluctuations can be described by functional Langevin equations
(including stochastic partial differential equations) [1,15,31,35–37,42–47], functional Fokker-Planck equations [1, 15, 31, 35–37, 45–51] and spatial master equations [1, 15, 51–53]. They have been used to study various problems in physics,
chemistry and biology [1,35–37,43–60], such as surface growth [58], electromagnetic field propagation [59], reaction diffusion of proteins in Drosophila embryo
development [51] and signal transduction of neurons [60].
For stochastic spatially inhomogeneous systems described by functional Langevin
2
equations and spatial master equations, standard procedures such as the JanssenDe Dominicis formalism [45–47] and Doi-Peliti formalism [61–65] have been developed to map these equations to field theory formalisms. Hence, field-theoretic
techniques developed in quantum field theory and statistical mechanics, such as
diagrammatic perturbation expansion and renormalization group [66–70], have
been employed to study these systems, in complementary to other methods such
as exact solutions [71–74], the Smoluchowski theory [75, 76] and computer simulations [77–80]. The standard and usually perturbative field-theoretic approaches,
although are able to incorporate the effect of fluctuations systematically, still cannot address the issues of global stability and global dynamics of the system.
An alternative approach is based on the concept of non-equilibrium potential.
This concept is pioneered by R. Graham and coworkers for stochastic systems described by Fokker-Planck equations in the small fluctuation limit [81, 82], further
developed by G. Hu for stochastic systems described by master equations in the
small fluctuation limit [83,84], and constructed alternatively by P. Ao for stochastic systems governed by Langevin equations (no requirement of small fluctuation
limit but with certain restrictions) [85]. The non-equilibrium potential in the small fluctuation limit is a Lyapunov function of the deterministic system, which
characterizes the global stability of the deterministic system in terms of the topography of the potential landscape (e.g., basin of attractions and barrier heights).
But it is not a Lyapunov function of the stochastic system with finite fluctuations
that quantifies the global stability of the stochastic system. The non-equilibrium
potential approach was extended and applied by R. Graham and H. S. Wio et al to
some specific spatially inhomogeneous systems, where the exact form of the nonequilibrium potential functional can be obtained [48, 86, 87]. This has proven to
be a useful complementary approach to other field-theoretic approaches. Recently, the non-equilibrium potential landscape framework has been established for
general spatially inhomogeneous systems (including general reaction diffusion
systems) with intrinsic fluctuations governed by spatial master equations [51].
What has also been realized for non-equilibrium systems is that the steady state probability distribution (closely related to the potential landscape) alone is not
complete in characterizing the non-equilibrium steady state [88], nor is it complete in determining the dynamics of the system [9, 81, 82, 89, 90]. The steady
state probability flux (also called the curl flux due to its divergence-free property)
must be considered together with the probability distribution (or equivalently the
potential landscape) to complete the whole picture for non-equilibrium systems. Energy input or pump to the system has been found to be one way to create
nonzero probability curl flux [91]. For spatially homogeneous non-equilibrium
3
systems, we have established a potential and flux landscape theory, which integrates the potential landscape and the curl flux together, to quantify the global
stability and global dynamics of the system [9–12, 14]. We have shown that the
effective driving force of the spatially homogeneous non-equilibrium dynamics
can be explicitly decomposed into a gradient-like term of the potential landscape
and a curl-like term of the probability flux. While the potential landscape attracts
the system down along the gradient similar to electrons moving in an electric field,
the probability flux drives the system in a curly way much like a magnetic field
acting on electrons. In the small fluctuation limit nonzero probability flux allows
for the existence of continuous attractors such as limit cycles and even possibly
chaotic strange attractors in the corresponding deterministic dynamics [9–14]. For
spatially inhomogeneous non-equilibrium systems, the indispensable probability
flux also needs to be brought out explicitly together with the potential landscape,
to give a complete characterization of the global stability and global dynamics of
spatially inhomogeneous non-equilibrium systems. This provides an alternative
and complementary approach to the current field-theoretic methods. It also forms
an extension of the non-equilibrium potential approach by explicitly introducing
and exploiting the probability curl flux, capable of studying the global dynamics of the system. Thus it may offer insights not accessible by other approaches.
In Table 1.1 we list the relevant methods and approaches mentioned and indicate
whether they are able to incorporate stochastic fluctuations and capable of studying the global stability and global dynamics of the system.
Fluctuations
Global
Stability
Global
Dynamics
Mean
Field
Theory
N
Perturbative
Field-Theoretic
Techniques
Y
Non-Equilibrium
Potential
Approach2
Y
Potential and
Flux Landscape
Theory
Y
N
N
Y
Y
N
N
N
Y
Table 1.1: Comparison of different approaches
2
The non-equilibrium potential is based on stochastic dynamics. Yet in the small fluctuation
limit stochastic fluctuations are completely suppressed. Thus the “Y(es)” for stochastic fluctuations is not clear-cut. Also, the “N(o)” for global dynamics is a reserved one, since there do exist
formulations in this respect even though it is not fully brought out in this approach.
4
The study of global stability and dynamics in the context of the potential and
flux landscape theory forms one major topic of this dissertation. Detailed explanations, discussions and illustrations are given in chapter 2. In particular, the
Brusselator reaction diffusion model is studied in Sec. 2.3 using the potential and
flux landscape theory, demonstrating its advantage over other approaches.
1.2
Non-Equilibrium Thermodynamics
Complementary to the dynamic aspect is the thermodynamic aspect of nonequilibrium systems. The relation between dynamics and thermodynamics has
been at the heart of controversy since the establishment of thermodynamics as
a scientific theory. Following Onsager’s seminal work [92], a theory of nonequilibrium thermodynamics in the linear regime based on the local equilibrium
hypothesis was systematically developed by Prigogine [93] and many other researchers [94], which was further applied and extended to a variety of directions
and areas [95–102]. In the past few decades a non-equilibrium thermodynamic framework based on stochastic Markovian dynamics, termed stochastic thermodynamics generally, has emerged [1, 2, 7, 11, 12, 103–137]. Much effort has
been devoted to constructing a consistent non-equilibrium thermodynamics, using methods of statistical mechanics, from the underlying stochastic dynamics,
described by master equations as well as Langevin and Fokker-Planck equations,
for steady states and also transient processes [1, 2, 7, 11, 12, 103–125]. The resulting work has been applied, for instance, in the study of gene network [126] in
biochemical systems [127]. The realization that thermodynamic quantities can be
defined for stochastic trajectories has directed the development of this field onto
the more refined single trajectory level [128–130]. (The term ‘stochastic thermodynamics’ also refers specifically to this refined non-equilibrium thermodynamics.) Its combination with the fluctuation theorem proves fruitful [131, 132].
It has been realized that the second law of thermodynamics has multiple facets
[114, 116–119]. Stochastic thermodynamics is now emerging as a powerful tool
for nanosciences [134], with applications in molecular motors [135] and chemical
oscillation systems [136] among others.
Stochastic thermodynamics is a work still in progress, with some remaining
issues to be addressed. We mention three of them that are relevant to the present
work. First, stochastic thermodynamics deals with both stochastic dynamics and
non-equilibrium thermodynamics. Yet the connection between these two levels
does not seem to be transparent. It would be nice to have a ‘bridge’ connect5
ing these two levels explicitly. Second, it has been noted that entropy production could be underestimated if physically different state transition mechanisms
(constituent processes, reservoirs, reaction channels etc.) are not identified correctly [116–118, 124]. Although a systematic presentation has been given for
Fokker-Planck equations with multiple (state) transition mechanisms on an essentially one-dimensional state space [118], a more general formulation is not yet
available. This is not just a trivial extension. Also, it is not yet clear under what
conditions the results for one transition mechanism may also apply to multiple
transition mechanisms. Third, spatially inhomogeneous systems have been largely unexplored in the context of stochastic thermodynamics, although there do exist
a few studies in this context [129, 137]. Since spatial inhomogeneity is typical of
non-equilibrium systems, this issue is obviously important.
This brings out the other major topic of this dissertation to be discussed in
chapter 3, where we address the above three issues related to non-equilibrium
thermodynamics within the context of the potential and flux landscape theory, by
synthesizing, extending and transcending some of the essential results developed previously in the framework of stochastic thermodynamics. The key feature in
our approach is that we use the potential and flux landscape framework as a bridge
to construct the non-equilibrium thermodynamics from the underlying stochastic
dynamics. From the potential and flux landscape and the associated dynamical decomposition equations, which are based on the stochastic dynamics, we construct
a set of non-equilibrium thermodynamic equations quantifying the relations of the
non-equilibrium entropy, entropy flow, entropy production, and other thermodynamic quantities. On both the dynamic level and thermodynamic level, we find
that the probability flux plays a central role. We will give detailed explanations,
discussions and illustrations with specific examples in chapter 3.
1.3
Scope and Structure
The systems considered in this dissertation are spatially homogeneous and
inhomogeneous systems, described by Langevin and Fokker-Plank dynamics (including their deterministic limit), on a one or multiple (including infinite) dimensional state space. We do not consider Fokker-Planck equations derived from master equations by truncating the Kramers-Moyal expansion [18,138] which usually
models systems with intrinsic fluctuations. We shall consider even state variables
under time reversal. Also, the non-equilibrium thermodynamics is covered on
the ensemble average level. These restrictions define a manageable scope of an
6
extensive subject for this dissertation to deal with and will be lifted in future work.
The rest of this dissertation is structured as follows. In chapter 2 we investigate the global stability and dynamics of non-equilibrium systems in terms of
the potential and flux landscape theory, which is a consolidation of the work in
Ref. [50]. In chapter 3 we address the thermodynamics of non-equilibrium systems in the context of the potential and flux landscape theory based on the work
of Ref. [139]. The conclusion of this dissertation is given in chapter 4.
7
Chapter 2
Global Stability and Dynamics of
Non-Equilibrium Systems
In this chapter we study the global stability and dynamics of spatially homogeneous and inhomogeneous non-equilibrium systems within the potential and
flux landscape framework.1 We first in Sec. 2.1 review and refine our previously
established potential and flux landscape theory for quantifying the global stability and dynamics of spatially homogeneous systems [9, 12]. Then in Sec. 2.2 we
extend the potential and flux landscape theory to spatially inhomogeneous nonequilibrium systems. Our extended theory is then applied to reaction diffusion
systems and, in particular, the Brusselator reaction diffusion model in Sec. 2.3.
Finally, we give a summary of this chapter in Sec. 2.4.
2.1
Global Stability and Dynamics of Spatially Homogeneous Non-Equilibrium Systems
First, we introduce the deterministic and stochastic dynamics spatially homogeneous non-equilibrium systems. Then we present and expand on the potential
and flux landscape framework developed previously for spatially homogeneous
non-equilibrium systems governed by Fokker-Planck equations [9, 10, 13, 14]. In
particular, we investigate the force decomposition equation in this framework and
1
Much of the material in this chapter was originally co-authored with Jin Wang. Reprinted with permission from W. Wu and J. Wang, The Journal of Chemical Physics, 139, 121920
(2013). Copyright 2013, AIP Publishing LLC. Yet the figures and tables in this chapter are all
new. Sec. 2.1.4 is new. Sec. 2.1.3 and Sec. 2.3 contain new materials.
8
reveal various other meanings of this equation by looking at it from multiple perspectives. Then we review and discuss a general method for uncovering the Lyapunov functions quantifying the global stability of the deterministic and stochastic
spatially homogeneous non-equilibrium dynamical systems described by FokkerPlanck equations [1–12]. Finally, we study a specific dynamical system in detail
to illustrate the general theory presented in this section.
2.1.1
Deterministic and Stochastic Dynamics of Spatially Homogeneous Systems
We discuss the deterministic dynamics first and then the stochastic Langevin
dynamics and the corresponding Fokker-Planck dynamics.
Deterministic Dynamics
The state of a spatially homogeneous system can usually be characterized by
a state vector ⃗q = (q1 , ..., qi , ..., qn ), representing the system’s degrees of freedom under study. The concrete meaning of the state vector and its components is
system-specific. For instance, the state vector of a spatially homogeneous (wellstirred) chemical reaction system may represent the collection of the concentrations of all the chemical species involved in the system at each moment; the concentration of each chemical species is a component of the state vector.
The deterministic dynamics of an autonomous spatially homogeneous system
in general can be described by an ordinary differential equation of the state vector:
d
⃗q = F⃗ (⃗q),
dt
(2.1)
where F⃗ (⃗q) is the deterministic driving force of the system. Mathematically, it is a
vector field on the state space of the system (i.e., the space of ⃗q). In general, F⃗ (⃗q)
may come from the contribution of several different sources. If those sources are
identifiable in the studied system, the driving force can be written in a decomposed
form in terms of different contributing sources (labeled below by the index r):
∑
F⃗ (⃗q) =
F⃗r (⃗q),
(2.2)
r
where F⃗r (⃗q) represents the deterministic driving force from source r. For example, the deterministic driving force of chemical-reaction systems can be decomposed in terms of different chemical reactions (labeled by r) [52]: F⃗ (⃗q) =
9
∑ ⃗
∑
q) =
νr wr (⃗q), where F⃗r (⃗q) = ⃗νr wr (⃗q) is the deterministic driving
r Fr (⃗
r⃗
force from chemical reaction r. Here, ⃗νr is the vector whose components are the
stoichiometric coefficients of chemical reaction r. And wr (⃗q) is the rate of chemical reaction r, dependent on the concentrations of the chemical species involved.
Langevin Dynamics
When stochastic fluctuations are present which change the state of the system
stochastically, the dynamics of the system becomes stochastic. In many cases, the
stochastic dynamics of the system is governed by a Langevin equation [15]:
d
⃗ q , t),
⃗q = F⃗ (⃗q) + ξ(⃗
dt
(2.3)
⃗ q , t) is the stochastic driving force in addition to the deterministic drivwhere ξ(⃗
⃗ q , t) can be decomposed according to statistically
ing force F⃗ (⃗q). In general, ξ(⃗
independent fluctuation sources (labeled by s) [15, 140] :
∑
⃗ q , t) =
⃗ s (⃗q)Γs ( t),
ξ(⃗
G
(2.4)
s
where Γs ( t) are Gaussian white noises with the following statistical property:
< Γs (t) >= 0,
< Γs (t)Γs′ (t′ ) >= δss′ δ(t − t′ ).
(2.5)
⃗ s (⃗q) characterizes the direction and strength of the sIn Eq. (2.4) the vector G
tochastic driving force from source s, while the Gaussian noise Γs (t) characterizes its stochastic nature. Stochastic fluctuations from different sources labeled
by different s are statistically independent as shown by the second equation in
Eq. (2.5). From Eqs. (2.4) and (2.5) we can derive the statistical property of the
total stochastic driving force from the contributions of all the fluctuation sources:
⃗ q , t) >= 0,
< ξ(⃗
⃗ q , t)ξ(⃗
⃗ q , t′ ) >= 2D(⃗q)δ(t − t′ ),
< ξ(⃗
(2.6)
where D(⃗q) as one half of the fluctuation correlator is the diffusion matrix (the
name is justified shortly in the Fokker-Planck dynamics) given by:
1∑⃗
⃗ s (⃗q).
D(⃗q) =
Gs (⃗q)G
(2.7)
2 s
D(⃗q) is a nonnegative-definite symmetric square matrix by construction. It accounts for the combined effects of stochastic fluctuations from all the fluctuation
sources as indicated by its expression in Eq. (2.7).
10
Fokker-Planck Dynamics
The Lagevin equation in Eq. (2.3) traces a single stochastic trajectory in the
state space ⃗qt for a given initial state, which represents the stochastic dynamical
evolution of the state of the system with time. It is useful to know how the probability distribution Pt (⃗q) for the random state vector ⃗qt evolves with time instead of
individual stochastic trajectories. The dynamical evolution of the probability distribution corresponding to the Langevin dynamics in Eq. (2.3) (interpreted as an
Ito stochastic differential equation) is governed by a partial differential equation
known as the Fokker-Planck equation [15, 16]:
(
)
∂
⃗
Pt (⃗q) = −∇ · F (⃗q)Pt (⃗q) + ∇ · ∇ · (D(⃗q)Pt (⃗q)) .
(2.8)
∂t
Here ∇ = ∂/∂⃗q is the n-dimensional vector differential operator in the state space. The notation ∇ · ∇ · (D(⃗q)Pt (⃗q)) means two∑successive operations of ∇·
on D(⃗q)Pt (⃗q), which in its component form reads ab ∂a ∂b (Dab (⃗q)Pt (⃗q)), with
the short notation ∂a ≡ ∂/∂qa . F⃗ (⃗q) in Eq. (2.8) is called the drift vector in
the Fokker-Planck equation and is given by the deterministic driving force in the
Langevin equation. D(⃗q) in Eq. (2.8) is the diffusion matrix in the Fokker-Planck
equation and is given by Eq. (2.7) accounting for the effects of stochastic fluctuations in the Langevin equation.
The Fokker-Planck equation in Eq. (2.8) can be identified as a continuity equation representing probability conservation:
∂
Pt (⃗q) = −∇ · J⃗t (⃗q).
∂t
The (transient) probability flux is given by
J⃗t (⃗q) = F⃗ ′ (⃗q)Pt (⃗q) − D(⃗q) · ∇Pt (⃗q),
(2.9)
(2.10)
where F⃗ ′ (⃗q) = F⃗ (⃗q) − ∇ · D(⃗q) is the effective drift vector (or effective driving force), which is the original drift vector F⃗ (⃗q) modified by a diffusion-induced
drift vector −∇ · D(⃗q). The Fokker-Planck equation in the form of a continuity
equation allows for an interpretation of the Fokker-Planck dynamics as a probability transport dynamics in the state space. Probability is transported in the state
space through the probability flux which, according to Eq. (2.10), comes from the
contribution of a drift process in the state space characterized by the drift velocity
field F⃗ ′ (⃗q) and a diffusion process in the state space characterized by the diffusion
matrix D(⃗q). This also justifies the appropriateness of the names, drift vector and
diffusion matrix, used in the Fokker-Planck equation.
11
2.1.2
Potential and Flux Landscape Theory for Spatially Homogeneous Non-Equilibrium Systems
We consider the steady state Ps (⃗q) of the Fokker-Planck equation which satisfies ∂Ps (⃗q)/∂t = 0. Then Eqs. (2.9) and (2.10) become
∇ · J⃗s (⃗q) = 0,
(2.11)
J⃗s (⃗q) = F⃗ ′ (⃗q)Ps (⃗q) − D(⃗q) · ∇Ps (⃗q).
(2.12)
Equation (2.11) states the steady state probability flux J⃗s is divergence-free everywhere in the state space. Thus it is a solenoidal curl vector field like the magnetic
field. Its field line has no beginning or end; it either extends to infinity or forms a
closed loop, since the field has no sinks or sources due to the divergence-free condition. The deterministic driving force F⃗ (⃗q) governing the deterministic dynamics and the diffusion matrix D(⃗q) encoding the effects of stochastic fluctuations
together through Eqs. (2.11) and (2.12) determine the steady state probability distribution Ps (⃗q) and the steady state probability flux J⃗s (⃗q). Ps (⃗q) characterizes the
global stochastic steady state of the system, while J⃗s (⃗q) governs the global steady
probability transport dynamics of the system in the state space. Although Ps does
not change with time, there is still probability transport going on in the state space
when J⃗s ̸= 0, which does not change Ps as long as J⃗s is divergence-free.
The detailed balance condition in the steady state as the equilibrium condition of the system characterizing microscopic reversibility is represented by zero
steady state probability flux J⃗s = 0 [15]. (This is for even state variables ⃗q under
time reversal as is what we consider in this dissertation. Also, there are subtleties
and extensions regarding this condition which will be discussed in Secs. 3.1.2 and
3.2.2 in the next chapter.) This means there is no probability transport dynamics in
the state space. The global stochastic steady state and steady probability transport
dynamics (vanish) in this case are characterized by Ps alone. Equation (2.12) in
this case becomes F⃗ ′ (⃗q)Ps (⃗q) − D(⃗q) · ∇Ps (⃗q) = 0. Dividing Ps on both sides
of the equation and rearranging it lead to the potential condition [15]:
F⃗ ′ (⃗q) = −D(⃗q) · ∇U (⃗q),
(2.13)
where U ≡ − ln Ps (up to an additive constant) is the (dimensionless) equilibrium
potential landscape. It plays a double role. On the one hand U connects to the
steady state probability distribution via Ps = e−U . Therefore U is the ‘weight’
of the probability distribution. Its topography reflects the shape of the steady
12
state probability distribution. We can say the potential landscape U characterizes
(as Ps does) the global stochastic steady state and steady probability transport
dynamics (vanish) for equilibrium systems. On the other hand, it connects to the
deterministic driving force F⃗ governing the deterministic dynamics of the system
and the diffusion matrix D governing the stochastic fluctuation dynamics through
Eq. (2.13). For equilibrium systems with detailed balance the effective driving
force F⃗ ′ has the form of the gradient of the potential landscape U with respect to
the diffusion matrix D. F⃗ and D are connected to each other by the potential U
through Eq. (2.13), which places a constraint on them. When the diffusion matrix
is invertible, the potential condition can be explicitly expressed as a constraint
equation on F⃗ and D :
[
]
∇ ∧ D −1 (⃗q) · F⃗ ′ (⃗q) = 0,
(2.14)
where the operator ∇∧ is the generalization of the curl operator ∇× in 3 di⃗ ab = ∂a Ab − ∂b Aa . For stochastic
mensions to n dimensions, defined as [∇ ∧ A]
systems, the constraint on F⃗ and D due to the detailed balance condition means
the deterministic dynamics of the system governed by F⃗ and the stochastic fluctuation dynamics represented by D have to cooperate with each other in a specific
way to balance each other out and produce zero probability flux, thus maintaining reversibility of the system, although each of them individually may have a
non-zero contribution to the probability flux.
In general, when the deterministic dynamics and the stochastic fluctuation dynamics of the system are not cooperating with each other in the specific way that
produce the detailed balance condition characterized by vanishing steady state
probability flux, the steady state probability flux J⃗s can be non-zero. The non-zero
flux J⃗s indicates microscopic irreversibility that breaks detailed balance and drives
the system away from equilibrium. Dividing Ps (⃗q) on both sides of Eq. (2.12) and
rearranging it gives the ‘force decomposition equation’ [9, 10, 13, 14]:
F⃗ ′ (⃗q) = −D(⃗q) · ∇U (⃗q) + V⃗s (⃗q),
(2.15)
where U (⃗q) ≡ − ln Ps (⃗q) (up to an additive constant) is the generalized dimensionless non-equilibrium potential landscape and V⃗s (⃗q) ≡ J⃗s /Ps is the steady state
probability flux velocity. They are related to each other by
V⃗s (⃗q) · ∇U (⃗q) = ∇ · V⃗s (⃗q),
(2.16)
which is obtained from Eq. (2.11), ∇ · J⃗s = ∇ · (e−U V⃗s ) = 0. The steady state
probability flux velocity V⃗s is related to J⃗s through V⃗s = J⃗s /Ps . In general V⃗s is
13
not divergence-free; thus it is generally not a solenoidal vector field. According
to Eq. (2.16) the necessary and sufficient condition for V⃗s to be divergence-free is
∇ · V⃗s (⃗q) = V⃗s (⃗q) · ∇U (⃗q) = 0, that is, V⃗s is perpendicular to the gradient of the
potential landscape U everywhere in the state space. When this condition is not
satisfied, V⃗s is not a solenoidal vector field while J⃗s still is. Yet since they are related to each other by J⃗s = Ps V⃗s , they are parallel to each other everywhere (except
for J⃗s = 0 or V⃗s = 0 where the concept of parallel fails). Therefore, their field
lines have the same curling shapes, though their field-line densities, conventionally representing their respective field magnitudes, may differ at different points
in the state space due to the factor Ps (⃗q). We summarize the (non-)equilibrium
dynamics in relation to flux and force in Table 2.1.
Detailed Balance
Detailed Balance Breaking
J⃗s
J⃗s = 0
J⃗s ̸= 0, ∇ · J⃗s = 0
V⃗s
V⃗s = 0
V⃗s ̸= 0, ∇ · V⃗s = V⃗s · ∇U
F⃗ ′ = −D · ∇U
F⃗ ′ = −D · ∇U + V⃗s
F⃗
′
Table 2.1: (Non-)equilibrium dynamics of spatially homogeneous systems2
For non-equilibrium systems, the effective driving force can be decomposed
into two terms according to Eq. (2.15). One term is the gradient (with respect to
the diffusion matrix D) of the potential landscape U (related to the steady state
probability distribution Ps = e−U ) which characterizes the global stochastic state
of the system. The other term is the curling steady state probability flux velocity
V⃗s (related to the steady state probability flux J⃗s = Ps V⃗s ) which represents nonequilibrium effects that break detailed balance. This is analogous to the motion of
an electron moving in a gradient electric field and a curl magnetic field. Alternatively, one may move the term ∇ · D within the effective driving force F⃗ ′ to the
right side of the equation and make a statement about the decomposition of the deterministic driving force F⃗ into three terms. In Sec. 2.1.3 we shall see the potential
landscape U in the small fluctuation limit is a Lyapunov function of the deterministic system, quantifying the global stability of the deterministic system. And the
potential landscape U and the flux velocity V⃗s together, in the small fluctuation
2
As has been noted previously, the relation between detailed balance (breaking) and probability
flux presented here in this chapter applies to even state variables and there are also subtleties and
extensions to be discussed in chapter 3.
14
limit, through the force decomposition equation, decide the deterministic driving force that governs the deterministic dynamics of the non-equilibrium system.
Therefore, for non-equilibrium spatially homogeneous systems, while the global
stability is quantified by the underlying potential landscape U closely related to
the steady state probability distribution, quantification of the global dynamics of
the non-equilibrium systems requires both the potential landscape U and the flux
velocity V⃗s , in contrast to equilibrium systems where U alone is enough.
The way Eq. (2.15) is written down and the above statements made in relation
to this equation is based on the perspective of seeing F⃗ (or F⃗ ′ ) as the force in this
equation, while seeing D, U and V⃗s as that which compose the force F⃗ (or F⃗ ′ ).
This perspective is natural when one is working in the small fluctuation limit and
connecting the stochastic dynamics to the deterministic dynamics of the system.
However, it is not the only perspective to look at this equation. Equation (2.15) on
its own is simply a statement of the relation between the four important quantities:
F⃗ (or F⃗ ′ ), D, U , and V⃗s . In this sense we can call it the Constitutive Equation of
the system. Grouping different quantities together allows one to look at Eq. (2.15)
from different perspectives. From a certain perspective, it is natural to group F⃗
(or F⃗ ′ ) and D together as they are both directly from the Langevin dynamics and
group U and V⃗s together as they are both quantities directly related to the characterization of the Fokker-Planck dynamics. The pair (U , V⃗s ) in a way serves as a
bridge between the pair (F⃗ , D) and the pair (Ps , J⃗s ). On the one hand, (U , V⃗s )
connects to (Ps , J⃗s ) through Ps = e−U and J⃗s = Ps V⃗s , which characterize the
global stochastic steady state and steady probability transport dynamics of nonequilibrium systems in the Fokker-Planck equation. On the other hand, (U , V⃗s )
connects through Eq. (2.15) to (F⃗ , D), which are the deterministic driving force
that governs the deterministic dynamics of the system and the diffusion matrix that
governs the stochastic fluctuation dynamics of the system, both coming from the
Langevin equation. Therefore, through the pair U and V⃗s , a bridge is built between
the Fokker-Planck dynamics governing the evolution of probability distributions and the Langevin dynamics governing the evolution of stochastic trajectories.
So from such a perspective, Eq. (2.15) is seen as the statement of the relation between two pairs of quantities: the pair (F⃗ , D) and the pair (U , V⃗s ). From another
perspective by grouping F⃗ , D and U together, Eq. (2.15) can be seen as a statement about the composition of V⃗s . It states that there two parts contributing to the
steady-state probability flux velocity that determines the steady-state probability
transport dynamics. One part is the drift process in the state space contributing
the drift velocityF⃗ ′ (⃗q). The other part is the diffusion process in the state space
contributing D(⃗q) · ∇U (⃗q), which specifies how the gradient of the potential U
15
and the diffusion matrix D together determine the diffusion contribution to the
probability flux velocity. Drift and diffusion together produce the total probability
flux velocity V⃗s that drives the probability transport dynamics in the state space. In
addition, in the context of non-equilibrium thermodynamics, Eq. (2.15) is closely
related to the entropy flow decomposition equation detailed in chapter 3. So it can
also be interpreted from that perspective. There could also be other possible perspectives. Therefore, the meaning of Eq. (2.15) is multi-dimensional, depending
on the perspective one looks at it and the context in which one studies it.
We can also derive a force decomposition equation in terms of transient quantities, from the definition of the transient probability flux in Eq. (2.10):
F⃗ ′ (⃗q) = −D(⃗q) · ∇S(⃗q, t) + V⃗t (⃗q),
(2.17)
where S(⃗q, t) ≡ − ln Pt (⃗q) and V⃗t (⃗q) ≡ J⃗t (⃗q)/Pt (⃗q). Subtracting Eq. (2.15) from
Eq. (2.17) leads to the following equation:
(
)
Pt (⃗q)
⃗
Vt (⃗q) = −D(⃗q) · ∇ ln
+ V⃗s (⃗q).
(2.18)
Ps (⃗q)
The meanings of two equations, Eq. (2.17) and Eq. (2.18), will be further investigated in chapter 3 and form part of an extension of the potential and flux landscape
framework to be presented in Sec. 3.1.2. Here we only mention that Eq. (2.18)
plays an important role in the proof of the Lyapunov property that quantifies the
global stability of the stochastic system with finite fluctuations.
2.1.3
Lyapunov Function Quantifying the Global Stability of
Spatially Homogeneous Non-Equilibrium Systems
We review and discuss a general method for uncovering Lyapunov functions quantifying the global stability of spatially homogeneous systems modeled by
Fokker-Planck equations [1–12]. We first give a brief review of the general Lyapunov function theory relevant to the current discussion.3 Then we discuss the
Lyapunov function for deterministic systems and finally for stochastic systems.
Lyapunov Function of Dynamical Systems
We first consider autonomous dynamical systems governed by an ordinary differential equation (ODE) ⃗q˙ = F⃗ (⃗q) on an n-dimensional Euclidean state space.
3
This material is new and does not exist in Ref. [50].
16
What is particularly interesting is the long-term behavior of a dynamical system
and the stability thereof. The simplest long-term behavior is a fixed point. The
stability of a fixed point can be studied by Lyapunov functions [141]. Without
loss of generality, assume the fixed point is at the origin ⃗q = 0, i.e., F⃗ (0) = 0.
A Lyapunov function is a continuous scalar function V(⃗q) defined on a region Ω
containing the fixed point with two additional properties. One is that V(⃗q) is positive definite, i.e., V(⃗q) ≥ 0 with equality if and only if ⃗q = 0. The other is that its
time derivative is negative semidefinite, i.e., V̇(⃗q) ≤ 0 for all ⃗q ∈ Ω\{0}. When
such a Lyapunov function exists, the fixed point ⃗q = 0 is (Lyapunov) stable, which
means a trajectory near the fixed point will indefinitely stay close to it. If the time
derivative of V(⃗q) is negative definite (stronger than negative semidefininte), i.e.,
V̇(⃗q) < 0 for all ⃗q ∈ Ω\{0}, then the fixed point ⃗q = 0 is asymptotically stable,
which means a trajectory near the fixed point will converge to it eventually. Further, if the Lyapunov function V(⃗q) is defined globally on the entire state space
Ω = Rn , radially unbounded (i.e., V(⃗q) → ∞ as ||⃗q|| → ∞), and its time derivative is globally negative definite (i.e., V̇(⃗q) < 0 for all ⃗q ∈ Rn \{0}), then the fixed
point ⃗q = 0 is globally asymptotically stable, which means all the trajectories will
converge to the fixed point eventually. We summarize these results in Table 2.2.
V(⃗q)
V̇(⃗q)
stable
V(⃗q) ≥ 0 (“=” iff ⃗q = 0)
V̇(⃗q) ≤ 0 ∀⃗q ∈ Ω\{0}
asymptotically
stable
V(⃗q) ≥ 0 (“=” iff ⃗q = 0)
V̇(⃗q) < 0 ∀⃗q ∈ Ω\{0}
globally
asymptotically
stable
V(⃗q) → ∞ (||⃗q|| → ∞),
V(⃗q) ≥ 0 (“=” iff ⃗q = 0)
V̇(⃗q) < 0 ∀⃗q ∈ Rn \{0}
Table 2.2: Lyapunov function and stability of a fixed point4
Fixed points are not the only type of long-term behavior a dynamical system
can exhibit. Other long-term behaviors include limit cycles and strange attractors.
For these more general behaviors, the LaSalle’s invariant set theorem generalizes
the above Lyapunov function theory [141]. An invariant set Ξ is a set that evolves
to itself under the dynamics, that is, ⃗q(0) ∈ Ξ implies ⃗q(t) ∈ Ξ for all t ≥ 0.5 The
4
5
The notation “iff” in the table is short for “if and only if”.
In some literatures this is defined as a positively invariant set.
17
LaSalle’s invariant set theorem is stated as follows [141].
LaSalle’s Theorem: Let Ω ⊂ Rn be a compact invariant set with respect to
the dynamics ⃗q˙ = F⃗ (⃗q). Let V(⃗q) be a continuously differentiable scalar function
defined on Ω such that V̇(⃗q) ≤ 0 in Ω. Let Λ be the set of all points in Ω where
V̇(⃗q) = 0. Let Θ be the largest invariant set in Λ. Then every trajectory starting in
Ω approaches Θ as t → +∞.
In the above theorem the function V(⃗q) is not required to be positive definite
as required for the Lyapunov function of a fixed point. It may still be called a
Lyapunov function in a generalized sense. The three sets in the theorem have the
relation Θ ⊆ Λ ⊆ Ω, where Θ and Ω are both invariant sets while Λ = {⃗q |V̇(⃗q) =
0} is not necessarily one. Also note that the set Ωc = {⃗q |V(⃗q) ≤ c} with V̇(⃗q) ≤ 0
in Ωc is an invariant set. Thus if we can find a function V(⃗q) with V̇(⃗q) ≤ 0 defined
on some domain not necessarily invariant, we can restrict that domain to Ωc for
some number c so that V(⃗q) is defined on an invariant set. If Ωc is also compact
(equivalent to bounded and closed for Euclidean space) then Ω in the theorem can
be taken as Ωc . In particular, if V(⃗q) is radially unbounded (with V̇(⃗q) ≤ 0), then
Ωc for any c can serve as Ω in the theorem.
In addition to finite dimensional dynamical systems governed by an ODE,
there are also dynamical systems described by partial differential equations or
more general functional differential equations on an infinite dimensional state
space. When the state space of an infinite dimensional dynamical system is a
Hilbert space (a linear space with an inner product that is complete) or more generally a Banach space (a complete normed linear space) which generalize Euclidean
space, the above stability results for finite dimensional dynamical systems in terms
of Lyapunov function (that for a fixed point and the LaSalle’s theorem) have been
extended in a pretty straightforward way to these infinite dimensional systems but
with some subtleties involved [142].6 We avoid these technical difficulties for infinite dimensional dynamical systems in this chapter and work on a more formal
level when dealing with these systems (more details given in Sec. 2.2.1).
Lyapunov Function of the Deterministic Dynamical System
It turns out that the leading order of the potential landscape of the stochastic
system in the small fluctuation limit gives a Lyapunov function of the corresponding deterministic system [81,82]. To investigate the system’s behavior under small
6
A subtle difference between an infinite dimensional linear space (e.g., Hilbert space or Banach
space) and a finite dimensional Euclidean space is that a subset of Euclidean space is compact if
and only if it is closed and bounded, while this is not true for an infinite dimensional linear space.
18
e q ), where
fluctuation conditions, we write the diffusion matrix as D(⃗q) = DD(⃗
the state-independent scale parameter D characterizes the magnitude of the diffue q ) is the re-scaled diffusion matrix.
sion matrix or the fluctuation strength and D(⃗
The corresponding steady-state Fokker-Planck equation then becomes
(
)
(
)
e q )Ps (⃗q) = 0.
− ∇ · F⃗ (⃗q)Ps (⃗q) + D∇ · ∇ · D(⃗
(2.19)
When D is small, the steady state probability distribution has the following asymptotic form [81, 82]:
[
]
∞
1
1
1 ∑ k (k)
Ps (⃗q, D) =
exp [−U (⃗q, D)] =
exp −
D U (⃗q) , (2.20)
Z(D)
Z(D)
D k=0
∫
where Z(D) = d⃗q exp [−U (⃗q, D)] is the normalization factor. Inserting Eq. (2.20) into Eq. (2.19) and keeping only the leading order of D leads to the
Fokker-Planck Hamilton-Jacobi equation [81, 82]:
e q ) · ∇U (0) (⃗q) = 0, (2.21)
HF P (∇U (0) , ⃗q) = F⃗ (⃗q) · ∇U (0) (⃗q) + ∇U (0) (⃗q) · D(⃗
where
U (0) (⃗q) = lim D U (⃗q, D) = lim {−D ln[Ps (⃗q, D)Z(D)]}
D→0
D→0
(2.22)
is the leading-order potential landscape called the intrinsic potential (also termed
the non-equilibrium potential). Equation (2.21) is of the same form as the HamiltonJacobi equation with zero energy in mechanical systems. The corresponding
Hamiltonian of the system reads:
e q ) · p⃗,
HF P (⃗p, ⃗q) = p⃗ · F⃗ (⃗q) + p⃗ · D(⃗
(2.23)
where p⃗ = ∇U (0) is the canonical momentum conjugate to ⃗q. The deterministic
dynamics ⃗q˙ = F⃗ (⃗q) forms a subdynamics (⃗p = 0) of the Hamilton’s canonical
equations determined by the Hamiltonian HF P in Eq. (2.23) [81, 82].
There does not always exist a global single-valued continuously-differentiable
function U (0) (⃗q) as a solution of the Fokker-Planck Hamilton-Jacobi equation. For
non-equilibrium systems U (0) (⃗q) may lose its differentiability under certain conditions though it is still continuous [81, 82]. When the continuously differentiable
solution U (0) of Eq. (2.21) exists, it gives a Lyapunov function of the corresponding deterministic system, which is shown in the following. Due to the relation
19
Ps = e−U and Eq. (2.22), U (0) is bounded from below if Ps is normalizable. Also,
if natural boundary conditions are used for Ps , then U (0) is radially unbounded.
Its time derivative is calculated as follows:
d
d (0)
e q )·∇U (0) (⃗q) ≤ 0,
U (⃗q) = ⃗q·∇U (0) (⃗q) = F⃗ (⃗q)·∇U (0) (⃗q) = −∇U (0) (⃗q)·D(⃗
dt
dt
where we have used the deterministic equation and the nonnegative-definite prope q ). Therefore, according to the discussion
erty of the scaled diffusion matrix D(⃗
of the LaSalle’s theorem, the intrinsic potential U (0) is a Lyapunov function of
the deterministic dynamical system. It can thus be used to study the global (also
e q ) is strictly positive-definite,
local) stability of the deterministic system. If D(⃗
(0)
then U (⃗q) will cease decreasing only when ∇U (0) (⃗q) = 0. Therefore the set
Λ = {⃗q |U̇ (0) (⃗q) = 0} is simply the set of the extremum points of U (0) (⃗q). The
asymptotic behaviors of the deterministic system (fixed points, limit cycles, or strange attractors) are thus contained in the set of the extremum points of U (0) (⃗q)
according to the LaSalle’s theorem. In particular, attractors correspond to minima of U (0) (⃗q). Local minima correspond to locally stable attractors, while global
minima correspond to globally stable attractors [81, 82].
Furthermore, the Fokker-Planck Hamilton-Jacobi equation in Eq. (2.21) is equivalent to the following two equations [81, 82]:
e q ) · ∇U (0) (⃗q) + V⃗ (0) (⃗q),
F⃗ (⃗q) = −D(⃗
s
(2.24)
V⃗s(0) (⃗q) · ∇U (0) (⃗q) = 0.
(2.25)
(0)
V⃗s
is a vector field to be
In fact, we can always write down Eq. (2.24) where
determined. Then we plug it into Eq. (2.21) and we can derive Eq. (2.25). We
remark here that Eq. (2.24) is simply the small fluctuation limit of the general
force decomposition equation in Eq. (2.15). Comparing Eq. (2.24) with Eq. (2.15),
we can see the diffusion-induced force ∇ · D in Eq. (2.15) vanishes in the small
fluctuation limit in Eq. (2.24) so that F⃗ ′ becomes F⃗ . The potential U in Eq. (2.15)
(0)
is only left with the lowest order U (0) in Eq. (2.24). V⃗s (⃗q) in Eq. (2.24) is able
to be identified as the zero-order steady state probability flux velocity in the small
fluctuation limit. It is named as the intrinsic steady state probability flux velocity
(or intrinsic flux velocity for short). Equation (2.25) is a result of Eq. (2.16) in the
(0)
small fluctuation limit. It states that the intrinsic flux velocity V⃗s is perpendicular
to the gradient of the intrinsic potential U (0) . This is because the lowest order of
the divergence term ∇ · V⃗s is one less than the term V⃗s · ∇U (notice there is a D−1
in U ). This property is not generally true for systems with finite fluctuations.
20
Therefore, in the small fluctuation limit, the potential landscape U and the
steady state probability flux velocity V⃗s that serve as a bridge between the FokkerPlanck dynamics (governing the global probability transport dynamics) and the
Langevin dynamics (governing the single stochastic trajectory dynamics) give a
(0)
global description of the deterministic system in terms of U (0) and V⃗s . The intrinsic potential landscape U (0) , related to the stochastic steady state probability
(0)
distribution in the small fluctuation limit through Ps ∼ exp[−U (0) /D], is the
Lyapunov function of the deterministic non-equilibrium system that quantifies its
global stability. The intrinsic potential landscape U (0) together with the intrinsic
(0)
curling probability flux velocity V⃗s , related to the intrinsic steady-state prob(0)
(0) (0)
ability flux by J⃗s = Ps V⃗s , determine through Eq. (2.24) the driving force
F⃗ (⃗q) which governs the deterministic dynamics of the non-equilibrium system.
While the topography of the intrinsic potential landscape guides the dynamics
down the gradient of the landscape, the intrinsic flux velocity drives the system to
flow around in a curling way. In particular, since the asymptotic dynamics of the
deterministic system takes place in the set of the extreme points of U (0) where the
gradient term vanishes ∇U (0) = 0, the driving force of the asymptotic dynamics
(0)
is thus purely the intrinsic flux velocity V⃗s as seen from Eq. (2.24). It is V⃗ (0) that
is responsible for the time-dependent asymptotic dynamics of, for instance, limit
cycles and strange attractors. For equilibrium systems, the intrinsic flux veloci(0)
ty V⃗s vanishes. The entire dynamics of the system is then determined only by
the gradient of the intrinsic potential landscape without the curling flux velocity
contribution. The asymptotic dynamics of equilibrium systems is therefore static
(0)
(fixed points) since it lacks the driving force V⃗s .
Lyapunov Function of the Stochastic Dynamical System
To investigate the Lyapunov function of spatially homogeneous dynamical
systems with finite fluctuations governed by Fokker-Planck equations, we introduce the concept of relative entropy (or Kullback-Leibler divergence) well known
in information theory [16, 115, 118, 143].7 The relative entropy of the stochastic
spatially homogeneous system with respect to the stationary probability distribu7
The relative entropy will also be discussed in chapter 3 in the context of non-equilibrium
thermodynamics, where it plays an integral role in the formulation of a set of non-equilibrium
thermodynamic equations. In this chapter, however, we are not concerned with its thermodynamic
meanings. We just remind the reader here to not confuse it with the regular entropy.
21
tion is defined as follows:
A[Pt (⃗q)] =
∫
(
)
Pt (⃗q)
d⃗q Pt (⃗q) ln
,
Ps (⃗q)
(2.26)
where Pt (⃗q) is the solution of the time-dependent Fokker-Planck equation (Eq. (2.8)) and Ps (⃗q) is the solution of the corresponding stationary Fokker-Planck
equation (Eqs. (2.11) and (2.12)). We assume Pt (⃗q) and Ps (⃗q) are both positive
functions normalized to one. The relative entropy quantifies the deviation of the
transient probability distribution Pt from the stationary probability distribution Ps .
It turns out to be a Lyapunov function(al) of the stochastic dynamical system.
We note that the stochastic dynamics here is governed by a linear partial differential equation (the Fokker-Planck equation) rather than an ordinary differential
equation. It can be regarded as an infinite dimensional dynamical system. The
(stochastic) state of the system is described by a probability distribution function
P (⃗q). The state space is thus an infinite dimensional function space, which can be
appropriately defined by specifying P (⃗q) to satisfy certain conditions (e.g., being
positive, normalized, smooth enough, with certain boundary conditions imposed,
with a metric or norm defined etc.). We assume the function state space forms a
Hilbert or Banach space so that the Lyapunov function theory can still be applied.
In this case, the Lyapunov function becomes a functional since it is a function of
the stochastic state P (⃗q) which in itself is a function. The stationary probability
distribution Ps (⃗q) satisfies the stationary Fokker-Planck equation. It is the analog
of a fixed point of an ODE. We study its stability by investigating the Lyapunov
property of the relative entropy defined in Eq. (2.26). We refer readers to Table
2.2 for the Lyapunov function theory in relation to a fixed point of an ODE.
To prove the Lyapunov property of A[Pt (⃗q)], we first show it is positive definite and then prove its time derivative is negative semidefinite [1, 2, 7, 12, 16, 100,
115, 118, 140, 144].
∫
Ps (⃗q)
A[Pt (⃗q)] = − d⃗q Pt (⃗q) ln
P (⃗q)
( t
)
∫
Ps (⃗q)
≥ − d⃗q Pt (⃗q)
−1
Pt (⃗q)
∫
∫
=
d⃗q Pt (⃗q) − d⃗q Ps (⃗q) = 0,
where we have used the inequality ln x ≤ x − 1 for x > 0 and that Ps (⃗q) and
Pt (⃗q) are both normalized to one. The equality holds if and only if x = 1, i.e.,
22
Pt (⃗q) = Ps (⃗q). Therefore, we have proven A[Pt (⃗q)] is positive definite. Then we
calculate its time derivative.
∫
∫
∂Pt (⃗q) Pt (⃗q)
∂Pt (⃗q)
d
A[Pt (⃗q)] =
d⃗q
ln
+ d⃗q
dt
∂t
Ps (⃗q)
∂t
∫
(
) ( P (⃗q) )
t
= − d⃗q ∇ · J⃗t (⃗q) ln
Ps (⃗q)
(
)
∫
P
(⃗
q
)
t
=
d⃗q J⃗t (⃗q) · ∇ ln
Ps (⃗q)
(
)
∫
P
(⃗
q
)
t
=
d⃗q Pt (⃗q)V⃗t (⃗q) · ∇ ln
P (⃗q)
]
(
[
(s
)
)
∫
Pt (⃗q)
P
(⃗
q
)
t
+ V⃗s (⃗q) · ∇ ln
=
d⃗q Pt (⃗q) −D(⃗q) · ∇ ln
Ps (⃗q)
Ps (⃗q)
(
)
(
)
∫
Pt (⃗q)
Pt (⃗q)
= − d⃗q Pt (⃗q)∇ ln
· D(⃗q) · ∇ ln
Ps (⃗q)
Ps (⃗q)
)
(
∫
Pt (⃗q)
,
+ d⃗q Pt (⃗q)V⃗s (⃗q) · ∇ ln
Ps (⃗q)
where we have used Eq. (2.18) and Gauss’s theorem. We assume boundary terms
vanish under appropriate boundary conditions. (This forms part of the definition
of the function state space.) The first term in the last equation is non-positive:
(
)
(
)
∫
Pt (⃗q)
Pt (⃗q)
− d⃗q Pt (⃗q)∇ ln
· D(⃗q) · ∇ ln
≤ 0,
(2.27)
Ps (⃗q)
Ps (⃗q)
since D(⃗q) is positive semidefinite and Pt (⃗q) is positive. The second term vanishes
because
(
)
∫
P
(⃗
q
)
t
d⃗q Pt (⃗q)V⃗s (⃗q) · ∇ ln
Ps (⃗q)
(
)
∫
Pt (⃗q)
⃗
=
d⃗q Ps (⃗q)Vs (⃗q) · ∇
Ps (⃗q)
)
(
∫
Pt (⃗q)
⃗
=
d⃗q Js (⃗q) · ∇
Ps (⃗q)
∫
(
) ( P (⃗q) )
t
⃗
= − d⃗q ∇ · Js (⃗q)
= 0,
Ps (⃗q)
where we have used ∇ · J⃗s = 0 and Gauss’ theorem. Again we assume certain
boundary conditions ensure the boundary terms vanish. Thus we have proven
23
the time derivative of A[Pt (⃗q)] is negative semidefinite (non-positive). Therefore
the relative entropy A[Pt (⃗q)] is a Lyapunov functional of the stochastic spatially
homogeneous dynamical system governed by the Fokker-Planck equation. The stationary distribution Ps (⃗q) is stable, which means a probability distribution P (⃗q)
near Ps (⃗q) will stay indefinitely close to it under the Fokker-Planck dynamics. If
the diffusion matrix D(⃗q) is positive definite, which is a stronger condition than
positive semidefinite, then the equality in dA[Pt (⃗q)]/dt ≤ 0 holds if and only if
Pt (⃗q) = Ps (⃗q) according to Eq. (2.27). This means the time derivative of A[Pt (⃗q)]
is negative definite. Therefore, in this case, Ps (⃗q) is asymptotically stable. In other
words, a probability distribution P (⃗q) near Ps (⃗q) will eventually evolve into the stationary distribution Ps . Further, P (⃗q) in the definition of A[P (⃗q)] in Eq. (2.26) is
arbitrary in an appropriately defined function state space. This suggests that Ps (⃗q)
is globally asymptotically stable,8 which would mean that starting from an arbitrary initial condition P (⃗q, t0 ) in the state space, the system will eventually evolve
into the stationary probability distribution Ps (⃗q) following the Fokker-Planck dynamics. This demonstrates how the relative entropy A[Pt (⃗q)] can be used to study
the stability of stochastic dynamical systems with finite fluctuations.
2.1.4
An Illustrative Example
To illustrate the general theory presented in previous sections, we study a two dimensional autonomous dynamical system, which is a particular case of the
normal form of the Hopf bifurcation [141]. Although the asymptotic behavior
of the deterministic dynamics of this system is known, the study of this system
(including the stochastic dynamics) using the potential and flux landscape theory
presented below is new.
Deterministic Dynamics
The state of the system is specified by the state vector (x, y)| . We assume the
deterministic dynamics of the system follows the following set of ODEs.
{
ẋ = (1 − x2 − y 2 )x − y
.
(2.28)
ẏ = (1 − x2 − y 2 )y + x
8
For a finite dimensional dynamical system described by an ODE, showing a fixed point to
be globally asymptotically stable requires the condition that the Lyapunov function is radially
unbounded. For an infinite dimensional dynamical system as is in this case, this condition may
become subtle and we leave it open.
24
In other words, the deterministic driving force is given by
(
)
2
2
(1
−
x
−
y
)x
−
y
F⃗ (x, y) =
.
(1 − x2 − y 2 )y + x
(2.29)
The stream lines of the vector field F⃗ (x, y) are plotted in Fig. 2.1. It shows quite
clearly the dynamical behavior of the system. As seen from Fig. 2.1, the driving
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Figure 2.1: Streamlines of the deterministic driving force
force flows out from the origin and approaches a circle around the origin from the
inside. The driving force outside the circle also flows towards and approaches the
circle from the outside. This suggests the system has an unstable fixed point at
the origin and a stable limit cycle around the origin. More rigorously, this system
has only one fixed point at the origin (x, y) = (0, 0), which is the only solution to
the set of algebraic equations obtained by setting the right side of Eq. (2.28) as 0.
Linear stability analysis shows this fixed point is unstable. In addition to the fixed
point, one can also prove there is a stable limit cycle on the unit circle:
{
x(t) = cos t
.
(2.30)
y(t) = sin t
This solution can be verified directly by plugging it into Eq. (2.28). Later on
we shall construct a Lyapunov function using the theory presented in previous
25
sections to identify and study the stability of the fixed point at the origin and the
limit cycle on the unit circle for this particular system.
Langevin Dynamics
We assume the system is influenced by external fluctuations such that the dynamics of the system can be modeled by the following Langevin equation:
{
ẋ = (1 − x2 − y 2 )x − y + ξ1 (t)
,
(2.31)
ẏ = (1 − x2 − y 2 )y + x + ξ2 (t)
where ξi (t) (i = 1, 2) are Gauss white noises with the statistical property:
< ξi (t) >= 0,
< ξi (t)ξj (t′ ) >= 2Dδij δ(t − t′ ),
(2.32)
for i, j = 1, 2. This means the two fluctuating forces ξ1 (t) and ξ2 (t) are statistically independent. Also, the fluctuation strength characterized by D is the same
for both fluctuating forces and is independent of the state of the system. These assumptions are reasonable for external fluctuations under certain conditions. The
diffusion matrix in this case is thus given by D(x, y) = DI, where I is 2×2 identity matrix. The solution of the Langevin equation for a specific initial condition
is a stochastic trajectory in the state space.
Fokker-Planck Dynamics
The Fokker-Planck equation governing the evolution of probability distributions which corresponds to the Langevin equation in Eq. (2.31) reads
)
∂
∂ (
Pt (x, y) = −
[(1 − x2 − y 2 )x − y]Pt (x, y)
∂t
∂x
)
∂ (
−
[(1 − x2 − y 2 )y + x]Pt (x, y)
∂y
)
( 2
∂
∂2
+D
+
Pt (x, y).
∂x2 ∂y 2
(2.33)
The steady state distribution Ps (x, y) satisfies the stationary Fokker-Planck equation which is obtained by setting the right side of Eq. (2.33) as zero. When the
deterministic driving force is a nonlinear function, in general, the Fokker-Planck
equation cannot be solved exactly, even for the steady state. However, for this
particular dynamical system we are studying, there is a rotational symmetry in the
26
system which allows us to obtain the exact form of the steady state distribution of
Eq. (2.33) [5, 81]. It is easy to verify directly that the probability distribution
{
}
)2
1
1 ( 2
2
Ps (x, y) =
exp −
x +y −1
(2.34)
Z(D)
4D
is the steady state of Eq. (2.33). Z(D) is the normalization factor given by
[
(
)]
3√
1
,
Z(D) = π 2 D 1 + erf √
2 D
∫z
2
where erf(·) is the error function defined as erf(z) = √2π 0 e−s ds. The steady
state probability distribution Ps (x, y) in Eq. (2.34) for D = 1 is plotted in Fig. 2.2.
Its shape is like a hat. The origin sinks in indicating a lower probability, while the
circle around the origin bulges up indicating a higher probability.
Figure 2.2: Steady state probability distribution for D = 1
Potential and Flux Landscape
With the steady state distribution Ps (x, y) obtained in Eq. (2.34), we now have
the expression of the potential landscape
U (x, y) = − ln Ps (x, y) =
)2
1 ( 2
x + y 2 − 1 + ln Z(D).
4D
27
(2.35)
Note that the potential landscape is defined up to a constant independent of the
state of the system. The term ln Z(D) only depends on D and does not depend
on x and y. Thus it has no essential influence on the potential landscape and
can be dropped. Fig. 2.3 shows the shape of the potential landscape. It is like a
Mexican hat. Contrary to the shape of the hat in Fig. 2.2, now the center bulges up
while the circle around the center sinks in. This is in agreement with the relation
U (x, y) = − ln Ps (x, y) and the shape of the steady state distribution shown in
Fig. 2.2.
Figure 2.3: Potential landscape for D = 1
Calculation of the gradient of U (x, y) gives
)
1 ( 2
∇U (x, y) =
x + y2 − 1
D
(
x
y
)
.
(2.36)
What is also convenient is to consider the negative gradient of U (x, y), i.e., −∇U .
For D = 1 its streamlines are plotted in Fig. 2.4. As can be seen in Fig. 2.4, −∇U
points in the radial direction. Inside the unit circle, it points outward away from
the origin and towards the unit circle. Outside the unit circle, it points inward and
towards the unit circle. Both regions are showing the behavior of pulling the state
of the system towards the unit circle.
Note that since D = DI, we have ∇ · D = 0. Thus the effective driving force
is simply the deterministic driving force in this case: F⃗ ′ = F⃗ − ∇ · D = F⃗ . The
28
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Figure 2.4: Streamlines of the negative gradient of potential landscape for D = 1
force decomposition equation in Eq. (2.15), F⃗ ′ = −D · ∇U + V⃗s , is then reduced
to F⃗ = −D∇U + V⃗s . Plugging in the expressions of F⃗ in Eq. (2.29) and ∇U in
Eq. (2.36), we then derive the expression of the steady state flux velocity
(
)
−y
⃗
⃗
Vs (x, y) = F (x, y) + D∇U (x, y) =
.
(2.37)
x
The fact that the flux velocity V⃗s (x, y) is not identically zero indicates that the
system is in non-equilibrium without detailed balance. This can be verified for
concrete systems that can be modeled by the dynamical system we are studying.
Further, one can prove directly that in this particular case the flux velocity V⃗s is
divergence-free (a curl field) and it is perpendicular to the gradient of the potential
landscape, that is, ∇ · V⃗s = V⃗s · ∇U = 0. (See Table 2.1.) The streamlines of
the flux velocity are plotted in Fig. 2.5. As we can see, the flux velocity points in
the transverse direction, perpendicular to the radial direction (the direction of the
gradient of the potential landscape). This agrees with the fact that V⃗s · ∇U = 0.
Also, the streamlines of V⃗s form closed circles. This is in agreement with the fact
that in this particular case V⃗s is a curl field satisfying ∇ · V⃗s = 0.
The force decomposition equation in this case F⃗ = −D∇U + V⃗s decomposes
the driving force of the system into two perpendicular parts. One part is proportional to the negative gradient of the potential landscape, which drives the system
29
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Figure 2.5: Streamlines of the flux velocity
in the radial direction. The other part is the flux velocity, which drives the system
in the transverse direction. These two parts together determine the dynamics of
the system.
Global Stability and Dynamics
We calculate the intrinsic potential landscape and verify its Lyapunov property.
According to the expression of in the potential landscape in Eq. (2.35), we can
obtain the expression of the intrinsic potential landscape by its definition:
U (0) (x, y) = lim DU (x, y) =
D→0
)2
1( 2
x + y2 − 1 .
4
(2.38)
Note that even if one keeps the term ln Z(D) in Eq. (2.35), limD→0 D ln Z(D) = 0
means it does not contribute to the final result. Also notice that in this particular
case U (0) can be obtained by simply setting D = 1 in the expression of U in
Eq. (2.35) and removing the additive constant that has no essential effects. This
means U and U (0) are proportional to each other in this case. But we should point
out this is a special situation; it does not have to be so in general. The shape of the
intrinsic potential landscape looks like a Mexican hat as shown in Fig. 2.6. The
summit of the mountain at the center corresponds to the origin of the state space,
30
Figure 2.6: The intrinsic potential landscape as the Lyapunov function
while the valley around the mountain corresponds to the unit circle on the state
space. The behavior of the negative gradient of U (0) is also depicted in Fig. 2.4.
We have shown in the general case that U (0) satisfies a Fokker-Planck HamiltonJacobi equation (Eq. (2.21)). For this particular system, this equation becomes
F⃗ · ∇U (0) + ∇U (0) · ∇U (0) = 0,
(2.39)
e = D/D = I in this case. Using the expressions of F⃗ in Eq. (2.29) and
since D
(0)
U in Eq. (2.38), we easily verify that this is indeed true.
Next we verify the Lyapunov property of U (0) . The expression of U (0) in
Eq. (2.38) shows that it is non-negative, i.e., U (0) (x, y) ≥ 0. We calculate its time
derivative as follows:
U̇ (0) =
=
+
=
≤
ẋ∂x U (0) + ẏ∂y U (0)
[
]
(1 − x2 − y 2 )x − y (x2 + y 2 − 1)x
[
]
(1 − x2 − y 2 )y + x (x2 + y 2 − 1)y
−(x2 + y 2 − 1)2 (x2 + y 2 )
0.
31
(2.40)
2
Thus we have proven directly that U (0) = (x2 + y 2 − 1) /4 is a Lyapunov function of the dynamical system in Eq. (2.28). Its extremum states, according to Eq. (2.40), are given by the set Λ = {(x, y)|(0, 0), x2 + y 2 = 1}, which is the origin
and the unit circle. These correspond to the asymptotic behaviors of the system
in the long term. It may be intuitive to see that the origin as a single point corresponds to a fixed point. But it may not be so obvious why the unit circle should
correspond to a limit cycle instead of a continuous set of fixed points which is not
impossible. This can be further understood by the following investigation.
The Fokker-Planck Hamilton-Jacobi equation in Eq. (2.39) is equivalent to the
following two equations:
F⃗ = −∇U (0) + V⃗s(0) ,
V⃗ (0) · ∇U (0) = 0.
s
(2.41)
(2.42)
Using the expressions of F⃗ in Eq. (2.29) and U (0) in Eq. (2.38), one can calculate
(0)
(0)
V⃗s from Eq. (2.41) and find that in this particular case V⃗s = V⃗s = (−y, x)| .
That is, the intrinsic flux velocity (in the small fluctuation limit) is equal to the
flux velocity without taking the small fluctuation limit in this particular case. Thus
Fig. 2.5 for flux velocity also applies to the intrinsic flux velocity. Also, in addition
to Eq. (2.42) which states that the intrinsic flux velocity is perpendicular to the
(0)
gradient of the intrinsic potential landscape, we also have ∇· V⃗s = 0 which states
that the intrinsic flux velocity is also a curl field in this particular case. However,
these are special properties for this particular system. Equation (2.41) is the force
decomposition equation in the small fluctuation limit. Now the driving force of
the system is decomposed into two parts. One part is the gradient of the intrinsic
potential landscape and the other part is the intrinsic flux velocity perpendicular
to the first part. The gradient part of the driving force guides the system down the
intrinsic potential landscape, while the intrinsic flux velocity drives the system in
the transverse direction on the level set of the intrinsic potential landscape. See
Fig. 2.4, Fig. 2.5 and Fig. 2.6 for perspectives. (The field lines of the gradient
of the intrinsic potential landscape are along the radial direction. The field lines
of the intrinsic flux velocity form circles around the origin.) When the system
(0)
reaches its asymptotic sets where ∇U (0) = 0, the intrinsic flux velocity V⃗s (if
nonzero) continues to drive the asymptotic dynamics. In this particular case, when
(0)
the system is at the origin, we have V⃗s (0, 0) = (0, 0)| . Thus the intrinsic flux
velocity vanishes at the origin and it will not drive the system away from the
origin. When the system is on the unit circle with x2 + y 2 = 1, the intrinsic flux
(0)
velocity V⃗s = (−y, x)| continues to drive the system along the unit circle. This
32
is why the unit circle forms a limit cycle rather than a continuous set of fixed
(0)
points. It is the intrinsic flux velocity V⃗s that connects the points on the unit
circle to form a limit cycle.
Further, the stability of the fixed point at the origin and the limit cycle on the
unit circle can be studied by investigating the property of the extremum states
(0)
of
√ the intrinsic potential landscape U . If we introduce the radius variable r =
x2 + y 2 , then U (0) (r) = (r2 −1)2 /4. It is easy to prove that r = 0 is a maximum
point of U (0) while r = 1 is a minimum point of U (0) . Thus the fixed point at the
origin is unstable, while the limit cycle on the unit circle is stable. This can also
be understood from the shape of U (0) in Fig. 2.6. When the system is at the origin
corresponding to the summit of the mountain, a small perturbation will drive the
system away from the summit down the hill. This shows the origin is an unstable
fixed point. When the system deviates from the unit circle corresponding to the
valley around the mountain, it will tend to go down the hill and return to the valley.
This shows that the limit cycle is stable.
For the stochastic system described
by the Fokker-Planck equation in Eq. (2.33),
∫
the relative entropy A[Pt ] = d⃗xPt ln(Pt /Ps ) is a Lyapunov functional of the
Fokker-Planck equation, showing that the probability distribution Pt eventually
converge to the stationary distribution Ps following the Fokker-Planck dynamics, since in this particular case D = DI is positive definite. However, since we
do not have the exact transient solution Pt (x, y) of this Fokker-Planck equation,
we cannot verify it explicitly. In Sec. 3.2.5 of chapter 3, we will study OrnsteinUhlenbeck processes which can be solved exactly for both the steady state Ps and
the transient state Pt . That allows one to verify the Lyapunov property of the
relative entropy A[Pt ] explicitly for these systems.
2.2
Global Stability and Dynamics of Spatially Inhomogeneous Non-Equilibrium Systems
We extend results in Sec. 2.1 for spatially homogeneous non-equilibrium systems to general spatially inhomogeneous non-equilibrium systems. In Sec. 2.2.1
we briefly show how to make systematic extensions from spatially homogeneous
systems to spatially inhomogeneous systems. In Sec. 2.2.2 we focus on the dynamical equations for spatially inhomogeneous systems. In Sec. 2.2.3 we extend
the potential and flux landscape framework to general spatially inhomogeneous
non-equilibrium systems described by functional Fokker-Planck equations. In
33
Sec. 2.2.4 we generalize the method of uncovering Lyapunov functions to deterministic and stochastic spatially inhomogeneous non-equilibrium systems described by functional Fokker-Planck equations.
2.2.1
The Method of Formal Extension
We give a short introduction to the formal extension method. For a more elaborate presentation, we refer to the supplementary material of Ref. [50]. This method
exploits the connection of formal notations between spatially homogeneous and
inhomogeneous systems. It facilitates systematic formal extensions from spatially
homogeneous systems to spatially inhomogeneous systems, without confronting
directly the mathematical difficulties involved in dealing with spatially inhomogeneous systems with infinite degrees of freedom. A more mathematically rigorous
treatment of spatially inhomogeneous systems based on stochastic dynamics on
Hilbert space will be presented in Sec. 3.3 of chapter 3. In this chapter we simply assume that the state space of spatially inhomogeneous systems we consider
forms a Hilbert or Banach space.
For a spatially homogeneous system the state of the system with n degrees of
freedom at any given moment can be specified by a single n-dimensional state
vector ⃗q = {q1 , ..., qa , ..., qn }. For a spatially inhomogeneous system (or spatially
extended system, field) we assume the state of the system at any given moment can
be specified by a complete set of local quantities, such as local concentrations or
local densities, that can be regarded as continuous functions of the physical space,
⃗ x) = {ϕ1 (⃗x), ..., ϕa (⃗x), ..., ϕn (⃗x)}. Thus
represented as a vector-valued field ϕ(⃗
the system has infinite degrees of freedom, labeled by the discrete vector index
a and the continuous space index ⃗x together. Formally, the extension of the state
of the system from a spatially homogeneous system to a spatially inhomogeneous
system is achieved through extending the discrete index a of ⃗q to a discrete index
⃗ x). To facilitate the extension initially, we
a and a continuous space index ⃗x of ϕ(⃗
can introduce a discrete implicit space index λ and a discrete explicit space index
⃗ x) is represented by the
⃗xλ . Then the formal extension procedure from ⃗q to ϕ(⃗
following flow chart.
componentize
explicitize
extend index
continuize
vectorize
⃗ x)
−
⇀
−
⇀
−
⇀
⃗q −
↽−
−−
−−
−−
−−
−⇀
− qa −
↽
−−
−−
−−
−−
−
− ϕ(aλ) −
↽
−−
−−
−−
−
− ϕa (⃗xλ ) −
↽
−−
−−
−−
−
− ϕa (⃗x) −
↽−
−−
−−
−−
−−
−⇀
− ϕ(⃗
vectorize
reduce index
implicitize
discretize
componentize
(2.43)
Notice the index λ is written as if it is a vector index as a is, while ⃗xλ is written
as if it is a continuous space function argument ⃗x. This procedure does not only
34
apply to extension of states but also applies to all state indexes, such as the index
of the driving force and the diffusion matrix. Furthermore, a state function for a
spatially homogeneous system O(⃗q) will become a functional O[ϕ] for a spatially
inhomogeneous system, where [ϕ] is a short notation indicating dependence on
⃗ x). Sum over the discrete index will become integral for the conthe function ϕ(⃗
tinuous index. Partial derivatives over the state variables will become functional
derivatives. Integrals over the state variables will become functional integrals.
Some of the most used extensions are summarized in the following.
extension
(⃗q) −
↽−
−−
−−
−−
−⇀
− [ϕ]
reduction
extension
F⃗ (⃗q) −
↽−
−−
−−
−−
−⇀
− F⃗ (⃗x)[ϕ]
reduction
extension
′
D(⃗q) −
↽−
−−
−−
−−
−⇀
− D(⃗x, ⃗x )[ϕ]
reduction
∑ continuize ∫ d⃗x
−−
⇀
↽
−−
−−
−−
−
−
∆V
discretize
λ
δλλ′
∂
∂ϕa (⃗xλ )
continuize
′
−
−
⇀
↽
−−
−−
−−
−
− ∆V δ(⃗x − ⃗x )
discretize
δ
−−
⇀
↽
−−
−−
−−
−
− ∆V
δϕa (⃗x)
discretize
∫
∏∫
continuize
⃗ xλ ) ↽
−−
⇀
dϕ(⃗
D[ϕ]
−−
−−
−−
−
−
λ
2.2.2
continuize
(2.44)
discretize
Deterministic and Stochastic Dynamics of Spatially Inhomogeneous Systems
We use the results summarized in Eqs. (2.43) and (2.44) to formally extend
some of the most relevant and essential dynamical equations of spatially homogeneous systems to spatially inhomogeneous systems. A more mathematically
rigorous treatment will be given in Sec. 3.3.2.
Deterministic Dynamics
The formal extension of the deterministic driving force to spatially inhomogeneous systems is the deterministic driving force field functional F⃗ (⃗x)[ϕ] (see
Eq. (2.44)). The deterministic dynamical equation for spatially inhomogeneous
35
systems (i.e., the deterministic field equation) as an extension of Eq. (2.1) is therefore [51]:
∂⃗
ϕ(⃗x, t) = F⃗ (⃗x)[ϕ].
(2.45)
∂t
In many cases F⃗ (⃗x)[ϕ] can be represented by (nonlinear) partial differential oper⃗ x). For reaction diffusion systems,
ators or integral operators acting on the field ϕ(⃗
⃗ x)) +
the deterministic driving force functional has the form F⃗ (⃗x)[ϕ] = f⃗(ϕ(⃗
2⃗
⃗ x). In
D · ∇ ϕ(⃗x), which is determined only by the local values of the field ϕ(⃗
general, the∫ driving force functional could have non-local forms, for instance,
⃗ x ′ ).
F⃗ (⃗x)[ϕ] = d⃗x ′ G(⃗x, ⃗x ′ ) · ϕ(⃗
Functional Langevin Equation
When stochastic fluctuations (intrinsic or extrinsic) are present, a stochastic
description is required. We assume the stochastic spatially inhomogeneous systems we are concerned with can be modeled by the following functional Langevin
equation [15, 45–47, 51]:
∂⃗
⃗ x, t)[ϕ],
ϕ(⃗x, t) = F⃗ (⃗x)[ϕ] + ξ(⃗
∂t
(2.46)
⃗ x, t)[ϕ] is the stochastic driving force field functional accounting for the
where ξ(⃗
effect of stochastic fluctuations that change the state of the system. This equation
⃗ x, t)[ϕ] in general can be decomposed in terms
is a formal extension of Eq. (2.3). ξ(⃗
of different sources generating statistically independent stochastic fluctuations.
We label these stochastic fluctuation sources by a discrete index s and a continuous
index ⃗y together. The discrete index s does not have to be the state space vector
index and the continuous index ⃗y does not have to be the space index. Then as a
formal extension of Eq. (2.4) we have
∑∫
⃗
⃗ (s⃗y) (⃗x)[ϕ] Γs (⃗y , t),
ξ(⃗x, t)[ϕ] =
d⃗y G
(2.47)
s
where Γs (⃗y , t) are Gaussian noises with the following statistical property:
< Γs (⃗y , t) >= 0, < Γs (⃗y , t)Γs′ (⃗y ′ , t) >= δss′ δ(⃗y − ⃗y ′ )δ(t − t′ ).
(2.48)
The statistical property of the stochastic driving force can be obtained by Eqs. (2.47) and (2.48):
⃗ x, t)[ϕ] >= 0, < ξ(⃗
⃗ x, t)[ϕ]ξ(⃗
⃗ x ′ , t′ )[ϕ] >= 2D(⃗x, ⃗x ′ )[ϕ]δ(t − t′ ), (2.49)
< ξ(⃗
36
where D(⃗x, ⃗x ′ )[ϕ] is the diffusion matrix functional accounting for the combined
effect of stochastic fluctuations from all the independent sources labeled by (s⃗y )
and is given by
∫
1∑
′
⃗ (s⃗y) (⃗x)[ϕ]G
⃗ (s⃗y) (⃗x ′ )[ϕ].
D(⃗x, ⃗x )[ϕ] =
d⃗y G
(2.50)
2 s
D(⃗x, ⃗x ′ )[ϕ] has the nonnegative-definite property by construction. For any real
vector-valued integrable functions f⃗(⃗x), the following inequality holds
∫∫
d⃗xd⃗x ′ f⃗(⃗x) · D(⃗x, ⃗x ′ )[ϕ] · f⃗(⃗x ′ ) ≥ 0.
(2.51)
Functional Fokker-Planck Equation
Once the functional Langevin equation for the spatially inhomogeneous systems is written down, we can investigate the evolution of individual stochastic
trajectories in the field configuration state space. Since the evolution of the state
of the system is stochastic, what is useful to know is how the probability distribution of the state of the system evolves with time. In other words, what would be the
corresponding Fokker-Planck equation governing the evolution of the probability
distribution for spatially inhomogeneous dynamical systems?
From the form of the Fokker-Planck equation for spatially homogeneous systems given by Eq. (2.8) and the method of formal extension as summarized in
Eqs. (2.43) and (2.44), we could expect the Fokker-Planck equation for spatially
inhomogeneous systems would become some form of functional Fokker-Planck
Equation involving functionals and functional derivatives. In the following we
give a formal derivation of the functional Fokker-Planck equation using the formal extension method. First we write Eq. (2.8) in its component form:
∑
∑
∂
Pt (⃗q) = −
∂a (Fa (⃗q)Pt (⃗q)) +
∂a ∂b (Dab (⃗q)Pt (⃗q)) . (2.52)
∂t
a
ab
Then we extend the input of the state of the spatially homogeneous system (⃗q) to
the functional dependence form of [ϕ] of spatially inhomogeneous systems:
∑
∑
∂
Pt [ϕ] = −
∂a (Fa [ϕ]Pt [ϕ]) +
∂a ∂b (Dab [ϕ]Pt [ϕ]) .
∂t
a
ab
Next we extend the space index a and b to the double discrete index (aλ) and (bλ′ )
and further explicitize the space index ⃗xλ and ⃗xλ′ . The extension of the derivative
37
∂
∂
to double index with the explicit space index ⃗xλ is
. So now
∂qa
∂ϕa (⃗xλ )
we have
∑
∂
∂
Pt [ϕ] = −
(Fa (⃗xλ )[ϕ]Pt [ϕ])
∂t
∂ϕa (⃗xλ )
aλ
∑
∂
∂
+
(Dab (⃗xλ , ⃗xλ′ )[ϕ]Pt [ϕ]) .
∂ϕa (⃗xλ ) ∂ϕb (⃗xλ ′ )
aλbλ′
∂a =
Then we use the rules of continuization for the sum of the discrete space index λ
and for the partial derivatives given in the Eq. (2.44):
∑ continuize ∫ d⃗x
∂
δ
continuize
−
−
⇀
−
−
⇀
,
.
↽
−−
−−
−−
−
−
↽
−−
−−
−−
−
− ∆V
∆V
∂ϕ
x
δϕ
x
)
discretize
a (⃗
λ ) discretize
a (⃗
λ
Combining them together gives
∑
λ
∂
continuize
−
−
⇀
↽
−−
−−
−−
−
−
∂ϕa (⃗xλ ) discretize
∫
d⃗x
δ
.
δϕa (⃗x)
We also replace Fa (⃗xλ ) and Dab (⃗xλ , ⃗xλ′ )[ϕ] with their continuous forms Fa (⃗x)
and Dab (⃗x, ⃗x ′ )[ϕ]. Doing all these together we obtain the following functional
Fokker-Planck equation in its component form:
∑∫
∂
δ
Pt [ϕ] = −
d⃗x
(Fa (⃗x)[ϕ]Pt [ϕ])
∂t
δϕ
(⃗
x
)
a
a
∑ ∫∫
δ
δ
d⃗xd⃗x ′
+
(Dab (⃗x, ⃗x ′ )[ϕ]Pt [ϕ]) .(2.53)
′)
δϕ
(⃗
x
)
δϕ
(⃗
x
a
b
ab
By using the vector and matrix form and the short notation for the functional
derivative, we have the following functional Fokker-Planck equation in its vectormatrix form:
∫
(
)
∂
⃗
⃗
Pt [ϕ] = − d⃗x δϕ(⃗
x)[ϕ]Pt [ϕ]
⃗ x) · F (⃗
∂t
∫∫
⃗⃗ ′ · (D(⃗x, ⃗x ′ )[ϕ]Pt [ϕ]) .
+
d⃗xd⃗x ′⃗δϕ(⃗
(2.54)
⃗ x) · δϕ(⃗
x )
Thus we have obtained the general form of the functional Fokker-Planck equation
governing the evolution of the probability distribution functional for general spatially inhomogeneous systems through applying the formal extension method. A
38
special form of the general functional Fokker-Planck equation given in Eq. (2.54)
has been derived before for reaction diffusion systems through the procedure of
discretizing space and taking the continuum limit [15]. Here we have given a formal derivation, equivalent to discretizing space and taking the continuum limit, of
the more general form of the equation. Functional Fokker-Planck equations have
been utilized, for instance, in studying biological systems [145]. Such functional
Fokker-Planck equations can also arise from functional master equations for spatially inhomogeneous systems under certain conditions [15, 51]. In that case the
stochastic fluctuations are intrinsic and D(⃗x, ⃗x ′ )[ϕ] has specific forms.
The functional Fokker-Planck equation in Eq. (2.54) can be interpreted as a
continuity equation in the field configuration state space representing probability
conservation:
∫
∂
⃗ x)[ϕ],
(2.55)
Pt [ϕ] = − d⃗x ⃗δϕ(⃗
⃗ x) · Jt (⃗
∂t
where the probability flux field functional is given by
∫
′
⃗
⃗
Jt (⃗x)[ϕ] = F (⃗x)[ϕ]Pt [ϕ] − d⃗x ′ D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) Pt [ϕ],
(2.56)
where we have introduced the effective driving force field functional
∫
′
⃗
⃗
F (⃗x)[ϕ] = F (⃗x)[ϕ] − d⃗x ′⃗δϕ(⃗
x, ⃗x ′ )[ϕ].
⃗ x ′ ) · D(⃗
(2.57)
The functional Fokker-Planck Equation in the form of a continuity equation indicates the dynamical evolution of the system governed by the functional FokkerPlanck equation is a probability transport dynamics in the field configuration state
space. Since the field configuration state space is an infinite dimensional function
space, it is much harder to ‘visualize’ it compared to spatially homogeneous systems (especially systems with low degrees of freedom). Yet some analogies are
still useful. Probability distribution in the field configuration state space is redistributed through probability transport process that is determined by the probability
flux field. Equation (2.56) indicates that there are two sources contributing to the
probability flux field. One source is through ‘drifting’ where the effective driving
force field functional F⃗ ′ (⃗x)[ϕ] represents a steady drift velocity field in the field
configuration state space, instructing how probability is transported through drifting. The other source is through ‘diffusing’ where the diffusion matrix functional
in the field configuration state space instructs how probability diffuses from high
39
density regions to low density regions. Thus the probability distribution functional generates a probability flux field functional in the field configuration state space
through the drift effect and the diffusion effect, which are instructed, respectively,
by the effective driving force field functional and the diffusion matrix functional. The generated probability flux field functional transports probability in the
field configuration state space and thus redistributes the probability distribution
functional. The redistributed probability distribution functional again generates
a probability flux field functional through drift and diffusion effects. That again
redistributes the probability distribution. Thus the probability transport dynamics
in the field configuration state space is embodied in the feedback loop between the
probability distribution functional and the probability flux field functional, which
is instructed by the deterministic driving force field functional and the diffusion
matrix functional. The probability flux field functional is the driving force of the
probability transport dynamics in the field configuration state space. The probability transport dynamics in the state space for both spatially homogeneous and
inhomogeneous systems can be represented by the following graph, where ‘PD’
represents ‘Probability Distribution’ and ‘PF’ represents ‘Probability Flux’.
.drift
.
.PD
.redistribute
.PF
.diffuse
Figure 2.7: Probability transport dynamics
2.2.3
Potential and Flux Field Landscape Theory for Spatially
Inhomogeneous Non-Equilibrium Systems
We extend the potential and flux landscape theory for spatially homogeneous
systems presented in Sec. 2.1.2 to spatially inhomogeneous systems. We also discuss some aspects specific to spatially inhomogeneous systems that was not revealed in spatially homogeneous systems.
For steady state probability distribution ∂Ps [ϕ]/∂t = 0, the functional Fokker-
40
Planck equation in the form of Eqs. (2.55) and (2.56) become
∫
⃗ x)[ϕ] = 0,
d⃗x ⃗δϕ(⃗
⃗ x) · Js (⃗
J⃗s (⃗x)[ϕ] = F⃗ ′ (⃗x)[ϕ]Ps [ϕ] −
∫
d⃗x ′ D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) Ps [ϕ].
(2.58)
(2.59)
Equation (2.58) is a global constraint which requires the integral over the whole
physical space to be 0 rather than requiring the integrand itself to be 0. The steady
state probability flux field functional J⃗s (⃗x)[ϕ] has vanishing functional divergence
everywhere in the field configuration state space. By analogy with spatially homogeneous systems, we may call it a ‘solenoidal’ or curl vector field in the field
configuration state space, as it has no sinks or sources and thus having to rotate
around. F⃗ (⃗x)[ϕ] governing the deterministic dynamics and D(⃗x, ⃗x ′ )[ϕ] accounting for the stochastic fluctuation dynamics together through Eq. (2.59) determine
the steady state probability distribution functional Ps [ϕ] and the curl steady state
probability flux field functional J⃗s (⃗x)[ϕ]. Ps [ϕ] characterizes the global stochastic
steady state of spatially inhomogeneous systems while J⃗s (⃗x)[ϕ] characterizes the
global steady probability transport dynamics in the field configuration state space.
Detailed Balance and Local Equilibrium Condition
The detailed balance condition as the equilibrium condition characterizing microscopic reversibility is represented by vanishing steady state probability flux:
J⃗s (⃗x)[ϕ] = 0.
(2.60)
(This is also for even state variables. Its further extension is given in chapter 3.)
This condition means there is no probability transport dynamics happening in the
field configuration state space. The global stochastic steady state and dynamics
of the system is characterized by Ps [ϕ] alone without J⃗s (⃗x)[ϕ]. For spatially inhomogeneous systems Eq. (2.60) requires the probability flux field J⃗s (⃗x)[ϕ] to be
zero in every physical space point (or spatial cell considering finite resolutions)
for every state the system is in. Thus the detailed balance condition for spatially inhomogeneous systems has a local nature (balanced on every physical space
point or in every spatial cell) which characterizes the local equilibrium condition
of the system. The detailed balance condition as the local equilibrium condition
41
for spatially inhomogeneous systems poses a very strong constraint on the system. Setting the left side of Eq. (2.59) as 0, dividing Ps [ϕ] on both sides of the
equation and rearranging it lead to the ‘potential condition’ for spatially inhomogeneous systems, which is an extension of the potential condition for spatially
homogeneous systems given in Eq. (2.13):
∫
′
⃗
F (⃗x)[ϕ] = − d⃗x ′ D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
(2.61)
⃗ x ′ ) U [ϕ],
where U [ϕ] ≡ − ln Ps [ϕ] is the equilibrium potential field functional. Hence, for
spatially inhomogeneous systems when the detailed balance condition as the local equilibrium condition is satisfied, the effective driving force field functional
has the form of the functional gradient of the the potential field landscape with respect to the diffusion matrix functional D(⃗x, ⃗x ′ )[ϕ]. U [ϕ] serves as a link between
F⃗ (⃗x)[ϕ] (determining the deterministic dynamics) and D(⃗x, ⃗x ′ )[ϕ] (characterizing the stochastic fluctuation dynamics), which places a constraint on these two.
When the diffusion matrix functional D(⃗x, ⃗x ′ )[ϕ] is non-singular ( ⃗x and ⃗x ′ are
regarded as two continuous matrix indexes together with two discrete matrix indexes within D ), Eq. (2.61) can be written explicitly as a constraint equation on
F⃗ ′ (⃗x)[ϕ] and D(⃗x, ⃗x ′ )[ϕ]:
]
[∫
′′ −1
′′
′′
δϕb (⃗x ′ )
d⃗x Dac (⃗x, ⃗x )[ϕ]F̃c (⃗x )[ϕ]
]
[∫
′′ −1
′
′′
′′
d⃗x Dbc (⃗x , ⃗x )[ϕ]F̃c (⃗x )[ϕ] ,
(2.62)
= δϕa (⃗x)
where the inverse of D(⃗x, ⃗x ′ )[ϕ] is defined by the following equation:
∑∫
−1
d⃗x ′ Dab (⃗x, ⃗x ′ )Dbc
(⃗x ′ , ⃗x ′′ ) = δac δ(⃗x − ⃗x ′′ ).
(2.63)
b
Equation (2.62) is the extension of Eq. (2.14) to spatially inhomogeneous systems.
Detailed Balance Breaking and Non-Equilibrium Condition
In general, the detailed balance as the local equilibrium condition may not
be satisfied, in which case the system can have non-zero probability flux field
J⃗s (⃗x)[ϕ] ̸= 0, which characterizes detailed balance breaking and indicates that the
system is not in local equilibrium. J⃗s (⃗x)[ϕ] ̸= 0 means it is not 0 all over the physical space for every state the system is in. Yet it can still be 0 in some regions in the
42
physical space. So there could be a distinction between partial non-equilibrium
and complete non-equilibrium situations in the physical space for spatially inhomogeneous systems. When the probability flux field is non-zero in some regions
in the physical space while still 0 in some other regions, some parts of the system
are in non-equilibrium while others are still in equilibrium locally. In this case
we can say the system is in partial non-equilibrium (or partial equilibrium) in the
physical space. When the probability flux field is non-zero globally at every space
point, we can say the system is in complete non-equilibrium in the physical space,
since all the parts of the system are in non-equilibrium characterized by non-zero
probability flux. This is a new feature manifested in spatially inhomogeneous
systems that was not revealed in spatially homogeneous systems.
Dividing Ps [ϕ] on both sides of Eq. (2.59) and rearranging the equation gives
the following functional force decomposition equation, which is an extension of
Eq. (2.15) to spatially inhomogeneous systems:
∫
′
⃗
⃗ x)[ϕ],
F (⃗x)[ϕ] = − d⃗x ′ D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
(2.64)
⃗ x ′ ) U [ϕ] + Vs (⃗
where U [ϕ] = − ln Ps [ϕ] is the generalized non-equilibrium potential field and
V⃗s (⃗x)[ϕ] = J⃗s (⃗x)[ϕ]/Ps [ϕ] is the steady state probability flux velocity field. They
are related by
∫
∫
⃗
⃗
⃗ x)[ϕ],
d⃗x Vs (⃗x)[ϕ] · δϕ(⃗
d⃗x ⃗δϕ(⃗
(2.65)
⃗ x) U [ϕ] =
⃗ x) · Vs (⃗
which is a result of Eq. (2.58) expressed in terms of U [ϕ] and V⃗s (⃗x)[ϕ]. Thus in
non-equilibrium spatially inhomogeneous systems with detailed balance breaking,
the effective driving force field can be decomposed into two terms. One term is
the functional gradient (with respect to the diffusion matrix functional) of the
potential field U , which characterizes the global stochastic state of the system.
The other term is the curling probability flux velocity field which represents the
effect of detailed balance breaking that drives the system away from equilibrium.
In Sec. 2.2.4 we will see that in the small fluctuation limit the zero-order potential field landscape becomes a Lyapunov functional of the deterministic spatially
inhomogeneous system quantifying the global stability of the system. In the small
fluctuation limit the functional force decomposition equation states how the zeroorder potential field landscape U [ϕ] and the probability flux velocity field functional V⃗s (⃗x)[ϕ] together determine the deterministic driving force field functional
of the system (governing the deterministic dynamics of the spatially inhomogeneous system). Therefore, for non-equilibrium spatially inhomogeneous systems,
43
while the global stability is quantified by the underlying potential field landscape
U [ϕ], quantification of the underlying global dynamics of the non-equilibrium spatially inhomogeneous systems requires both the potential field landscape U [ϕ]
and the probability flux velocity field V⃗s (⃗x)[ϕ], in contrast to equilibrium systems
where U [ϕ] alone is sufficient. As discussed in Sec. 2.1.2 for spatially homogeneous systems, the force decomposition equation is only one perspective of many
to look at that equation. Other perspectives discussed there may also apply here
similarly and we shall not repeat again. We summarize the equilibrium and nonequilibrium dynamics of spatially inhomogeneous systems in Table 2.3.
Detailed Balance
J⃗s (⃗x)[ϕ]
J⃗s (⃗x)[ϕ] = 0
V⃗s (⃗x)[ϕ]
V⃗s (⃗x)[ϕ] = 0
Detailed Balance Breaking
∫
⃗ x)[ϕ] = 0
J⃗s (⃗x)[ϕ] ̸= 0, d⃗x ⃗δϕ(⃗
⃗ x) · Js (⃗
∫
V⃗s (⃗x)[ϕ] ̸= 0, d⃗x V⃗s (⃗x)[ϕ] · ⃗δϕ(⃗
⃗ x) U [ϕ] =
∫
d⃗x ⃗δ⃗ · V⃗s (⃗x)[ϕ]
ϕ(⃗
x)
′
′
F⃗ (⃗x)[ϕ]
⃗
∫ F ′(⃗x)[ϕ] =′
− d⃗x D(⃗x, ⃗x )[ϕ]·
⃗δ⃗ ′ U [ϕ]
ϕ(⃗
x )
−
∫
F⃗ ′ (⃗x)[ϕ] =
⃗ x)[ϕ]
d⃗x ′ D(⃗x, ⃗x ′ )[ϕ]·⃗δϕ(⃗
⃗ x ′ ) U [ϕ]+ Vs (⃗
Table 2.3: (non-)equilibrium dynamics of spatially inhomogeneous systems9
Similar to spatially homogeneous systems, a time-dependent force decomposition equation can also be derived from the definition of the time-dependent
probability flux field in Eq. (2.56):
∫
′
⃗
⃗ x)[ϕ],
F (⃗x)[ϕ] = − d⃗x ′ D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
(2.66)
⃗ x ′ ) S[ϕ] + Vt (⃗
where S[ϕ] ≡ − ln Pt [ϕ] and V⃗t (⃗x)[ϕ] ≡ J⃗t (⃗x)[ϕ]/Pt [ϕ]. From Eq. (2.64) and
Eq. (2.66) we also have the following equation:
(
)
∫
P
[ϕ]
t
′
′
V⃗t (⃗x)[ϕ] = − d⃗x D(⃗x, ⃗x )[ϕ] · ⃗δϕ(⃗
+ V⃗s (⃗x)[ϕ],
(2.67)
⃗ x ′ ) ln
Ps [ϕ]
which will used in the study of the Lyapunov function(al) of spatially inhomogeneous stochastic dynamics systems.
9
These relations are further generalized in chapter 3.
44
2.2.4
Lyapunov Functional Quantifying the Global Stability of
Spatially Inhomogeneous Non-Equilibrium Systems
We extend the methods that are used to uncover Lyapunov functions of spatially homogeneous systems discussed in Sec. 2.1.3 to spatially inhomogeneous
systems to quantify the global stability of the system. Similar to spatially homogeneous systems, the intrinsic potential field landscape of the spatially inhomogeneous system in the small fluctuation limit is a Lyapunov functional of the deterministic system quantifying its global stability. The relative entropy functional
of the spatially inhomogeneous system is a Lyapunov functional of the stochastic
system with finite fluctuations which can be used to explore its global stability.
Lyapunov Functional of Deterministic Spatially Inhomogeneous Systems
To be able to adjust the strength of fluctuation, which usually applies to syse x, ⃗x ′ )[ϕ]. Then the
tems with extrinsic fluctuations, we write D(⃗x, ⃗x ′ )[ϕ] = DD(⃗
corresponding stationary functional Fokker-Planck equation becomes
∫
(
)
⃗
−
d⃗x ⃗δϕ(⃗
·
F
(⃗
x
)[ϕ]P
[ϕ]
⃗ x)
s
∫∫
(
)
′⃗
′
⃗
e
+ D
d⃗xd⃗x δϕ(⃗
x, ⃗x )[ϕ]Ps [ϕ]
⃗ x) · δϕ(⃗
⃗ x ′ ) · D(⃗
= 0.
(2.68)
When D is small, assume the steady state distribution has the asymptotic form:
]
[
∞
exp {−U (D)[ϕ]}
1
1 ∑ k (k)
D U [ϕ] . (2.69)
Ps (D)[ϕ] =
=
exp −
Z(D)
Z(D)
D k=0
Plugging it into Eq. (2.68) and keeping only the leading order of D, we derive the
functional Fokker-Planck Hamilton-Jacobi equation:
H
[ δ U (0) , ϕ ]
∫ FFP ϕ
(0)
=
d⃗x F⃗ (⃗x)[ϕ] · ⃗δϕ(⃗
[ϕ]
⃗ x) U
∫∫
(
)
(
)
(0)
e x, ⃗x ′ )[ϕ] · ⃗δ⃗ ′ U (0) [ϕ]
[ϕ] · D(⃗
+
d⃗xd⃗x ′ ⃗δϕ(⃗
⃗ x) U
ϕ(⃗
x )
= 0,
(2.70)
45
where U (0) [ϕ] = lim D U (D)[ϕ] = lim [−D ln (Ps (D)[ϕ]Z(D))] is the zeroD→0
D→0
order potential field landscape of spatially inhomogeneous systems. We call it
the intrinsic potential field landscape. Note that δϕ U (0) is a short notation for
⃗δ⃗ U (0) [ϕ]. Equation (2.70) has the form of a Hamilton-Jacobi equation for fields
ϕ(⃗
x)
with zero energy. The corresponding Field Hamiltonian of the system is
∫
∫∫
e x, ⃗x ′ )[ϕ] · ⃗π (⃗x ′ ),
⃗
HF F P [ π, ϕ ] = d⃗x F (⃗x)[ϕ] · ⃗π (⃗x) +
d⃗xd⃗x ′⃗π (⃗x) · D(⃗
(2.71)
(0)
⃗
where ⃗π (⃗x) = ⃗δϕ(⃗
U
[ϕ]
is
the
canonical
momentum
field
conjugate
to
ϕ(⃗
x).
⃗ x)
As the solution of the functional Fokker-Planck Hamilton-Jacobi equation,
(0)
U [ϕ] can be proven to be the Lyapunov functional of the corresponding de⃗ x, t)/∂t =
terministic system governed by the deterministic field equation ∂ ϕ(⃗
(0)
F⃗ (⃗x)[ϕ]. The time derivative of U [ϕ] is non-negative as shown below:
∫
⃗ x, t)
d (0)
∂ ϕ(⃗
(0)
U [ϕ] =
d⃗x
· ⃗δϕ(⃗
[ϕ]
⃗ x) U
dt
∂t
∫
(0)
=
d⃗x F⃗ (⃗x)[ϕ] · ⃗δϕ(⃗
[ϕ]
⃗ x) U
∫∫
(
)
(
)
(0)
e x, ⃗x ′ )[ϕ] · ⃗δ⃗ ′ U (0) [ϕ]
= −
d⃗xd⃗x ′ ⃗δϕ(⃗
[ϕ] · D(⃗
⃗ x) U
ϕ(⃗
x )
≤ 0.
Within the proof we have used the chain rule of taking derivatives involving functionals, the deterministic field equation, and the nonnegative-definite property of
the diffusion matrix functional given by Eq. (2.51). If we require D(⃗x, ⃗x ′ )[ϕ]
e x, ⃗x ′ )[ϕ] to be positive definite, then the equality holds if and only
and thus D(⃗
(0)
if ⃗δϕ(⃗
[ϕ] = 0. Therefore the intrinsic potential field landscape U (0) [ϕ] is
⃗ x) U
a Lyapunov functional of the spatially inhomogeneous deterministic dynamical
systems. It decreases monotonically in time while the corresponding deterministic system reaches its asymptotic behaviors, which are contained within the set
⃗ x)|⃗δ⃗ U (0) [ϕ] = 0}. Therefore
of the extremum states of U (0) [ϕ], i.e., Λ = {ϕ(⃗
ϕ(⃗
x)
we can use the intrinsic potential field landscape to quantify the global stability of
spatially inhomogeneous deterministic dynamical systems.
Similar to spatially homogeneous systems, Eq. (2.70) is equivalent to the following two equations:
∫
(
)
e x, ⃗x ′ )[ϕ] · ⃗δ⃗ ′ U (0) [ϕ] + V⃗ (0) (⃗x)[ϕ],
F⃗ (⃗x)[ϕ] = − d⃗x ′ D(⃗
(2.72)
s
ϕ(⃗
x )
46
∫
(0)
d⃗x V⃗s(0) (⃗x)[ϕ] · ⃗δϕ(⃗
[ϕ] = 0.
⃗ x) U
(2.73)
The first equation is the force decomposition equation in the small fluctuation
(0)
limit (see Eq. (2.64)), where V⃗s (⃗x)[ϕ] is the lowest order of V⃗s (⃗x)[ϕ] in the small fluctuation limit (the intrinsic probability flux velocity field functional). The
(0)
second equation is the orthogonality condition stating that V⃗s (⃗x)[ϕ] is perpendicular to the functional gradient of U (0) [ϕ] in the field configuration state space.
According to Eq. (2.72), when the system reaches its asymptotic behaviors where
⃗δ⃗ ′ U (0) [ϕ] = 0, the driving force field of the asymptotic dynamic is V⃗s(0) (⃗x)[ϕ].
ϕ(⃗
x )
The intrinsic potential field landscape U (0) [ϕ] and the intrinsic probability
(0)
flux velocity field functional V⃗s (⃗x)[ϕ], as the small fluctuation limit of U [ϕ]
and V⃗s (⃗x)[ϕ], provides a global description in the field configuration state space
of the deterministic spatially inhomogeneous non-equilibrium system. U (0) [ϕ]
is the Lyapunov functional of the deterministic spatially inhomogeneous non(0)
equilibrium system characterizing its global stability. U (0) [ϕ] and V⃗s (⃗x)[ϕ] together through Eq. (2.72) determine F⃗ (⃗x)[ϕ] that governs the deterministic dynamics of the system. The intrinsic potential field landscape guides the dynamics
down the direction of its functional gradient, while the intrinsic probability flux
velocity field drives the system in a curling way. For equilibrium spatially inhomo(0)
geneous systems, the intrinsic probability flux velocity field functional V⃗s (⃗x)[ϕ]
vanishes. In this case, the dynamics of the system is determined by the functional
gradient of the intrinsic potential field landscape alone. This has been shown to be
true for the well stirred spatially homogeneous systems. Here we see a generalized
duality law in terms of potential field landscape and flux velocity field functional
for spatially inhomogeneous non-equilibrium systems. In this way, we realize the
force field decomposition in the field configuration state space that determines the
non-equilibrium dynamics of spatially inhomogeneous systems.
Lyapunov Functional of Stochastic Spatially Inhomogeneous Systems
Now we turn to the Lyapunov functional of stochastic spatially inhomogeneous systems with finite fluctuations. Similar to spatially homogeneous systems,
the relative entropy for the functional Fokker-Planck equation is defined as:
(
)
∫
Pt [ϕ]
A[Pt [ϕ]] = D[ϕ]Pt [ϕ] ln
.
(2.74)
Ps [ϕ]
47
We assume Pt [ϕ] and Ps [ϕ] are both positive and normalized to the same constant.
We show that A[Pt [ϕ]] has the Lyapunov property. However, we must mention that
the following ‘proof’ only has a formal nature. (The space of P [ϕ] is a space of
functionals; the property of such a space is unclear.) We first prove that A[Pt [ϕ]] is
non-negative and then prove its time derivative is non-positive. The formal proof
is similar to that for spatially homogeneous systems, just in functional languages.
)
(
∫
Ps [ϕ]
A[Pt [ϕ]] = − D[ϕ]Pt [ϕ] ln
Pt [ϕ]
(
)
∫
Ps [ϕ]
≥ − D[ϕ]Pt [ϕ]
−1
Pt [ϕ]
∫
∫
=
D[ϕ]Pt [ϕ] − D[ϕ]Ps [ϕ] = 0,
where we have used ln x ≤ x − 1 for x > 0 and Ps [ϕ] and that Pt [ϕ] are both normalized to the same constant. Then we calculate the time derivative of A[Pt [ϕ]].
(
)
∫
d
Pt [ϕ]
d
A[Pt [ϕ]] =
D[ϕ] Pt [ϕ] ln
dt
dt
Ps [ϕ]
(
) ∫
∫
∂Pt [ϕ]
Pt [ϕ]
∂Pt [ϕ]
=
D[ϕ]
ln
+ D[ϕ]
∂t
Ps [ϕ]
∂t
( ∫
)
)(
∫
Pt [ϕ]
⃗
⃗
=
D[ϕ] − d⃗x δϕ(⃗
x)[ϕ]
ln
⃗ x) · Jt (⃗
Ps [ϕ]
[
(
)]
∫
∫
Pt [ϕ]
⃗
⃗
=
D[ϕ] d⃗x Jt (⃗x)[ϕ] · δϕ(⃗
⃗ x) ln
Ps [ϕ]
[
(
)]
∫
∫
P
[ϕ]
t
=
D[ϕ] d⃗x Pt [ϕ]V⃗t (⃗x)[ϕ] · ⃗δϕ(⃗
⃗ x) ln
Ps [ϕ]
[ ∫
(
)
∫
∫
P
[ϕ]
t
′
′
=
D[ϕ] d⃗x Pt [ϕ] − d⃗x D(⃗x, ⃗x )[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) ln
Ps [ϕ]
[
(
)]
]
Pt [ϕ]
+V⃗s (⃗x)[ϕ] · ⃗δϕ(⃗
⃗ x) ln
Ps [ϕ]
[
)]
(
∫
∫∫
Pt [ϕ]
′ ⃗
= − D[ϕ] Pt [ϕ]
d⃗xd⃗x δϕ(⃗
⃗ x) ln
Ps [ϕ]
[
(
)]
Pt [ϕ]
·D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) ln
Ps [ϕ]
[
(
)]
∫
∫
Pt [ϕ]
⃗
⃗
+ D[ϕ] d⃗x Pt [ϕ]Vs (⃗x)[ϕ] · δϕ(⃗
,
⃗ x) ln
Ps [ϕ]
48
where we have plugged in Eq. (2.67). The first term in the last equation is nonpositive, because the diffusion matrix functional within the integrand has the
nonnegative-definite property represented by Eq. (2.51) and Pt [ϕ] is positive. The
second term in the last line vanishes because
(
)
∫
∫
P
[ϕ]
t
D[ϕ] d⃗x Pt [ϕ]V⃗s (⃗x)[ϕ] · ⃗δϕ(⃗
⃗ x) ln
Ps [ϕ]
(
)
∫
∫
P
[ϕ]
P
[ϕ]
s
t
⃗δ⃗
=
D[ϕ] d⃗x Pt [ϕ]V⃗s (⃗x)[ϕ] ·
Pt [ϕ] ϕ(⃗x) Ps [ϕ]
(
)
∫
∫
Pt [ϕ]
⃗
⃗
=
D[ϕ] d⃗x Js (⃗x)[ϕ] · δϕ(⃗
⃗ x)
Ps [ϕ]
∫
∫
(
) ( P [ϕ] )
t
⃗
⃗
= − D[ϕ] d⃗x δϕ(⃗
x)[ϕ]
= 0.
⃗ x) · Js (⃗
Ps [ϕ]
Therefore, dA[Pt [ϕ]]/dt ≤ 0. If the diffusion matrix functional is positive definite, the equality holds only when Pt [ϕ] = Ps [ϕ]. In the above formal proof
we have used the continuity equation of the functional Fokker-Planck equation
and the Gauss’ theorem in the field configuration state space several times. We
assume certain boundary conditions, if applicable, are imposed so that boundary
terms within applying Gauss’ theorem vanish. Thus the relative entropy A[Pt [ϕ]]
has the Lyapunov property of decreasing monotonically in time, while the timedependent probability distribution functional approaches the steady state probability distribution functional showing its asymptotic stability. The relative entropy
quantifies the global stability of stochastic spatially inhomogeneous systems.
2.3
Reaction Diffusion Systems
We apply the general framework developed in Sec. 2.2 to reaction diffusion
systems [15, 51–53, 146–148] and in particular the Brusselator reaction diffusion
model [1, 93].10
2.3.1
Dynamics of Reaction Diffusion Systems
We first consider the deterministic dynamics of reaction diffusion sytems and
then the stochastic dynamics.
10
The material of the Brusselator model presented here is new and does not exist in Ref. [50].
49
Deterministic Dynamics
The macroscopic state of multi-species chemical reaction diffusion systems
can be characterized by the set of local concentrations of each chemical species
⃗ x) = {ϕ1 (⃗x), ..., ϕm (⃗x), ...ϕn (⃗x)}. The macroscopic
over the physical space ϕ(⃗
equation governing the deterministic dynamics of reaction diffusion systems is
the reaction diffusion equation [15, 51, 52]:
∂⃗
⃗ x, t)) + D · ∇2 ϕ(⃗
⃗ x, t),
ϕ(⃗x, t) = f⃗(ϕ(⃗
∂t
(2.75)
where D is a diagonal matrix with the diagonal element Dm (m = 1, 2, ..., n)
as the diffusion constant of chemical species m. The RHS of the equation is the
deterministic driving force of the system. The first term is the chemical reaction
force, coming from the contribution of local chemical reactions. The second term
is the the diffusion force from local diffusion processes.
The chemical reaction force has the following decomposed form in terms of
different chemical reactions labeled by the index r:
∑
⃗ x, t)) =
⃗ x, t)).
f⃗(ϕ(⃗
⃗νr wr (ϕ(⃗
(2.76)
r
The vector ⃗νr , with its m-th component being the stoichiometric coefficient νmr ,
characterizes the number of molecules involved within chemical reaction r and
thus which direction the state of the system changes in the state space due to
⃗ x, t)) is the rate of chemical reaction r which characchemical reaction r. wr (ϕ(⃗
terizes how fast the state of the system changes due to chemical reaction r. Its
expression can be inferred by the law of mass action under certain conditions. ⃗νr
⃗ x, t)) together characterize chemical reaction r.
and wr (ϕ(⃗
The diffusion force can also be written alternatively in its decomposed form
in terms of different diffusion processes of different chemical species labeled by
the index m [51]:
∑
⃗ x, t) =
D · ∇2 ϕ(⃗
⃗em Dm ∇2 ϕm (⃗x, t),
(2.77)
m
where ⃗em = (0, ..., 1, ...0)| with value 1 at the mth component and 0 elsewhere
is the m-th base vector in the concentration space. The vector ⃗em characterizes
which direction the state of the system changes due to the diffusion of species
m. Dm ∇2 ϕm (⃗x, t) characterizes how fast the state of the system changes due
50
to the diffusion of species m. Combining Eq. (2.76) and Eq. (2.77) we have the
following decomposed form of the deterministic reaction diffusion equation in
terms of elementary state transitions [51]:
∑
∑
∂⃗
⃗ x, t)) +
ϕ(⃗x, t) =
⃗νr wr (ϕ(⃗
⃗em Dm ∇2 ϕm (⃗x, t).
∂t
r
m
(2.78)
To illustrate the above ideas, we consider the Brusselator model with oscillatory behaviors of chemical reactions [1,93]. It consists of four chemical reactions:
k
1
A −
→
X
k2
2X + Y −
→ 3X
k
3
B+X −
→
Y +D
k
4
X −
→
E
The two chemical species of interest are X and Y . The concentrations of A and
B are kept constant, while D and E are removed immediately upon production.
kr (r = 1, 2, 3, 4) are the respective reaction rate constants of these four chemical
reactions. The state of the system is characterized by the concentrations of X and
Y denoted by [X] and [Y ], respectively. When the chemical species are not well
stirred as these chemical reactions take place, local concentrations of X and Y are
used to characterize the state of the system. That means [X] and [Y ] become local
quantities (concentration fields) dependent on the physical space coordinate ⃗x. We
⃗ x) = {ϕ1 (⃗x), ϕ2 (⃗x)} as
identify the two components of the state of the system ϕ(⃗
follows: ϕ1 (⃗x) = [X] and ϕ2 (⃗x) = [Y ]. Each reaction is characterized by a pair
⃗νr and ωr . For the first reaction, the number of X increases by 1 while the number
of Y does not change if the reaction takes place once. Thus we have ⃗ν1 = (1, 0)| .
According to the law of mass action, the reaction rate of the first reaction is given
by ω1 = k1 [A]. For the second reaction, the net increase of the number of X is
1 (decreased by 2 and increased by 3) while the number of Y decreases by 1 if
the reaction happens once. That means ⃗ν2 = (1, −1)| . The reaction rate is given
by ω2 = k2 [X]2 [Y ] according to the law of mass action. Similarly, for the third
reaction we have ⃗ν3 = (−1, 1)| and ω3 = k3 [B][X]. For the fourth reaction we
have ⃗ν4 = (−1, 0)| and ω4 = k4 [X]. We summarize these results in Table 2.4.
According to Eq. (2.76), the chemical reaction force for the Brusselator model is
(
)
4
2
∑
k
[A]
+
k
[X]
[Y
]
−
k
[B][X]
−
k
[X]
1
2
3
4
f⃗([X], [Y ]) =
⃗νr ωr =
(2.79)
2
−k2 [X] [Y ] + k3 [B][X]
r=1
51
Reaction 1
Reaction 2
Reaction 3
Reaction 4
⃗νr
(1, 0)|
(1, −1)|
(−1, 1)|
(−1, 0)|
ωr
k1 [A]
k2 [X]2 [Y ]
k3 [B][X]
k4 [X]
Table 2.4: Characterization of chemical reactions in the Brusselator model
When the chemical species are not well stirred, there will be diffusion processes
across the physical space. Denote the diffusion constants of X and Y as D1 and
D2 , respectively. The diffusion force, according to Eq. (2.77), is given by
(
)
2
2
∑
D
∇
[X]
1
⃗ x, t) =
D · ∇2 ϕ(⃗
⃗em Dm ∇2 ϕm (⃗x, t) =
.
(2.80)
2
D
∇
[Y
]
m=1
2
Combining Eq. (2.79) and Eq. (2.80) together, we have the deterministic reaction
diffusion equation of the Brusselator model as follows:

∂[X]



= k1 [A] + k2 [X]2 [Y ] − k3 [B][X] − k4 [X] + D1 ∇2 [X]

 ∂t
(2.81)


 ∂[Y ]


= −k2 [X]2 [Y ] + k3 [B][X] + D2 ∇2 [Y ]
∂t
By making the following scale transformations:
√
√
k2 /k4 [Y ] → ϕ2
k2 /k4 [X] → ϕ1
k4 t → t
√
k12 k2 /k43 [A] → A
Dm /k4 → Dm
(2.82)
(k3 /k4 ) [B] → B
we can rewrite Eq. (2.81) in dimensionless quantities:

∂ϕ1


= A − (B + 1)ϕ1 + ϕ12 ϕ2 + D1 ∇2 ϕ1

 ∂t



 ∂ϕ2 = Bϕ − ϕ 2 ϕ + D ∇2 ϕ
2
2
1
1 2
∂t
52
(2.83)
Functional Langevin Dynamics
When there are stochastic fluctuations influencing the dynamics of the system, the dynamics becomes stochastic. For simplicity, we only consider stochastic
fluctuations from external environmental stochastic influences, or extrinsic fluctuations. In addition to the deterministic dynamics of reaction diffusion systems in
Eq. (2.78), there will also be stochastic fluctuations influencing the dynamics of
the system. We assume the stochastic reaction diffusion system can be described
by the following functional Langevin equation:
∑
∑
∂⃗
⃗ x, t)) +
⃗ x, t)[ϕ],
ϕ(⃗x, t) =
⃗νr wr (ϕ(⃗
⃗em Dm ∇2 ϕm (⃗x, t) + ξ(⃗
∂t
r
m
(2.84)
⃗ x, t)[ϕ] is the stochastic driving force field functional representing spatialwhere ξ(⃗
temporal environmental random influences, with the following statistical property:
⃗ x, t)[ϕ] >= 0, < ξ(⃗
⃗ x, t)[ϕ]ξ(⃗
⃗ x ′ , t′ )[ϕ] >= 2D(⃗x, ⃗x ′ )[ϕ]δ(t − t′ ). (2.85)
< ξ(⃗
Equation (2.84) is a special case of the general functional Langevin equation in
Eq. (2.46), with the deterministic driving force field functional F⃗ (⃗x)[ϕ] adapted
to reaction diffusion systems given by the RHS of Eq. (2.78). Its solution is a
stochastic trajectory in the concentration field configuration state space.
For the Brusselator model with spatial-temporal stochastic fluctuations, the
functional Langevin equation becomes:

∂ϕ1


= A − (B + 1)ϕ1 + ϕ12 ϕ2 + D1 ∇2 ϕ1 + ξ1

 ∂t
(2.86)



 ∂ϕ2 = Bϕ − ϕ 2 ϕ + D ∇2 ϕ + ξ
1
2
2
2
1 2
∂t
where ξ1 and ξ2 are two random field functionals, written more specifically as
ξ1 (⃗x, t)[ϕ1 , ϕ2 ] and ξ2 (⃗x, t)[ϕ1 , ϕ2 ]. They have the following statistical properties:
< ξi (⃗x, t)[ϕ1 , ϕ2 ] >= 0,
< ξi (⃗x, t)[ϕ1 , ϕ2 ]ξj (⃗x ′ , t′ )[ϕ1 , ϕ2 ] >= 2Dij (⃗x, ⃗x ′ )[ϕ1 , ϕ2 ]δ(t − t′ ),
where i, j = 1, 2. The specific forms of Dij (⃗x, ⃗x ′ )[ϕ1 , ϕ2 ] (i, j = 1, 2) are dependent on the properties of the random environmental fluctuations. If these fluctuations are independent of the state of the system, which is usually the case for many
53
environmental fluctuations approximately, we can drop the functional dependence
of ϕ1 and ϕ2 and write Dij (⃗x, ⃗x ′ ). If these fluctuations are spatially uncorrelated,
then Dij (⃗x, ⃗x ′ ) (i, j = 1, 2) have the form Dij (⃗x)δ(⃗x − ⃗x ′ ). If the fluctuation
strengths are also uniform across space, we have Dij δ(⃗x − ⃗x ′ ). Further, if the two
random fields ξ1 and ξ2 are uncorrelated and have equal fluctuation strength, the
diffusion matrix field functional would have the simple form Dδij δ(⃗x − ⃗x ′ ). We
assume this is the case for simplicity.
Functional Fokker-Planck Dynamics
The evolution of the probability distribution functional corresponding to the
functional Langevin equation for reaction diffusion systems in Eq. (2.84) is governed by the functional Fokker-Planck equation for reaction diffusion systems:
([
]
)
∫
∑
∑
∂
⃗ x)) +
Pt [ϕ] = − d⃗x ⃗δϕ(⃗
⃗νr wr (ϕ(⃗
⃗em Dm ∇2 ϕm (⃗x) Pt [ϕ]
⃗ x) ·
∂t
r
m
∫∫
⃗⃗ ′ · (D(⃗x, ⃗x ′ )[ϕ]Pt [ϕ]) ,
+
d⃗xd⃗x ′ ⃗δϕ(⃗
(2.87)
⃗ x) · δϕ(⃗
x )
This equation is a special case of the general functional Fokker-Planck equation
in Eq. (2.54), where the deterministic driving force field functional F⃗ (⃗x)[ϕ] is
adapted to reaction diffusion systems.
For the Brusselator model with spatial-temporal stochastic fluctuations characterized by the diffusion matrix field functional Dδij δ(⃗x − ⃗x ′ ), the functional
Fokker-Planck equation corresponding to its functional Langevin equation in Eq. (2.86) is then given by:
∫
]
)
∂
δ ([
Pt [ϕ] = − d⃗x
A − (B + 1)ϕ1 + ϕ12 ϕ2 + D1 ∇2 ϕ1 Pt [ϕ]
∂t
δϕ1 (⃗x)
∫
]
)
δ ([
− d⃗x
Bϕ1 − ϕ12 ϕ2 + D2 ∇2 ϕ2 Pt [ϕ]
δϕ2 (⃗x)
[(
)2 (
)2 ]
∫
δ
δ
+D d⃗x
+
(2.88)
Pt [ϕ].
δϕ1 (⃗x)
δϕ2 (⃗x)
Unfortunately, we do not know how to solve this functional differential equation
exactly, not even for the steady state.
54
2.3.2
Potential and Flux Field Landscape for Reaction Diffusion Systems
We apply the functional force decomposition equation developed for general
spatially inhomogeneous systems in Eq. (2.64) to reaction diffusion systems.
[
] ∫
∑
∑
2
⃗ x)) +
⃗νr wr (ϕ(⃗
⃗em Dm ∇ ϕm (⃗x) − d⃗x ′⃗δ⃗ ′ · D(⃗x, ⃗x ′ )[ϕ]
ϕ(⃗
x )
∫r
= −
m
⃗ x)[ϕ].
d⃗x ′ D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) U [ϕ] + Vs (⃗
(2.89)
In other words, the effective driving force field for non-equilibrium reaction diffusion dynamics (the LHS of the above equation) can be decomposed into two
parts. One part is the functional gradient of the potential field landscape U [ϕ],
with respect to the diffusion matrix field D(⃗x, ⃗x ′ )[ϕ]. The other part is the steady
state probability velocity field V⃗s (⃗x)[ϕ], which breaks detailed balance and indicates ∫the reaction diffusion system is away from local equilibrium. (Note that the
term d⃗x ′⃗δϕ(⃗
x, ⃗x ′ )[ϕ] is the force field from inhomogeneous diffusion in
⃗ x ′ ) · D(⃗
the field configuration state space, which vanishes if stochastic fluctuations are
independent of the state of the system.)
The detailed balance condition as the local equilibrium condition or the potential condition is given by the absence of V⃗s (⃗x)[ϕ] in Eq. (2.89):
[
] ∫
∑
∑
⃗ x)) +
⃗νr wr (ϕ(⃗
⃗em Dm ∇2 ϕm (⃗x) − d⃗x ′⃗δ⃗ ′ · D(⃗x, ⃗x ′ )[ϕ]
ϕ(⃗
x )
∫
r
= −
m
d⃗x ′ D(⃗x, ⃗x ′ )[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) U [ϕ].
(2.90)
This is a special form of the general detailed balance condition or potential condition for spatially inhomogeneous systems in Eq. (2.61) adapted to reaction diffusion systems. It says the effective driving force field for equilibrium reaction
diffusion systems has the form of a functional gradient of the potential field landscape (with respect to D(⃗x, ⃗x ′ )[ϕ]). When this condition is satisfied, the reaction
diffusion system will be in detailed balance in all the spatial points, i.e., in local equilibrium. When it is not satisfied, non-vanishing V⃗s (⃗x)[ϕ] will then break
detailed balance and indicates non-equilibrium conditions of the system.
For the Brusselator reaction diffusion model, the functional force decomposition equation as a special case of Eq. (2.89). Noticing that the diffusion
55
matrix field functional is assumed to have the form Dδij δ(⃗x − ⃗x ′ ), the term
∫
d⃗x ′⃗δϕ(⃗
x, ⃗x ′ )[ϕ] vanishes and the spatial integral in the functional gra⃗ x ′ ) · D(⃗
dient term can be carried out. We finally reach the following functional force
decomposition equation for the Brusselator reaction diffusion model:
{
A − (B + 1)ϕ1 + ϕ12 ϕ2 + D1 ∇2 ϕ1 = −Dδϕ1 U [ϕ] + V1s [ϕ]
(2.91)
Bϕ1 − ϕ12 ϕ2 + D2 ∇2 ϕ2 = −Dδϕ2 U [ϕ] + V2s [ϕ]
The notation δϕm (m = 1, 2) is short for the functional derivative δ/δϕm (⃗x)
(m = 1, 2). The non-zero flux velocity field functionals V1s [ϕ] and V2s [ϕ] indicate
detailed balance breaking and thus non-equilibrium condition for this particular
system. If the system satisfies the detailed balance condition, they vanish from
these two equations. We would then have the following potential condition for the
Brusselator reaction diffusion model, indicating the system is in equilibrium:
{
A − (B + 1)ϕ1 + ϕ12 ϕ2 + D1 ∇2 ϕ1 = −Dδϕ1 U [ϕ]
.
(2.92)
Bϕ1 − ϕ12 ϕ2 + D2 ∇2 ϕ2 = −Dδϕ2 U [ϕ]
We investigate whether this potential condition can be satisfied. Note that the
above potential condition implies δϕ1 (⃗x) δϕ2 (⃗x ′ ) U [ϕ] = δϕ2 (⃗x ′ ) δϕ1 (⃗x) U [ϕ]. This
means in Eq. (2.92) the functional derivative δϕ2 of the left side of the first equation
and the functional derivative δϕ1 of the left side of the second equation should be
equal to each other. We thus have ϕ12 (⃗x)δ(⃗x − ⃗x ′ ) = [B − 2ϕ1 (⃗x)ϕ2 (⃗x)]δ(⃗x − ⃗x ′ ),
which further implies ϕ12 (⃗x) = B − 2ϕ1 (⃗x)ϕ2 (⃗x). This should hold for all allowed ϕ1 (⃗x) and ϕ2 (⃗x). But it is not possible as B is a a parameter independent
of the state of the system. This means for stochastic fluctuations characterized by
the diffusion matrix field Dδij δ(⃗x − ⃗x ′ ), the stochastic Brusselator reaction diffusion model cannot satisfy the potential condition (detailed balance) in Eq. (2.92),
i.e., the system cannot exist in equilibrium. Thus the steady state probability flux
velocity field functional Vms [ϕ] (m = 1, 2) must be non-zero, breaking detailed
balance and indicating the system is in non-equilibrium. This seems to agree with
the fact that the Brusselator system has non-equilibrium chemical oscillations.
2.3.3
Lyapunov Functional Quantifying the Global Stability of
Reaction Diffusion Systems
We investigate the Lyapunov functions that quantify the global stability of
reaction diffusion systems. The intrinsic potential field landscape U (0) [ϕ] (see Eq. (2.69)) is a Lyapunov functional of the deterministic reaction diffusion system.
56
According to the discussions in Sec. 2.2 for general spatially inhomogeneous systems, U (0) [ϕ] satisfies a functional Fokker-Planck Hamilton-Jacobi equation (Eq. (2.70)). Adapting this equation to reaction diffusion systems gives:
HF F P [δϕ U (0) , ϕ]
)
(
∫
∑
∑
(0)
⃗ x)) +
⃗em Dm ∇2 ϕm (⃗x) · ⃗δϕ(⃗
=
d⃗x
⃗νr wr (ϕ(⃗
[ϕ]
⃗ x) U
∫∫
+
m
r
(
)
(
)
′ ⃗
(0)
′
(0)
⃗
e
d⃗xd⃗x δϕ(⃗
[ϕ] · D(⃗x, ⃗x )[ϕ] · δϕ(⃗
[ϕ]
⃗ x) U
⃗ x ′)U
= 0,
(2.93)
e x, ⃗x ′ )[ϕ] is the rescaled diffusion matrix field functional. The Lyapunov
where D(⃗
property of U (0) [ϕ] for reaction diffusion systems can also be proven by adapting
the general proof to reaction diffusion systems:
∫
⃗ x, t)
∂ ϕ(⃗
(0)
· ⃗δϕ(⃗
[ϕ]
⃗ x) U
∂t
(
)
∫
∑
∑
2
(0)
⃗ x)) +
=
d⃗x
⃗νr wr (ϕ(⃗
⃗em Dm ∇ ϕm (⃗x) · ⃗δϕ(⃗
[ϕ]
⃗ x) U
d (0)
U [ϕ] =
dt
d⃗x
∫∫
= −
r
m
(
)
(
)
(0)
′
(0)
⃗
e
d⃗xd⃗x ′ ⃗δϕ(⃗
U
[ϕ]
·
D(⃗
x
,
⃗
x
)[ϕ]
·
δ
U
[ϕ]
⃗ x)
⃗ x ′)
ϕ(⃗
≤ 0.
Therefore, U (0) [ϕ] quantifies the global stability of the deterministic reaction diffusion system. The asymptotic dynamics of the deterministic reaction diffusion
system in the long term takes place in the region of the extremum states of U (0) [ϕ]
in the concentration field configuration state space.
The functional Fokker-Planck Hamilton-Jacobi equation for reaction diffusion
systems in Eq. (2.93) is equivalent to the following two equations:
∑
∑
⃗ x)) +
⃗νr wr (ϕ(⃗
⃗em Dm ∇2 ϕm (⃗x)
r
= −
∫
m
(
)
e x, ⃗x ′ )[ϕ] · ⃗δ⃗ ′ U (0) [ϕ] + V⃗ (0) (⃗x)[ϕ],
d⃗x ′ D(⃗
s
ϕ(⃗
x )
(2.94)
∫
(0)
[ϕ] = 0.
d⃗x V⃗s(0) (⃗x)[ϕ] · ⃗δϕ(⃗
⃗ x) U
57
(2.95)
According to Eq. (2.94), for non-equilibrium reaction diffusion systems without
detailed balance, the intrinsic potential field landscape U (0) [ϕ] and the intrinsic
(0)
flux velocity field V⃗s (⃗x)[ϕ] together determine the driving force field and thus
the dynamics of reaction diffusion systems. Also, the asymptotic dynamics of
reaction diffusion systems, where δϕ U (0) [ϕ] = 0, is driven purely by the intrin(0)
sic flux velocity field V⃗s (⃗x)[ϕ]. For equilibrium reaction diffusion systems with
(0)
detailed balance, V⃗s (⃗x)[ϕ] = 0. The dynamics of the equilibrium reaction diffusion system is then only determined by the intrinsic potential field functional
U (0) [ϕ]. And the asymptotic dynamics is static due to the lack of asymptotic driv(0)
ing force field V⃗s (⃗x)[ϕ].
For reaction diffusion system with finite stochastic fluctuations described by
the functional
∫ Fokker-Planck equation (Eq. (2.87)), the relative entropy functional
A[Pt [ϕ]] = D[ϕ]Pt [ϕ] ln (Pt [ϕ]/Ps [ϕ]) has the Lyapunov property of decreasing
monotonically in time. It can be used to study the global stability of the stochastic
reaction diffusion system.
Then we further apply the above results of general reaction diffusion systems
to the Brusselator reaction diffusion model. The intrinsic potential field landscape
U (0) [ϕ] of this particular system satisfies the following functional Fokker-Planck
Hamilton-Jacobi equation:
H
[δ U (0) , ϕ]
∫ FFP ϕ
(
)
=
d⃗x A − (B + 1)ϕ1 + ϕ12 ϕ2 + D1 ∇2 ϕ1 δϕ1 U (0) [ϕ]
∫
(
)
+
d⃗x Bϕ1 − ϕ12 ϕ2 + D2 ∇2 ϕ2 δϕ2 U (0) [ϕ]
∫
(
)2 (
)2
+
d⃗x δϕ1 U (0) [ϕ] + δϕ2 U (0) [ϕ]
= 0.
(2.96)
Unfortunately, we do not know whether this equation can be solved exactly either. What we do know is that if we can find a solution U (0) [ϕ] of this equation,
then U (0) [ϕ] is a Lyapunov functional of the deterministic Brusselator reaction
diffusion model decreasing monotonically dU (0) [ϕ]/dt ≤ 0. And the asymptotic behaviors of the system is contained within the set of the extremum states of
U (0) [ϕ] defined as Λ = {ϕ|δϕ U (0) [ϕ] = 0}.
Applying Eq. (2.94) to the Brusselator reaction diffusion model, we have its
58
functional force decomposition equation in the small fluctuation limit:
{
(0) s
A − (B + 1)ϕ1 + ϕ12 ϕ2 + D1 ∇2 ϕ1 = −δϕ1 U (0) [ϕ] + V1 [ϕ]
(0) s
Bϕ1 − ϕ12 ϕ2 + D2 ∇2 ϕ2 = −δϕ2 U (0) [ϕ] + V2
(2.97)
[ϕ]
This means the driving force field of the deterministic Brusselator reaction diffusion model can be decomposed into two parts. One part is the functional gradient
of the intrinsic potential field landscape U (0) [ϕ] and the other part is the intrinsic
(0)
flux velocity field V⃗s [ϕ]. U (0) [ϕ] as the Lyapunov functional of the deterministic
(0)
system guides the system down its functional gradient, while V⃗s [ϕ] drives the
system in the transverse direction, perpendicular to the direction of the functional gradient of U (0) [ϕ]. This last statement comes from Eq. (2.95), which for the
Brusselator reaction diffusion model becomes
∫
(0) s
(0) s
d⃗x V⃗1 [ϕ]δϕ1 U (0) [ϕ] + V⃗2 [ϕ]δϕ2 U (0) [ϕ] = 0.
(2.98)
When the system reaches its asymptotic behavior in the long term, the functional
gradient of U (0) [ϕ] vanishes. Thus the asymptotic dynamics is purely driven by
(0)
the transverse component V⃗s [ϕ].
For the stochastic Brusselator reaction diffusion model governed by the functional Fokker-Planck equation in Eq.
∫ (2.88), we can use the relative entropy functional of the system A[Pt [ϕ]] = D[ϕ]Pt [ϕ] ln (Pt [ϕ]/Ps [ϕ]) to investigate the
stochastic system’s global stability since it has the Lyapunov property of decreasing monotonically in time. Since we did not (do not know how to) solve the functional Fokker-Planck equation for the Brusselator reaction diffusion model, we
do not have the explicit expressions of Pt [ϕ] and Ps [ϕ] to calculate A[Pt [ϕ]] and
thus verify its Lyapunov property directly. In chapter 3, we study the stochastic
neuronal model which can be solved exactly with analytical expressions obtained.
The relative entropy A(t) is calculated explicitly for a specific initial condition.
And we can verify explicitly that A(t) decreases with time, which is also illustrated in Fig. 3.1.
2.4
Summary
In this chapter, we established a potential and flux field landscape theory
to quantify the global stability and dynamics of spatially inhomogeneous nonequilibrium field systems. We extended our potential and flux landscape theory
59
for spatially homogeneous systems to spatially inhomogeneous non-equilibrium
field systems described by functional Fokker-Planck equations. Within this extended potential and flux field landscape framework, we found that for equilibrium spatially inhomogeneous systems in detailed balance, the potential field alone
determines both the global stability and dynamics of the system. The topography of the potential field landscape in terms of the basin of attraction and barrier
heights can quantify the global stability of the system. The functional gradient of the potential field landscape alone gives the effective driving force field of
the system. However, for non-equilibrium spatially inhomogeneous systems with
detailed balance broken, although the topography of the potential field landscape
characterizes the global stability of the system, the dynamics of the system cannot
be determined by the potential field landscape alone. Both the functional gradient
of the potential field and the non-zero curl probability flux are required to determine the non-equilibrium dynamics of the spatially inhomogeneous system. This
mimics the dynamics of a charge particle in both an electric field and a magnetic
field. The non-zero flux characterizes the non-equilibrium nature of the system
and implies the system is an open system exchanging matter, energy and information with the environment.
In the small fluctuation limit, the intrinsic potential field as the leading order
of the potential field, closely related to the steady state probability distribution,
is a Lyapunov functional of the deterministic spatially inhomogeneous system
that characterizes the global stability of the deterministic system. In the small
fluctuation limit, the driving force governing the deterministic dynamics of the
spatially inhomogeneous system is determined by both the functional gradient of
the intrinsic potential field and the curling intrinsic probability flux velocity for
non-equilibrium systems with detailed balance broken. The relative entropy functional of the stochastic spatially inhomogeneous non-equilibrium system is found
its Lyapunov functional, quantifying the global stability of the stochastic spatially
inhomogeneous non-equilibrium system with finite fluctuations. We also applied
our general extended framework for spatially inhomogeneous systems to the more
specific reaction diffusion systems with extrinsic fluctuations and illustrated the
theory further with the Brusselator reaction diffusion model.
Our potential and flux field landscape theory offers an alternative general
approach to other field-theoretic techniques. It is also an extension of the nonequilibrium potential approach by including the indispensable curl flux field, suitable for studying the global stability and dynamics of spatially inhomogeneous
non-equilibrium field systems. In the next chapter we shall investigate the nonequilibrium thermodynamics in the potential and flux landscape framework.
60
Chapter 3
Non-Equilibrium Thermodynamics
of Stochastic Systems
In this chapter, we explore the non-equilibrium thermodynamics of spatially homogenous and inhomogeneous non-equilibrium systems.1 The key feature
in our approach is to use the potential-flux landscape framework as a bridge to
construct the non-equilibrium thermodynamics from the underlying stochastic dynamics. This chapater is structured as follows. In Sec. 3.1 we construct the nonequilibrium thermodynamics for spatially homogeneous systems with one state
transition mechanism. In Sec. 3.2 we generalize the non-equilibrium thermodynamic framework to spatially homogeneous systems with multiple state transition
mechanisms. In Sec. 3.3 we formulate the non-equilibrium thermodynamic framework for spatially inhomogeneous systems. In Sec. 3.4 we give a summary of this
chapter.
3.1
Non-Equilibrium Thermodynamics for Spatially Homogeneous Stochastic Systems with One State Transition Mechanism
We consider spatially homogeneous systems described by Langevin and FokkerPlanck equations with one state transition mechanism. By state transition mecha1
Most of the material in this chapter (including all the appendices at the end of this dissertation)
was originally co-authored with Jin Wang. Reprinted with permission from W. Wu and J. Wang,
The Journal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIP Publishing LLC.
61
nism we mean mechanisms that are responsible for transitions of the state of the
system, which can take on different forms in different contexts. For a system of
interacting particles coupled to several heat reservoirs with distinct temperatures,
each heat reservoir represents a different state transition mechanism. For chemical reaction systems, each chemical reaction channel is a different state transition
mechanism. The non-equilibrium conditions of a system is usually sustained by
coupling the system to multiple reservoirs, thus involving multiple (state) transition mechanisms. Therefore the study of systems with one transition mechanism
may seem to be of very limited application. Yet we shall show in Sec. 3.2 under
certain conditions the results for one transition mechanism also apply to multiple
transition mechanisms that are equivalent to one effective transition mechanism.
We first introduce the Langevin and Fokker-Planck stochastic dynamics. Then
we present the extended potential-flux landscape framework. The stochastic dynamics is then placed into the general non-equilibrium thermodynamic context.
The more specific non-equilibrium isothermal process is then studied, with the
first and second laws of thermodynamics uncovered together with a set of nonequilibrium thermodynamic equations. After a summary and discussion, we conclude this section with an extension of the results to systems with one general
transition mechanism.
3.1.1
Stochastic Dynamics
We consider spatially homogeneous systems governed by the following Langevin
equation [15, 16, 50, 140]:
∑
⃗ s (⃗q, t)dWs (t).
d⃗q = F⃗ (⃗q, t)dt +
G
(3.1)
s
Here ⃗q is the state vector representing the dynamical variables of the system.
Ws (t) (s = 1, 2, ...) are statistically independent standard Wiener processes. The
index s labels fluctuation sources, which all come from one transition mechanism
(e.g., one heat reservoir). For generality, time-dependence of the deterministic and
⃗ s (⃗q, t) are considered, which accounts for
stochastic driving forces F⃗ (⃗q, t) and G
changing external conditions that can be represented by a set of time-dependent
external control parameters {λ} [114, 129, 130, 133]. The solution of Eq. (3.1)
traces a stochastic trajectory.
When Eq. (3.1) is interpreted as an Ito stochastic differential equation, the evolution of the probability distribution is governed by the following Fokker-Planck
62
equation [15, 16]:
(
)
∂
Pt (⃗q, t) = −∇ · F⃗ (⃗q, t)Pt (⃗q, t) − ∇ · (D(⃗q, t)Pt (⃗q, t)) ,
(3.2)
∂t
where the drift vector F⃗ (⃗q, t) is given by the deterministic driving force and the d∑ ⃗
⃗ s (⃗q, t)/2 characterizes the stochastic fluciffusion matrix D(⃗q, t) = s G
q , t)G
s (⃗
tuating forces. D(⃗q, t) is nonnegative definite symmetric by construction, but we
require it to be positive definite symmetric. Equation (3.2) is to be solved with a
given initial distribution P (⃗q, t0 ) in a given region of the state space (call it the
accessible state space) with proper boundary conditions imposed (e.g., reflective,
periodic or natural boundary conditions) [16]. Pt (⃗q, t) as the solution of Eq. (3.2)
may be called the transient probability distribution, with the subscript t indicating
‘transient’. We assume certain conditions are satisfied so that Pt (⃗q, t) is always
positive and normalized to 1 in the accessible state space [15, 16]. Equation (3.2)
has the form of a continuity equation ∂t Pt + ∇ · J⃗t = 0, representing probability
conservation. The transient probability flux J⃗t is identified as:
J⃗t (⃗q, t) = F⃗ ′ (⃗q, t)Pt (⃗q, t) − D(⃗q, t) · ∇Pt (⃗q, t),
(3.3)
where F⃗ ′ (⃗q, t) = F⃗ (⃗q, t) − ∇ · D(⃗q, t) is the effective drift vector (or effective
driving force).
At each instant of time, with the time t in F⃗ (⃗q, t) and D(⃗q, t) fixed (equivalently, with the external control parameters {λ} fixed), we define an instantaneous stationary distribution Ps (⃗q, t) by setting the right side of Eq. (3.2) as zero [114,129]:
(
)
⃗
∇ · F (⃗q, t)Ps (⃗q, t) − ∇ · (D(⃗q, t)Ps (⃗q, t)) = 0.
(3.4)
Mathematically, this is a family of stationary Fokker-Planck equations parameterized by t (or {λ}). Each one of them is just the usual stationary Fokker-Planck
equation; the existence and uniqueness of its solution is guaranteed by certain
conditions on the drift vector, diffusion matrix and boundary conditions [16]. We
shall thus assume that at each instant of time Ps (⃗q, t) is unique, positive and normalized. Equation (3.4) also has the form ∇ · J⃗s = 0, with the instantaneous
stationary probability flux given by:
J⃗s (⃗q, t) = F⃗ ′ (⃗q, t)Ps (⃗q, t) − D(⃗q, t) · ∇Ps (⃗q, t).
(3.5)
∇ · J⃗s = 0 means J⃗s is divergence-free. There is no sink for the flux to go into
or source for it to come out from; it has to rotate and thus has a curl nature (a
solenoidal vector field).
63
The time variable t in Ps (⃗q, t) and J⃗s (⃗q, t) solely comes from the time dependence of F⃗ (⃗q, t) and D(⃗q, t) that represents changing external conditions. Although Ps (⃗q, t) is time-dependent when external conditions change, it is generally
not a solution of the transient (dynamical) Fokker-Planck equation (Eq. (3.2)).
In particular, Ps (⃗q, t) does not depend on the initial distribution P (⃗q, t0 ), while
the transient distribution Pt (⃗q, t) does [15, 16, 129]. Therefore, Ps (⃗q, t) should
not be confused with Pt (⃗q, t). When the external conditions do not change with
time, which means F⃗ (⃗q) and D(⃗q) are time-independent, the transient distribution
Pt (⃗q, t) will relax to the supposedly unique stationary distribution Ps (⃗q) in the
long time limit, and the relaxation process can be characterized by a relaxation
time scale (determined by the eigenvalues of the Fokker-Planck operator) [16].
When external conditions change with time, meaning F⃗ (⃗q, t) and D(⃗q, t) are
time-dependent, the instantaneous stationary distribution Ps (⃗q, t) plays the role of
the reference distribution at each moment which the transient distribution Pt (⃗q, t)
would try to relax to at that moment. If the external conditions change on a time
scale much slower than the relaxation time scale (relatively speaking, relaxation
happens very fast), the system would approximately stay in the instantaneous stationary distribution at each moment and thus Pt (⃗q, t) ≈ Ps (⃗q, t) all the time. In the
context of thermodynamics this describes quasi-static processes (not necessarily
reversible) [100]. In a comparable situation in quantum mechanics, this defines
the so-called adiabatic processes, where the word adiabatic has no direct relation
with heat exchange, though.
3.1.2
Generalized Potential-Flux Landscape Framework
The potential-flux landscape framework presented in the following is an extension of the previous framework on non-equilibrium steady state processes [9,
10, 12–14, 50] and that presented in Sec. 2.1.2 of chapter 2, by accommodating
transient processes and time-dependent external conditions. It will facilitate the
establishment of the non-equilibrium thermodynamic formalism later on.
The definition of the stationary probability flux in Eq. (3.5) can be reformulated as a stationary dynamical decomposition equation (also called force decomposition equation in previous work [9, 10, 13, 14]) and chapter 2:
F⃗ ′ (⃗q, t) = −D(⃗q, t) · ∇U (⃗q, t) + V⃗s (⃗q, t),
(3.6)
where U (⃗q, t) = − ln Ps (⃗q, t) is the stationary potential landscape and V⃗s (⃗q, t) =
J⃗s (⃗q, t)/Ps (⃗q, t) is the stationary flux velocity. We shall simply refer to flux velocity as flux when its meaning is clear from the notation and context. The ‘time’
64
t in the above equation is not of a dynamical nature; it can be replaced by external
control parameters {λ}. In the stationary dynamical decomposition equation, the
effective driving force is decomposed into a gradient-like part of the stationary
potential landscape and a curl-like part of the stationary flux velocity. The word
gradient-like refers to the fact that there is a diffusion matrix D before the gradient, while curl-like means J⃗s = Ps V⃗s is divergence-free although V⃗s itself is
generally not. For even state variables, the detailed balance condition characterizing equilibrium steady state is indicated by vanishing stationary flux, V⃗s = 0, or
equivalently, J⃗s = 0. Equation (3.6) is then reduced to F⃗ ′ = −D · ∇U , which is
the potential condition characterizing detailed balance [15, 16]. Thus the dynamics of the equilibrium steady state is determined solely by the gradient-like part
of the stationary potential landscape, without the contribution from the curl-like
part of the stationary flux. Non-zero stationary flux V⃗s or J⃗s indicates detailed
balance breaking in the steady state, where the dynamics of the non-equilibrium
steady state is determined by both the gradient-like part of the stationary potential landscape and the curl-like part of the stationary flux. In this sense, these
two parts represent, respectively, the detailed balance preserving and detailed balance breaking parts of the steady state. In chapter 2 we have elaborated on the
application of the stationary dynamical decomposition equation in the study of
the system’s global stability and dynamics; we shall not repeat this subject here
further again.
The definition of the transient flux in Eq. (3.3) can be reformulated as a transient dynamical decomposition equation:
F⃗ ′ (⃗q, t) = −D(⃗q, t) · ∇S(⃗q, t) + V⃗t (⃗q, t),
(3.7)
where S(⃗q, t) = − ln Pt (⃗q, t) may be termed the transient potential landscape and
V⃗t (⃗q, t) = J⃗t (⃗q, t)/Pt (⃗q, t) the transient flux velocity. In Eq. (3.7), the effective
driving force is decomposed into a gradient-like part of the transient potential
landscape and a part that is the transient flux velocity V⃗t , which is no longer curllike since the transient flux J⃗t (= Pt V⃗t ) is generally not divergence-free. However,
V⃗t and J⃗t still have significant physical meanings, which will be clarified later.
From the stationary and transient dynamical decomposition equations in Eqs. (3.6) and (3.7), we can derive a ‘relative’ dynamical constraint equation:
V⃗r (⃗q, t) = −D(⃗q, t) · ∇A(⃗q, t),
(3.8)
where A(⃗q, t) = U (⃗q, t) − S(⃗q, t) = ln[Pt (⃗q, t)/Ps (⃗q, t)], as the difference between the stationary and transient potential landscapes, can be termed the relative
65
potential landscape and V⃗r (⃗q, t) = V⃗t (⃗q, t) − V⃗s (⃗q, t), as the difference between
the transient and stationary flux velocities, may be called the relative flux velocity. Equation (3.8) expresses a constraint between A and V⃗r . Since D(⃗q, t) is
positive definite, Eq. (3.8) implies that the necessary and sufficient condition for
V⃗r = 0 (i.e., V⃗t = V⃗s ) is A = 0 (i.e., Pt = Ps ). In other words, Pt = Ps is
equivalent to V⃗t = V⃗s . When the transient distribution reaches the (instantaneous)
stationary distribution, the transient flux also coincides with the (instantaneous)
stationary flux. Deviation of the transient distribution from the stationary distribution indicates deviation of the transient flux from the stationary flux and vice
versa. Therefore, the relative potential landscape A and the relative flux velocity V⃗r directly characterize the non-stationary condition, namely deviation of the
transient state from the instantaneous steady state. What is inherent in such a deviation is the tendency to reduce the deviation through a relaxation process, an
irreversible non-equilibrium process, until or unless there is no more deviation.
Thus non-zero V⃗r (or A) is also an indicator of the irreversible non-equilibrium
relaxation process inherent in the non-stationary condition.
There are two typical situations where deviation of the transient state from the
steady state can be created, that is, by displacing either the transient state or the
steady state. When external conditions are constant, the steady state is fixed. If
the system is prepared in a state different from the steady state, a non-stationary
condition is created and the induced relaxation process towards the fixed steady
state follows. We can call it the state-preparation induced relaxation. When external conditions change with time, so is the (instantaneous) steady state. Even
if the system is initially prepared in a steady state, the change of external conditions, if fast enough (compared to the relaxation time scale), can displace the system out of the instantaneous steady state. This way a non-equilibrium relaxation
process is induced by changing external conditions. We can call it the externaldriving induced relaxation. If the external conditions change constantly and so
fast that relaxation cannot keep up, the system will be constantly displaced out
of the instantaneous steady state. (A systematic investigation on relevant issues
in chemical reaction systems based on non-Markovian dynamics can be found in
Ref. [34].) In the opposite scenario, which is the quasi-static limit, the steady state changes so slowly due to slow external driving that it cannot shake off the fast
relaxation process; the system will be approximately sticking to the instantaneous
steady state all the time and no deviation is created. In general, deviation of the
transient state from the steady state may result from both of the following factors:
(1) initial preparation of the transient state; (2) external driving of the steady state.
Yet we also remark that external driving does not always drive the steady state
66
away from the transient state; it can also drive them closer to each other if there is
already a deviation. In other words, external driving can work both ways of creating and increasing a deviation or reducing and eliminating an existent deviation.
The relative flux velocity V⃗r (or the relative potential landscape A) characterizes
the ‘relative’ deviation of the transient state from the steady state and indicates
the non-equilibrium relaxation process that always tries to reduce and eliminate
the deviation; whether that attempt is successful or not depends on which direction external driving works towards and their relative ‘speed’. We note that the
qualifier ‘induced’ in the term ‘induced relaxation’ is a matter of perspective; if
deviation from the steady state is given, then the relaxation process attempting to
reduce that deviation may as well be labeled as ‘spontaneous’ as it is driven by an
inherent tendency.
Then we come back to the meaning of the transient flux V⃗t and J⃗t . In fact, vanishing transient flux V⃗t = 0 (J⃗t = 0) implies the detailed balance condition in the
steady state V⃗s = 0 (J⃗s = 0). This is because V⃗t = 0 (J⃗t = 0) implies ∇ · J⃗t = 0,
which is the same equation as the instantaneous stationary Fokker-Planck equation in Eq. (3.4), just with Ps replaced by Pt . By the assumption (justified under
reasonable conditions) that the instantaneous stationary solution is unique at each
moment, we have Pt = Ps and thus V⃗s = V⃗t = 0 (also J⃗s = 0). The reverse
statement, however, is not true. V⃗s = 0 (J⃗s = 0) does not imply V⃗t = 0 (J⃗t = 0).
Even if the steady state obeys detailed balance V⃗s = 0 (J⃗s = 0), the system can
still be in the process of relaxing from the transient state to the steady state (with
detailed balance); thus V⃗t = V⃗s + V⃗r = V⃗r ̸= 0 according to the discussion under
Eq. (3.8). The condition V⃗t = 0 (J⃗t = 0) is equivalent to the following two conditions both satisfied: (1) detailed balance in the steady state: V⃗s = 0 (J⃗s = 0);
(2) stationarity: V⃗r = 0 (A = 0). In other words, there are two basic ways to
generate a non-zero transient flux V⃗t or J⃗t . One way is detailed balance breaking
in the steady state, indicated by non-zero stationary flux V⃗s (or J⃗s ), which characterizes the irreversible non-equilibrium nature of the steady state. The other way
is non-stationarity, indicated by non-zero relative flux V⃗r (or A), which characterizes the irreversible non-equilibrium nature of the relaxation process from the
transient state to the (instantaneous) steady state. These are the two fundamental
aspects of non-equilibrium processes [115]; together they are captured in the transient flux V⃗t (or J⃗t ). (Since the relaxation process contains two different facets,
state preparation and external driving, we may as well say there are three basic
aspects of non-equilibrium processes [129]). Therefore, V⃗t (or J⃗t ) characterizes
the combined non-equilibrium effects from both the detailed balance breaking in
the steady state characterized by V⃗s (or J⃗s ) and the relaxation process induced by
67
deviation from the steady state characterized by V⃗r (or A).
Hence, we reinterpret the following equation:
V⃗t (⃗q, t) = V⃗s (⃗q, t) + V⃗r (⃗q, t)
(3.9)
as a flux decomposition equation. It represents a fundamental decomposition of
the total non-equilibrium irreversible process characterized by V⃗t , into the stationary (steady-state) non-equilibrium irreversible process characterized by V⃗s and the
relaxational non-equilibrium irreversible process characterized by V⃗r . (Note that
the subscript t in V⃗t can represent both ‘transient’ and ‘total’, the subscript s in V⃗s
can represent both ‘stationary’ and ‘steady-state’, and the subscript r in V⃗r can represent both ‘relative’ and ‘relaxation(al)’.) The equilibrium condition is characterized by the vanishing of all the three fluxes V⃗t = V⃗s = V⃗r = 0 (with V⃗s = V⃗r = 0
implied by V⃗t = 0). Non-zero fluxes V⃗t , V⃗s and V⃗r represent the essential characteristics of non-equilibrium processes. Thus by digging out the physical meaning
of the fluxes as characterizing different aspects of non-equilibrium processes, we
have elevated Eq. (3.9) from merely an equation defining V⃗r into an equation that
represents a fundamental decomposition of non-equilibrium processes.
The flux decomposition equation in Eq. (3.9) can also be regarded as a form of
dynamical decomposition equation as the stationary dynamical decomposition equation in Eq. (3.6) and the transient dynamical decomposition in Eq. (3.7). Strictly speaking, Eq. (3.8) is a dynamical constraint equation as there is no decomposition. Yet collectively we shall simply refer to Eqs. (3.6)-(3.9) as the dynamical
decomposition equations. The quantities involved in all these four equations are
all dynamical quantities that can be defined and constructed directly from the stochastic dynamical equation. Yet there is more. In addition to the dynamical
aspects, these equations can also be understood in the context of non-equilibrium
thermodynamics, where they will take on thermodynamic meanings with their
thermodynamic aspects revealed. The dynamical decomposition equations serve
as a bridge connecting stochastic dynamics with non-equilibrium thermodynamics, within which the fluxes V⃗t , V⃗s and V⃗r play an essential role in characterizing
the non-equilibrium nature of the (thermo)dynamics of the system. Therefore, it
can be expected that these equations and in particular the fluxes will also manifest
themselves on the non-equilibrium thermodynamic level.
Before discussing the non-equilibrium thermodynamics, we remark that the
stationary potential landscape U = − ln Ps and the transient potential landscape
S = − ln Pt are only defined up to a common additive constant (corresponding to
a common multiplicative constant of probability distributions), which leaves unchanged the definition of the relative potential landscape A = U − S = ln(Pt /Ps )
68
as well as the dynamical decomposition equations in Eqs. (3.6)-(3.9). In other
words, only their difference (i.e., relative values) rather than absolute values is
physically meaningful; the common additive constant means a freedom to choose
the reference point to measure them. This issue becomes important when it comes
to spatially inhomogeneous systems.
3.1.3
Non-Equilibrium Thermodynamic Context
Now we place the stochastic dynamics into the non-equilibrium thermodynamic context. We discuss three related description levels, namely the microscopic level, the macroscopic level and the ensemble level.
We assume the microscopic state of the system is specified by a microstate
vector ⃗q. The space of the microstate ⃗q is called the microstate space. The microstate ⃗q evolves according to a stochastic dynamics described by the Langevin
equation (Eq. (3.1)), tracing a stochastic trajectory on the microstate space. In general, the microscopic stochastic dynamics is regulated by macroscopic external
conditions (e.g., mechanical constraints, the temperatures of the heat reservoirs
the system is coupled to), which enter the microscopic dynamical equation in the
form of external control parameters [129, 130]. Such dynamics may emerge as an
effective internal description of open systems, which can be derived from the larger dynamics of the system and the environment by eliminating the environment’s
degrees of freedom [149]. (For a concrete example based on master equations and
investigations on the consistency of the internal dynamics and the larger dynamics, see Ref. [150].) We note that the word ‘microscopic’ used here is based on the
premise that the stochastic dynamics is applicable, which may require the description of the system to be on a certain level of coarse-graining. Thus the description
may actually be on a ‘mesoscopic’ level in terms of its physical scale. We shall,
however, proceed with the word ‘microscopic’, with its implication in mind.
We define the macroscopic external conditions that the system is subject to
as a macrostate, which is assumed to be specified by a set of macrostate variables, denoted by {λ} ≡ {..., λi , ...}, regulating the microscopic internal dynamics in the form of external control parameters. The space of the macrostate {λ}
is referred to as the macrostate space. The macrostate defined this way, however, does not have to represent the properties of the system itself; it may reflect
relations between the system and the environment. In particular, it can represent non-equilibrium constraints imposed on the system by the environment or
sustained non-equilibrium conditions through constant system-environment interaction and exchange of matter and energy [1, 100, 105, 108, 150]. Therefore,
69
in addition to equilibrium macrostates, {λ} may also represent non-equilibrium
macroscopic steady states [1, 100]. An extension of the equilibrium state space to
accommodate non-equilibrium steady states has been proposed in a phenomenological thermodynamic framework [100]. The idea of incorporating matter and
energy fluxes as state variables has also been suggested in ‘extended irreversible
thermodynamics’ [102]. Generally, equilibrium macrostates form a subspace of
the extended space of macroscopic steady states, with those parameters characterizing non-equilibrium macroscopic conditions vanishing. We also consider timedependent external conditions which can be represented by {λ(t)}. To simplify
⃗ s (⃗q, t) and
notation, we often just write {λ}. The time dependence in F⃗ (⃗q, t), G
that which originates from them can now be replaced by {λ}, taking on the follow⃗ s (⃗q, {λ}), D(⃗q, {λ}), Ps (⃗q, {λ}), J⃗s (⃗q, {λ}), U (⃗q, {λ})
ing form: F⃗ (⃗q, {λ}), G
and V⃗s (⃗q, {λ}).
An ensemble of systems compatible with the macrostate {λ} (i.e., all under the
same macroscopic conditions specified by {λ}), existing in different microstates
with certain probabilities, can be described by an ensemble probability distribution P (⃗q, {λ}). The Fokker-Planck equation (Eq. (3.2)) is interpreted as the dynamical equation governing the evolution of an ensemble of systems, initially
prepared in a particular ensemble distribution, each of which is independently evolving according to the Langevin equation (Eq. (3.1)). We denote the transient
ensemble distribution by Pt (⃗q, {λ}; t). (The time variable t behind the semicolon
represents the ‘total’ time variable, defined by the transient Fokker-Planck equation (Eq. (3.2)), with ∂t P (⃗q, {λ}; t) understood as the derivative with respect to
the total time variable.) The stationary ensemble distribution Ps (⃗q, {λ}) is the
distribution that an ensemble of systems, regardless of their initial preparation,
eventually settle in when the macrostate {λ} is fixed. However, there could also be a non-zero stationary probability flux J⃗s (⃗q, {λ}). Therefore, corresponding
to each macroscopic steady state {λ}, there is a stationary pair Ps (⃗q, {λ}) and
J⃗s (⃗q, {λ}) of the ensemble, which is the microscopic statistical characterization
of the macrostate {λ}. (Equivalently, {λ} can be characterized by the stationary potential landscape U (⃗q, {λ}) and the stationary flux velocity V⃗s (⃗q, {λ}) since U = − ln Ps and V⃗s = J⃗s /Ps [50]). Equilibrium macrostates {λ}eq can be
identified with those ensembles satisfying the detailed balance condition, characterized by J⃗s (⃗q, {λ}eq ) = 0 (or V⃗s (⃗q, {λ}eq ) = 0). Thus equilibrium macrostates
can be described micro-statistically by the stationary equilibrium ensemble distribution Ps (⃗q, {λ}eq ) alone. Non-equilibrium macroscopic steady states {λ}neq
can be identified with those ensembles with detailed balance broken, indicated
70
by J⃗s (⃗q, {λ}neq ) ̸= 0 (or V⃗s (⃗q, {λ}neq ) ̸= 0). Therefore, they must be described
micro-statistically by the pair Ps (⃗q, {λ}neq ) and J⃗s (⃗q, {λ}neq ) together. The stationary probability flux on the microstate space connects different microstates into
a coherent global state, representing the non-equilibrium macroscopic steady state
⃗ q , {λ}) together may be
on the micro-statistical level. The pair P (⃗q, {λ}) and J(⃗
more appropriately called an ensemble ‘state’ [88], though P (⃗q, {λ}) alone is also
referred to as that. (A relevant fact is that given the dynamical equation, the probability flux and probability distribution are related via, e.g., Eqs. (3.3) and (3.5).)
When the macrostate {λ} changes with time (external driving), Ps , J⃗s , U and V⃗s
also change accordingly, since they all directly depend on {λ}. Thus external
driving drives the steady state, where ‘steady state’ means both the macroscopic
steady state and the ensemble steady state and these two meanings are consistent.
For a discussion on the micro-macro correspondence of non-equilibrium states in
terms of master equations, see Ref. [2].
Three types of state have been introduced, namely the microstate ⃗q, the macrostate
⃗ q , {λ})). Accordingly, there are
{λ} and the ensemble state P (⃗q, {λ}) (and J(⃗
three types of state functions. A function that depends on the microstate ⃗q, which
may also depend on the macrostate {λ}, since the microstate and its dynamics are
subject to macroscopic conditions, is called a microstate function. A function that
does not depend on the microstate ⃗q, but only depends on the macrostate {λ}, is
called a macrostate function. We use calligraphy letters (e.g., S, A, U) to denote
macrostate functions. A macrostate function can usually be expressed as the ensemble average of a microstate function; if so, it is also an ensemble state function
as it is a function(al) of the ensemble distribution. To specify such a macrostate
function, we need to specify both the microstate function and the ensemble distribution in taking the ensemble average. In particular, whether the transient or
the stationary ensemble is used need to be indicated, in contrast with equilibrium
statistical mechanics where only the equilibrium ensemble is of concern.
3.1.4
State Functions of Non-Equilibrium Isothermal Processes
In the following we focus on systems in an environment with a constant temperature T , which at the same time may still be subject to other non-equilibrium
conditions, such as non-equilibrium steady states sustained by chemical potential
difference or nonconservative forces [105,129,150], non-equilibrium transient relaxation processes due to time-dependent external driving [34, 129, 130] or initial
71
state preparation [109, 115]. The environment in this situation serves as both a
heat reservoir providing a constant temperature and a source of non-equilibrium
conditions through system-environment interaction and exchange of matter and
energy. Such non-equilibrium processes taking place at a constant temperature
are referred to as non-equilibrium isothermal processes.
For some systems modeled by Langevin and Fokker-Planck dynamics, the diffusion matrix D characterizing the fluctuating forces is, under certain conditions,
proportional to the temperature of the heat reservoir (environment), as a result of
the fluctuation-dissipation theorem [15, 16, 151]. We shall assume that the systems we consider also satisfy this condition D ∝ T . It means the fluctuation
strength (the magnitude of D) is proportional to T . Intuitively, higher temperature corresponds to stronger fluctuations. Thus the temperature plays the role of
a fluctuation strength parameter. In general, D ∝ T breaks down when the temperature becomes too low as quantum effects dominate. We define the re-scaled
e q , {λ′ }) = T −1 D(⃗q, {λ}), where
temperature-independent diffusion matrix D(⃗
{λ′ } represents {λ} with T excluded.
Then we consider the state functions for non-equilibrium isothermal processes. For such processes three non-equilibrium (macro)state functions, namely the
non-equilibrium entropy, internal energy and free energy have been introduced
formally for master equations as well as for Fokker-Planck equations [11, 12, 114,
115]. Here we investigate these quantities in connection with the potential-flux
landscape framework. We also discuss some subtleties involved in their definitions, especially regarding the indeterminacy of the microscopic non-equilibrium
energy function and its connection to macroscopic external conditions, and we
propose possible solutions.
Non-Equilibrium Entropy
The non-equilibrium entropy of the system, more specifically, the non-equilibrium
macroscopic transient entropy, is defined by applying the Gibbs entropy formula
to the transient ensemble distribution (Boltzmann constant kB is set to 1 throughout this article) [11, 12, 109, 112, 118, 120, 125]:
∫
S({λ}; t) = ⟨S(⃗q, {λ}; t)⟩t = − Pt (⃗q, {λ}; t) ln Pt (⃗q, {λ}; t)d⃗q,
(3.10)
where ⟨·⟩t represents average over the transient ensemble Pt . The microstate function S(⃗q, {λ}; t) = − ln Pt (⃗q, {λ}; t) is the transient potential landscape introduced in Eq. (3.7). Now its thermodynamic meaning is the microscopic transient
72
entropy; its average over the transient ensemble gives the macroscopic transient
entropy. We note that S = − ln Pt is also referred to as the ‘stochastic entropy’
if it is defined on a stochastic trajectory [128–130]. As mentioned before, S is
defined up to an additive constant, so is its ensemble average S = ⟨S⟩t , which
means we are only concerned with entropy differences.
If the transient distribution Pt in Eq. (3.10) is replaced by the stationary distribution Ps , the resulting entropy Ss = ⟨− ln Ps ⟩s may be called the (non-equilibrium
macroscopic) stationary entropy. In the equilibrium limit Pt = Ps = Pe , Eq. (3.10) recovers the equilibrium entropy Se = ⟨− ln Pe ⟩e . Thus Eq. (3.10) seems
to be a natural extension of the equilibrium entropy to non-equilibrium systems.
Yet there are also other definitions of the non-equilibrium entropy in the literature
based on the Gibbs entropy postulate [101], which is more relevant to the concept
of relative entropy [115, 118, 143] that will also be discussed later.
Non-Equilibrium Internal Energy and Cross Entropy
We know that for equilibrium systems in contact with a heat reservoir, with
the microscopic energy E given a priori (e.g., the Hamiltonian of the conservative system), the equilibrium distribution is the canonical ensemble distribue
tion Pe = Z −1 e−E/T = e(Ae −E)/T , where Z is the partition function and Aee =
−T ln Z is the equilibrium free energy. The equilibrium internal energy is given
by Uee = ⟨E⟩e , where ⟨·⟩e represents the ensemble average over Pe . For isothermal non-equilibrium systems without detailed balance (e.g., a dissipative system),
a microscopic energy function connected to the non-equilibrium stationary distribution in a similar fashion as the equilibrium canonical ensemble is not given a
priori. Yet one may still ask whether it is possible to identify or construct such
a microscopic energy function, if the non-equilibrium stationary distribution is
solved [9, 10, 13, 14, 82, 85, 115, 152].
For general Fokker-Planck dynamics describing non-equilibrium isothermal
processes, there is one condition D ∝ T we can employ. In the study of small
fluctuation problems of the Fokker-Planck equation, the stationary distribution
can be expanded into an asymptotic series in terms of the fluctuation strength
parameter using the WKB method [12, 50, 82, 153]. Given that D ∝ T means
the temperature T plays the role of a fluctuation strength parameter, the stationary
73
distribution can be assumed to have the following asymptotic form for small T :
]
[
]
[
∑ k e (k)
1 +∞
1e
′
T U (⃗q, {λ })
exp −
exp − U (⃗q, {λ})
T k=0
T
Ps (⃗q, {λ}) =
=
, (3.11)
Z({λ})
Z({λ})
[
]
∫
e
where Z({λ}) = exp −U (⃗q, {λ})/T d⃗q is the normalization factor; {λ′ } is
e (⃗q, {λ}) = ∑+∞ T k U
e (k) (⃗q, {λ′ }) is an asymptotic series
{λ} with T excluded; U
k=0
e (k) (⃗q, {λ′ }) (k = 0, 1, ...) can be obtained by plugfor small T . The equation for U
ging the above ansatz into the stationary Fokker-Planck equation (Eq. (3.4)) and
match each order of T . Formally, Eq. (3.11) is an extension of the equilibrium
canonical ensemble distribution to non-equilibrium isothermal processes, where
e (⃗q, {λ}) serves as the miZ({λ}) is the analog of the partition function and U
croscopic non-equilibrium energy. However, there are some subtleties, which we
discuss below.
e (⃗q, {λ}) may depend on the temperature T (although this
First, in general, U
is not always the case), in contrast with equilibrium systems whose microscopic
energy (e.g., the system’s Hamiltonian) is generically independent of temperae (⃗q, {λ}) (if well defined) is an effective microscopic
ture. This indicates that U
energy, which may (partially) have a macroscopic origin. Its dependence on the
(non-equilibrium) external conditions {λ} suggests that it may also have included
the system-environment interaction energy. For perspectives on the issues around
including or not the interaction energy in the microscopic energy function and
thus also the internal energy, see Ref. [154]. In our case, since {λ} can also describe non-equilibrium steady states sustained by constant system-environment
interaction, it does not seem to be so unreasonable that interaction energy need
to be accounted for, in the description of an effective internal dynamics that only
involves the system’s degrees of freedom, in order to have a consistent thermodynamic description.
e (⃗q, {λ}) and Z({λ}) are not uniquely determined by Ps (⃗q, {λ}).
Second, U
e (⃗q, {λ}) is only determined up to an additive macrostate function C({λ}). AcU
cordingly, Z({λ}) is determined up to a multiplicative macrostate function in the
form of exp(−C({λ})/T ). This indeterminacy can be seen in another form. We
introduce the non-equilibrium stationary free energy (the name will be justified
later), motivated by the relation of the partition function with equilibrium free
energy [151]:
Aes ({λ}) = −T ln Z({λ}).
(3.12)
74
Note that Aes is a macrostate function, independent of ⃗q. Then Eq. (3.11) can be
written as:
[ (
)]
1 e
e
Ps (⃗q, {λ}) = exp
As ({λ}) − U (⃗q, {λ}) .
(3.13)
T
Further, we have
e (⃗q, {λ}) − Aes ({λ}) = T U (⃗q, {λ}),
U
(3.14)
where the dimensionless microstate function U (⃗q, {λ}) = − ln Ps (⃗q, {λ}) is the
stationary potential landscape introduced in Eq. (3.6); in the context of nonequilibrium thermodynamics, it is the microscopic stationary entropy (compare
with the microscopic transient entropy S = − ln Pt ). Equation (3.14) shows that
e (⃗q, {λ}) − Aes ({λ})
the stationary distribution can only determine the difference U
rather than each of them individually. There is a common additive macrostate
e (⃗q, {λ}) and Aes ({λ})
function C({λ}) in their expressions undetermined. If U
have thermodynamic meanings individually (as an extension of their equilibrium counterparts we expect so), the macrostate function C({λ}) not fixed by the
stochastic dynamical equation (which determines the stationary distribution) indicates that the microscopic internal dynamics alone is not sufficient to give a
complete description of the system’s thermodynamics [121, 130, 135]. Hence, ade (⃗q, {λ}) and Aes ({λ}), up to an
ditional information is needed to fix (one of) U
additive constant independent of {λ}.
One way is to fix Aes ({λ}). We know from equilibrium thermodynamics that
if the equations of state for the equilibrium free energy Aee ({λ}) are known, i.e.,
Λie ({λ}) = ∂ Aee ({λ})/∂λi (i = 0, 1, ...) are given, where λi ’s are the natural variables of the equilibrium free energy and Λie is the variable conjugate to λi , then
∫∑ i
the free energy can be found by integration Aee ({λ}) =
i Λe ({λ})dλi , determined up to an additive integration constant. This suggests that the ‘equations of
state’ for the non-equilibrium stationary free energy, namely
Λis ({λ}) = ∂ Aes ({λ})/∂λi ,
(3.15)
(with i = 0, 1, ... and λ0 ≡ T ) can serve as the additional information required
to supplement the thermodynamic description of the system apart from the microscopic internal dynamics. With Λis ({λ}) given, Aes ({λ}) is determined up to
∫∑ i
an additive constant by integration: Aes ({λ}) =
i Λs ({λ})dλi . Accordingly,
e
U (⃗q, {λ}) is also determined up to an additive constant by Eq. (3.13) or (3.14). In
the discussion of the second law of thermodynamics in terms of free energy, we
will also propose an operational definition of Aes consistent with this approach.
75
e (⃗q, {λ}). Since U
e (⃗q, {λ}) as a function of ⃗q has
The other way is to fix U
already been partially determined by the microscopic internal dynamics, the freedom that is left to be fixed is a macrostate function C({λ}). Thus we consider
e . We define the non-equilibrium stationdetermining the ensemble average of U
e (⃗q, {λ})⟩s , where ⟨·⟩s means ensemble average
ary internal energy Ues ({λ}) = ⟨U
e is then transferred to Ues . This
over Ps . The undetermined function C({λ}) in U
shows the non-equilibrium stationary internal energy Ues cannot be derived from
the microscopic internal dynamics; it need to be supplemented to complete the
e is determined.
thermodynamic description of the system. If Ues is given, then U
Accordingly, Aes is also determined. In the discussion of the first law of thermodynamics, we will propose an operational definition of the non-equilibrium
stationary internal energy Ues .
The equivalence of these two approaches, determining Aes ({λ}) or Ues ({λ}),
can be seen more directly from Eq. (3.14), by taking the ensemble average over Ps ,
which gives Ues = Aes + T Ss , with the stationary entropy Ss = ⟨U ⟩s = ⟨− ln Ps ⟩s
determined by the microscopic internal dynamics. Thus with either Aes ({λ}) or
e (⃗q, {λ}) is then determined. This also validates our statemenUes ({λ}) given, U
e (⃗q, {λ}) as an effective microscopic energy partially has a macroscopt that U
ic origin; it is determined by both the microscopic internal dynamics and the
macroscopic external conditions. In fact, there exists a family of non-equilibrium
systems, all compatible with the same microscopic internal dynamics regulated
by macroscopic external conditions {λ}, which are not thermodynamically equivalent, as they are characterized by different stationary free energy functions
Aes ({λ}), or equivalently stationary internal energy functions Ues ({λ}). This agrees with the perspective that the subsystem dynamics is not sufficient to handle
the full first law of thermodynamics [121, 130, 135].
e addressed,
With the issues around the microscopic non-equilibrium energy U
e
e over the
we define the non-equilibrium internal energy U as the average of U
e ⟩t . In the steady state Pt = Ps , it coincides with
transient ensemble, i.e., Ue = ⟨U
e ⟩s already introduced. In
the non-equilibrium stationary internal energy Ues = ⟨U
e
the equilibrium state Pt = Ps = Pe , the microscopic non-equilibrium energy U
ee ; then Ue and Ues
reduces to the equilibrium microscopic energy of the system U
e
e
recover the equilibrium internal energy Ue = ⟨Ue ⟩e . According to Eq. (3.14), the
non-equilibrium internal energy is also given by:
e
e (⃗q, {λ})⟩t = T U({λ}; t) + Aes ({λ}),
U({λ};
t) = ⟨U
76
(3.16)
where
∫
U({λ}; t) = ⟨U (⃗q, {λ})⟩t = −
Pt (⃗q, {λ}; t) ln Ps (⃗q, {λ})d⃗q.
(3.17)
The dimensionless macrostate function U = ⟨− ln Ps ⟩t is the average of the microscopic stationary entropy − ln Ps over the transient ensemble distribution Pt .
Thus it does not have the exact form of the Gibbs entropy formula. It is neither
the transient entropy St = ⟨− ln Pt ⟩t , nor the stationary entropy Ss = ⟨− ln Ps ⟩s .
In the language of information theory [143], U = ⟨− ln Ps ⟩t is the cross entropy,
of the stationary distribution Ps with respect to the transient distribution Pt . The
word ‘cross’ refers to the fact that its definition ⟨− ln Ps ⟩t involves two different
probability distributions Ps and Pt . In the steady state Pt = Ps , the transient entropy, cross entropy and stationary entropy coincide St = U = Ss = ⟨− ln Ps ⟩s .
e ⟩t reduces to Ues = ⟨U
e ⟩s in the steady state. Accordingly, Eq. (3.16)
Also, Ue = ⟨U
reads Ues = T Ss + Aes ; equivalently, Aes = Ues − T Ss , which is the relation of internal energy, entropy and free energy in non-equilibrium steady states of isothermal
processes. In the equilibrium state Pt = Ps = Pe , it recovers the equilibrium
relation Aee = Uee − T Se .
The non-equilibrium internal energy Ue in isothermal processes is related to
the cross entropy U by Eq. (3.16). If in the isothermal process external conditions do not change (no external driving), then Aes ({λ}) remains constant. The
e according to Eq. (3.16), is then proportional to the
change of internal energy U,
change of cross entropy U, i.e., ∆Ue = T ∆U. If external conditions also change,
then ∆Ue = T ∆U + ∆Aes . Internal energy Ue and cross entropy U have different
physical meanings; yet their thermodynamic equations turn out to have similar
structures. (That is also why they are denoted with similar symbols.) In nonequilibrium isothermal processes, the non-equilibrium internal energy Ue is what
we mainly work with. For more general non-equilibrium processes, even if there
is no obvious way to introduce the non-equilibrium internal energy, the cross entropy U = ⟨− ln Ps ⟩t is still readily defined by the probability distributions Ps and
Pt .
Non-Equilibrium Free Energy and Relative Entropy
Motivated by the relation of free energy, internal energy and entropy in equilibrium thermodynamics, Aee = Uee − T Se , we introduce the microscopic nonequilibrium free energy
e q , {λ}; t) = U
e (⃗q, {λ}) − T S(⃗q, {λ}; t).
A(⃗
77
(3.18)
The macroscopic non-equilibrium free energy is defined as the ensemble average
e
of A:
e
e q , {λ}; t)⟩t = U({λ};
e
A({λ};
t) = ⟨A(⃗
t) − T S({λ}; t).
(3.19)
In the equilibrium situation Pt = Ps = Pe , the non-equilibrium free energy Ae recovers the equilibrium free energy, as the non-equilibrium internal energy Ue and
non-equilibrium entropy S reduce to their respective equilibrium counterparts Uee
and Se . Therefore, the non-equilibrium free energy is an extension of the equilibrium free energy to non-equilibrium isothermal processes. From Eqs. (3.14) and
(3.18), we also have
e q , {λ}; t) = T A(⃗q, {λ}; t) + Aes ({λ}),
A(⃗
(3.20)
where
A(⃗q, {λ}; t) = U (⃗q, {λ}; t)−S(⃗q, {λ}; t) = ln[Pt (⃗q, {λ}; t)/Ps (⃗q, {λ})]. (3.21)
The microstate function A = U − S = ln(Pt /Ps ) was introduced in Eq. (3.8)
as the relative potential landscape. In the context of non-equilibrium thermodynamics, it is, according to Eq. (3.21), the microscopic relative entropy, between
the microscopic stationary entropy U and the microscopic transient entropy S.
Taking the transient ensemble average of Eqs. (3.20) and (3.21), we have
e
A({λ};
t) = T A({λ}; t) + Aes ({λ}),
(3.22)
A({λ}; t) = U({λ}; t) − S({λ}; t)
∫
=
Pt (⃗q, {λ}; t) ln[Pt (⃗q, {λ}; t)/Ps (⃗q, {λ})]d⃗q.
(3.23)
The macrostate function A = ⟨A⟩t = ⟨ln(Pt /Ps )⟩t in Eq. (3.23) is known as relative entropy (Kullback-Leibler divergence) in information theory [16, 115, 118,
143]. It quantifies the deviation of the (transient) distribution Pt from the (stationary) distribution Ps . According to Eq. (3.23), when Pt = Ps , the relative entropy
A vanishes. Thus Eq. (3.22) shows Ae = Aes when Pt = Ps , justifying the name
‘non-equilibrium stationary free energy’ given to Aes as it was introduced in Eq. (A.1). Equation (3.22) relates the non-equilibrium free energy in isothermal
processes to relative entropy [115, 118]. (We remark that this is not the same as
the postulate in Ref. [101], which defines the non-equilibrium entropy in terms of
relative entropy. The implication of these different postulates need further investigation.) The change of these quantities in isothermal processes are related by
78
∆Ae = T ∆A + ∆Aes . If external conditions do not change, we have ∆Ae = T ∆A.
For more general non-equilibrium processes, the relative entropy is still readily
defined by A = ⟨ln(Pt /Ps )⟩t , even if the non-equilibrium free energy Ae is not
easy to introduce.
The non-equilibrium internal energy Ue and non-equilibrium free energy Ae in
isothermal processes are related, respectively, to the cross entropy U and relative
entropy A, as shown in Eqs. (3.16) and (3.22) as well as their microscopic counterparts in Eqs. (3.14) and (3.20). These relations are summarized below:
e = T U + Aes ,
U
Ue = T U + Aes ,
(3.24)
e = T A + Aes ,
A
Ae = T A + Aes .
(3.25)
Note that macrostate functions are denoted by calligraphy letters. The definitions
e , Ue = ⟨U
e ⟩t ,
of the quantities on the left side of these equations are given by U
e = U
e − T S, Ae = ⟨A⟩
e t = Ue − T S; those on the right side are given by
A
U = − ln Ps , U = ⟨U ⟩t = ⟨− ln Ps ⟩t , A = U − S = ln(Pt /Ps ), A = ⟨A⟩t =
U − S = ⟨ln(Pt /Ps )⟩t . Due to the same mathematical structure in Eqs. (3.24) and
e , U,
e A,
e Ae and U , U, A, A, satisfy similar
(3.25), these two sets of quantities, U
e , S and
equations as their counterparts do. In particular, the microstate functions U
e also satisfy a set of dynamical decomposition equations, corresponding to those
A
for the potential landscapes (microscopic entropies) U , S and A in Eqs. (3.6)(3.8). They can be written in the following alternative form that is more convenient
for application in non-equilibrium thermodynamics:


e −1 ⃗
e −1 ⃗ ′
e

 ∇U = D · Vs − D · F
(3.26)
∇S = D −1 · V⃗t − D −1 · F⃗ ′


 e
e −1 · V⃗r
∇ A = −D
e = T −1 D is the rescaled diffusion matrix independent of T . These three eD
quations in Eq. (3.26) are, respectively, the alternative forms of the stationary,
transient and relative dynamical decomposition equations. We demonstrate in the
e S and Ae and the
following that from the three non-equilibrium state functions U,
three dynamical decomposition equations (Eq. (3.26)) together with the flux decomposition equation V⃗t = V⃗s + V⃗r (Eq. (3.9)), an entire set of thermodynamic equations, representing the thermodynamic laws governing non-equilibrium
isothermal processes, can be constructed, together with the expressions of the
thermodynamic quantities involved.
79
3.1.5
Thermodynamic Laws of Non-Equilibrium Isothermal Processes
We study the first law of thermodynamics in terms of the non-equilibrium
internal energy and then the second law of thermodynamics in terms of the nonequilibrium entropy as well as the non-equilibrium free energy. We shall calculate
the rate of change of these thermodynamic functions. It is convenient to first
consider more general cases. To simplify notation, ⟨·⟩ represents average over
the transient ensemble. Let O be a general microstate function. Its (transient)
ensemble average is the macrostate function O = ⟨O⟩. The rate of change of O is
calculated as follows:
∫
dO
d
=
OPt d⃗q
dt
dt
∫
∫
(
)
=
(∂t O) Pt d⃗q − O ∇ · J⃗t d⃗q
= ⟨∂t O⟩ + ⟨V⃗t · ∇O⟩,
(3.27)
where we have used ∂t Pt = −∇ · J⃗t and integration by parts with vanishing
boundary terms under appropriate boundary conditions. By introducing the advective derivative as in fluid dynamics:
DO
∂O ⃗
=
+ Vt · ∇O,
Dt
∂t
(3.28)
Eq. (3.27) can also be written as d⟨O⟩/dt = ⟨DO/Dt⟩.
The First Law of Non-Equilibrium Thermodynamics
The rate of change of the non-equilibrium internal energy, according to Eq. (3.27), is given by:
e ⟩ + ⟨V⃗t · ∇U
e ⟩.
Uė = ⟨∂t U
(3.29)
e ⟩ is generally non-zero if {λ} change with time
e is a function of {λ}, ⟨∂t U
Since U
e ⟩ = ∑ ⟨∂λ U
e ⟩λ̇i .
(time-dependent external conditions). More explicitly, ⟨∂t U
i
i
Temperature T ≡ λ0 is excluded here since T is constant in isothermal processes. Motivated by equilibrium statistical mechanics [151], we introduce the
e ⟩ (i = 1, 2, ...), conjugate to the
(non-equilibrium) generalized force Λi ≡ ⟨∂λi U
(non-equilibrium) generalized coordinate λi . Λi and λi form a conjugate pair.
80
Λi dλi is the product of the generalized force and the generalized displacement, which represents the thermodynamic work
through changing the external
∑ done
e ⟩λ̇i = ∑ Λi dλi /dt can be idencondition represented by λi . Therefore, i ⟨∂λi U
i
tified as the total thermodynamic work per unit time (i.e., power) done through
changing the external conditions specified by {λ}, a process referred to as external driving. Thus we define the external-driving power as follows:
∑
∑
e ⟩λ̇i =
e⟩ =
Λi λ̇i ,
(3.30)
⟨∂λi U
Ẇed = ⟨∂t U
i
i
where the subscript ‘ed’ of Ẇed represents external driving and the sum excludes
λ0 = T . The sign convention is that Ẇed is positive (negative) when the environment is doing work on the system (the system is doing work on the environment)
through changing external conditions. For a careful account of the concepts of
work, especially when interaction energy is involved, see Ref. [154]. Since {λ}
can represent non-equilibrium external conditions, the thermodynamic work done
through changing {λ} may also incorporate non-equilibrium effects. Thus Eq.
(3.30) is an extension of the equilibrium thermodynamic work to non-equilibrium
isothermal processes.
When discussing the equations of state of the stationary free energy, we introduced Λis = ∂λi Aes (i = 0, 1, ...) in Eq. (3.15). Here we introduced the generalized
e ⟩t (i = 1, 2, ...), with the subscript t spelled out for clarity. These
force Λit ≡ ⟨∂λi U
two sets of quantities are related, due to the following results (proven in Appendix
A):
e ⟩s (i = 1, 2, ...),
Λis = ∂λi Aes = ⟨∂λi U
e ⟩s − Ss .
Λ0s = ∂T Aes = ⟨∂T U
(3.31)
According to the first equation, Λit coincide with Λis in the steady state for i =
1, 2, ..., thus justifying the notation used. For i = 0 (λ0 = T ), the second equation differs from the equilibrium relation ∂T Aee = −Se by an additional term
e ⟩s , due to the possible temperature dependence of the effective microscopic
⟨∂T U
e.
non-equilibrium energy U
Comparing with the first law of equilibrium thermodynamics, the second term in Eq. (3.29) can be interpreted as the rate of heat transfer. It will become
clear in the discussion of the second law that this heat is only part of the total
heat transferred between the environment and the system. There is another part
called the housekeeping heat [100, 108] due to the dissipation in maintaining the
non-equilibrium steady state, which is constantly produced within the system and
81
constantly expelled into the environment, leaving the non-equilibrium internal energy unchanged. The part of heat left after removing the housekeeping heat is
called excess heat [100, 108]. That is what we are dealing with here. Thus we
define the second term in Eq. (3.29), with a negative sign, as the excess heat flow
rate:
e ⟩.
Q̇ex = −⟨V⃗t · ∇U
(3.32)
The adopted sign convention is that heat is positive (negative) when transferred
from the system to the environment (from the environment to the system), which
seems natural considering that the house-keeping heat can only flow from the
system to the environment.
Now Eq. (3.29) can be written as a balance equation for the non-equilibrium
internal energy:
Uė = Ẇed − Q̇ex .
(3.33)
This is an extension of the first law of equilibrium thermodynamics to non-equilibrium
isothermal processes, as a statement of energy conservation. It states there are two
different ways to change the non-equilibrium internal energy. One way is doing
work through changing external conditions. The other way is transferring excess
heat, the heat in addition to the housekeeping heat that is generated and expelled
constantly in maintaining non-equilibrium steady states. For equilibrium systems
the housekeeping heat vanishes and the excess heat and the total heat coincide
with the equilibrium heat; the non-equilibrium internal energy and work also reduces to their equilibrium counterparts. Then Eq. (3.33) becomes a statement of
the first law in equilibrium thermodynamics .
The first law of thermodynamics suggests an operational definition of the nonequilibrium stationary internal energy Ues , considering how equilibrium internal
energy can be operationally defined by work in adiabatic processes. Consider processes that are quasi-static (in the sense that {λ} change so slowly that Pt ≈ Ps all
the time) as well as adiabatic to the excess heat (i.e., Qex = 0 during the process).
For perspectives on the constantly generated house-keeping heat in the steady state
as well as some other technical issues involved in such processes, we refer to Ref. [100]. In such quasi-static adiabatic processes, we can use the thermodynamic
work Wed performed to determine the non-equilibrium stationary internal energy, according to ∆Ues = Wed . With Ues ({λ}) operationally defined, the effective
e (⃗q, {λ}) can then be constructed
microscopic non-equilibrium energy function U
from the microscopic internal dynamics. Strictly speaking, the first law of nonequilibrium thermodynamics in Eq. (3.33) is not ‘derived’; the determination of
e is dependent on the application of the quasi-static behavior of Eq. (3.33) in the
U
82
e
first place. Yet the transient behavior of Eq. (3.33) is not arbitrarily assumed, if U
has already been determined.
The Second Law of Non-Equilibrium Thermodynamics in terms of Entropy
The rate of change of the non-equilibrium entropy is also given by Eq. (3.27).
Yet it is not difficult to prove that ⟨∂t S⟩t = 0 with S = − ln Pt , due to probability
conservation. Thus we actually have
Ṡ = ⟨V⃗t · ∇S⟩.
(3.34)
Plugging in Eq. (3.26) for ∇S, we have
Ṡ = ⟨V⃗t · D −1 · V⃗t ⟩ − ⟨V⃗t · D −1 · F⃗ ′ ⟩.
(3.35)
Thus the rate of change of the non-equilibrium entropy is split into two parts. The
first term is always nonnegative and can be identified as the entropy production
rate within the system [2, 11, 12, 112, 118, 120]:
Ṡpd = ⟨V⃗t · D −1 · V⃗t ⟩,
(3.36)
where the subscript ‘pd’ represents production. The entropy production rate Ṡpd ,
as a macrostate function, measures the extent of non-equilibrium irreversibility
of the system on the macroscopic level. The transient flux velocity V⃗t , as a microstate function, indicates the non-equilibrium irreversibility of the system on the
microscopic level. Thus it is not unexpected that they are related to each other in
Eq. (3.36). Since D −1 is positive definite and the transient ensemble distribution
Pt is positive in the accessible state space, the necessary and sufficient condition
for Ṡpd ≡ 0 is V⃗t ≡ 0, that is, the probability flux V⃗t (or J⃗t ) vanishes all the time
and everywhere in the accessible (micro)state space. The second term (without
the minus sign) in Eq. (3.35) represents the rate of entropy flow from the system
to the environment [2, 11, 12, 112, 118, 120]:
Ṡf l = ⟨V⃗t · D −1 · F⃗ ′ ⟩,
(3.37)
where the subscript ‘f l’ represents flow. Therefore Eq. (3.35) can be identified as
the (transient) entropy balance equation:
Ṡ = Ṡpd − Ṡf l .
83
(3.38)
It states that the entropy of the system is increased when there is entropy production within the system and decreased when there is entropy flow from the
system into the environment. This transient entropy balance equation is a direct
reflection and manifestation of the transient dynamical decomposition equation
F⃗ ′ = −D · ∇S + V⃗t , as seen from Eqs. (3.34)-(3.38).
For non-equilibrium isothermal processes, the entropy flow is related to the
total heat transfer according to dS
¯ f l = dQ
¯ tot /T , where d¯ represents inexact
differentials. Thus we define the total heat flow rate:
e −1 · F⃗ ′ ⟩,
Q̇tot = T Ṡf l = ⟨V⃗t · D
(3.39)
where we have used Eq. (3.37). The sign convention for heat is as before; heat
is positive when flowing from the system to the environment. The non-negativity
of the entropy production rate Ṡpd ≥ 0 allows the entropy balance equation (Eq. (3.38)) to be written as an inequality:
dQ
¯ tot
dS ≥ −
,
T
(3.40)
which has the conventional form of the second law of thermodynamics, except for
a sign due to the adopted sign convention of heat already explained.
Comparing Eq. (3.32) and Eq. (3.39), we see that in non-equilibrium isothermal processes the heat that changes the non-equilibrium internal energy (i.e., the
excess heat) and the heat that contributes to the entropy flow (i.e., the total heat)
do not match each other generally. This difference indicates that there is another
part of heat (the house-keeping heat) which contributes to the entropy flow yet
does not change the non-equilibrium internal energy. In other words, the total
heat is composed of the excess heat and the housekeeping heat. Therefore, a single concept of heat in equilibrium thermodynamics has differentiated into three
distinct yet related concepts of heat in non-equilibrium thermodynamics, namely
the total heat, the excess heat and the housekeeping heat. In terms of the rate of
heat transfer, we have the following heat flow decomposition equation [100, 108]:
Q̇tot = Q̇hk + Q̇ex .
(3.41)
From Eqs. (3.32), (3.39), (3.41) and the dynamical decomposition equation (Ee and ∇A,
e we derive the expression of the housekeeping heat
q. (3.26)) for ∇U
dissipation rate (‘dissipation’ indicates its dissipative nature):
e −1 · V⃗s ⟩ = ⟨V⃗s · D
e −1 · V⃗s ⟩,
Q̇hk = ⟨V⃗t · D
84
(3.42)
which is always nonnegative according to the last equation. The equality of the
e = 0, proven in Aptwo expressions of Q̇hk in Eq. (3.42) is a result of ⟨V⃗s · ∇A⟩
pendix A. The sign property Q̇hk ≥ 0 means that the housekeeping heat generated
within the system is always transferred from the system into the environment and
not in the opposite direction. The excess heat and thus also the total heat can be
transferred in both directions.
Since the housekeeping heat is constantly generated within the system due to
dissipation (an irreversible process) and expelled into the environment in sustaining non-equilibrium steady states, we can expect that it does not only contribute to
the entropy flow from the system to the environment, but also the entropy production within the system. In fact, from the second expression of Q̇hk in Eq. (3.42),
Q̇hk is intimately related to the stationary flux V⃗s . The necessary and sufficient
condition for Q̇hk ≡ 0 is V⃗s ≡ 0, namely the stationary probability flux vanishes all the time and everywhere in the accessible state space. V⃗s as a microstate
function is a microscopic indicator of detailed balance breaking and thus nonequilibrium irreversibility of the steady state. Hence, Q̇hk , as a macrostate function, also indicates on the macroscopic level the non-equilibrium irreversibility
of the steady state. This suggests the housekeeping heat contributes to the entropy production in the system. This part of entropy production associated with
detailed balance breaking in the steady state has been termed adiabatic entropy
production [116–118]. (Note that the word ‘adiabatic’ here has no direct relation
with heat transfer. In the thermodynamic context quasi-static entropy production
or steady-state entropy production may be more appropriate, given that there are
also processes adiabatic to heat. In this work we still use the name initially given
to it.) In non-equilibrium isothermal processes the adiabatic entropy production
rate is the housekeeping heat dissipation rate divided by temperature, which, using
Eq. (3.42), is given by
Ṡad = T −1 Q̇hk = ⟨V⃗t · D −1 · V⃗s ⟩ = ⟨V⃗s · D −1 · V⃗s ⟩.
(3.43)
The subscript ‘ad’ of Ṡad represents ‘adiabatic’. Ṡad is also nonnegative as Q̇hk is,
which agrees with the requirement that entropy production is nonnegative. Similar
to Q̇hk , the necessary and sufficient condition for Ṡad ≡ 0 is V⃗s ≡ 0. According to
the relation between entropy flow and heat flow dS
¯ f l = dQ/T
¯
, the same Ṡad given
in Eq. (3.43) is also equal, in quantity, to the entropy flow rate associated with
the housekeeping heat transferred from the system to the environment (it can be
referred to as the housekeeping entropy flow rate). Thus we use the same notation
Ṡad to represent both the adiabatic entropy production rate and the housekeeping
85
entropy flow rate. Their physical meanings, however, are different (one is entropy
production and the other is entropy flow).
There is also entropy flow associated with the excess heat, called the excess
entropy flow [116–118]. Using the expression of Q̇ex in Eq. (3.32), the excess
entropy flow rate is given by:
Ṡex = T −1 Q̇ex = −⟨V⃗t · ∇U ⟩,
(3.44)
e = T ∇U derived from Eq. (3.24). Therefore, correspondwhere we have used ∇U
ing to the heat flow decomposition equation in Eq. (3.41), there is also an entropy
flow decomposition equation [116–118]:
Ṡf l = Ṡad + Ṡex .
(3.45)
In words, the total entropy flow is composed of the housekeeping entropy flow and
the excess entropy flow, just as the total heat is composed of the housekeeping heat
and the excess heat. This non-equilibrium thermodynamic entropy flow decomposition equation (as well as the heat flow decomposition equation) is a direct reflection of the stationary dynamical decomposition equation F⃗ ′ = −D · ∇U + V⃗s ,
as seen from the expressions in Eqs. (3.37), (3.43) and (3.44). Since Ṡad ≥ 0,
Eq. (3.45) can also be written as an inequality:
dS
¯ fl ≥
dQ
¯ ex
.
T
(3.46)
Equations (3.45) and (3.46) represent another facet of the second law of nonequilibrium thermodynamics, in addition to what is revealed in (3.38) and (3.40)
[116–118].
The difference in the (total) entropy production rate in Eq. (3.36) and the adiabatic entropy production rate in Eq. (3.43) indicates there is another part of
entropy production, termed nonadiabatic entropy production [116–118]. This is
expressed as an entropy production decomposition equation:
Ṡpd = Ṡad + Ṡna ,
(3.47)
which states the total entropy production is composed of the adiabatic entropy
production (associated with the non-equilibrium irreversibility in the steady state)
and the nonadiabatic entropy production (associated with the non-equilibrium irreversibility in the relaxation process from the transient state to the steady state).
86
Using the expression of Ṡpd in Eq. (3.36) and that of Ṡad in Eq. (3.43), we derive
the following equivalent expressions of the nonadiabatic entropy production rate:
Ṡna = ⟨V⃗t · D −1 · V⃗r ⟩ = −⟨V⃗t · ∇A⟩
= ⟨V⃗r · D −1 · V⃗r ⟩ = −⟨V⃗r · ∇A⟩ = ⟨∇A · D · ∇A⟩,
(3.48)
which is always nonnegative. The first expression is a result of Eqs. (3.9), (3.36)
and (3.43). The third expression is derived from the first one using the property
⟨V⃗s · D −1 · V⃗r ⟩ = 0, which is equivalent to ⟨V⃗s · ∇A⟩ = 0 (already proven in Appendix A) considering the relative dynamical constraint equation (Eq. (3.8)). The
other three expressions can also be derived using these equations. According to Eq. (3.48), the necessary and sufficient condition for Ṡna ≡ 0 is V⃗r ≡ 0 (or A ≡ 0).
As discussed in the potential-landscape framework, these are the stationary conditions, which means the system is staying in the (instantaneous) steady state all
the time. Therefore, non-zero Ṡna is produced by the irreversible relaxation process from the transient state to the steady state due to the non-stationary condition
characterized by non-zero V⃗r (or A). For non-equilibrium isothermal processes
we will see later that the nonadiabatic entropy production rate is proportional to
the free energy spontaneous dissipation rate (i.e., dissipative work [131, 132] per
˙
unit time), Ṡna = Aesd /T .
The non-equilibrium thermodynamic entropy production decomposition equation (Eq. (3.47)) is a direct reflection of the dynamical decomposition equation of
the flux: V⃗t = V⃗s + V⃗r . Within Eq. (3.47) the nonnegative total entropy production
rate Ṡpd is decomposed into two parts that are individually nonnegative, the adiabatic entropy production rate Ṡad associated with the stationary flux V⃗s (Eq. (3.43))
and the nonadiabatic entropy production rate Ṡna associated with the relative flux
V⃗r (Eq. (3.48)). The adiabatic entropy production rate Ṡad , determined by the stationary flux V⃗s , characterizes the irreversibility of maintaining a non-equilibrium
steady state. The nonadiabatic entropy, determined by the relative flux V⃗r , characterizes the irreversibility of the non-equilibrium spontaneous relaxation from
the transient state to the steady state. These two sources of non-equilibrium irreversibility constitute the total entropy production rate, determined by the transient
(total) flux V⃗t , which characterizes the total non-equilibrium irreversibility of the
system. For the total entropy production to be identically zero, the adiabatic and
nonadiabatic entropy productions must vanish individually, which is the equilibrium condition Pt = Ps = Pe characterized by V⃗t = V⃗s = V⃗r = 0. Since Ṡad ≥ 0
87
and Ṡna ≥ 0, Eq. (3.47) can also be written as inequalities:
dS
¯ pd ≥
d¯Aesd
,
T
dS
¯ pd ≥
dQ
¯ hk
.
T
(3.49)
Equations (3.47) and (3.49) reveal another face of the second law of non-equilibrium
thermodynamics.
Furthermore, plugging the entropy flow decomposition equation (Eq. (3.45))
and the entropy production decomposition equation (Eq. (3.47)) into the entropy
balance equation (Eq. (3.38)), we can derive another form of the entropy balance
equation [116–118]:
Ṡ = Ṡna − Ṡex .
(3.50)
This is a result of canceling the same quantity Ṡad with two different meanings in
those two decomposition equations. On the one hand, Ṡad has the meaning of the
adiabatic entropy production rate in the entropy production decomposition equation. On the other hand, it has the meaning of the housekeeping entropy flow rate
in the entropy flow decomposition equation. For non-equilibrium isothermal processes, this is related to the double role the housekeeping heat plays as explained
before. The adiabatic entropy production (the housekeeping heat) generated within the system is completely transferred out of the system into the environment as a
part of the total entropy flow (heat flow). Therefore the adiabatic entropy production (the housekeeping heat) does not accumulate in the system and thus does not
contribute to the net change of the system entropy (the internal energy). That is
why the change in the system entropy can be expressed alternatively as a tradeoff
between the nonadiabatic entropy production and the excess entropy flow given
by Eq. (3.50). Since Ṡna ≥ 0, Eq. (3.50) can also be written as an inequality:
dS ≥ −
dQ
¯ ex
.
T
(3.51)
According to dQ
¯ tot = dQ
¯ hk + dQ
¯ ex and dQ
¯ hk ≥ 0, we have −¯
dQex /T ≥
−¯
dQtot /T . Hence Eq. (3.51) is a stronger statement of the second law of thermodynamics than that given by Eq. (3.40) [100,108]. Equations (3.50) and (3.51) represent another dimension of the second law of thermodynamics for non-equilibrium
isothermal processes. There is actually more. Another aspect of the second law
will be revealed by investigating the non-equilibrium free energy.
88
The Second Law of Non-Equilibrium Thermodynamics in terms of Free Energy
The rate of change of the non-equilibrium free energy is given by
˙
Ae = Uė − T Ṡ = (Ẇed − Q̇ex ) − T (Ṡna − Ṡex ) = Ẇed − T Ṡna , (3.52)
where we have used Ae = Ue − T S, Eqs. (3.33) and (3.50) as well as Ṡex = Q̇ex /T .
Ẇed is the external driving power defined in Eq. (3.30). The term T Ṡna , proportional to the nonadiabatic entropy production rate (notice the minus sign before
it), represents the amount of free energy that is irreversibly lost (i.e., dissipated)
per unit time, due to the spontaneous relaxation process. Thus we refer to it as the
˙
free energy spontaneous dissipation rate, denoted by Aesd , with the subscript ‘sd’
representing ‘spontaneous dissipation’. Using Eq. (3.48), we have:
˙
e −1 · V⃗r ⟩ = −⟨V⃗t · ∇A⟩
e
Aesd = T Ṡna = ⟨V⃗t · D
e −1 · V⃗r ⟩ = −⟨V⃗r · ∇A⟩
e = ⟨∇A
e · D · ∇A⟩.
e
= ⟨V⃗r · D
(3.53)
˙
Aesd is also nonnegative as with Ṡna . The necessary and sufficient condition for
˙
Aesd ≡ 0 is also the stationary condition: V⃗t = V⃗s (Pt = Ps ) characterized by
˙
V⃗r = 0 (A = 0). Non-zero Aesd is a result of the non-stationary condition, under
which the system is in the non-equilibrium irreversible process of spontaneously relaxing from the transient state to the (instantaneous) steady state. Now Eq. (3.52) becomes the non-equilibrium free energy balance equation:
˙
˙
Ae = Ẇed − Aesd .
(3.54)
˙
˙
It can also be written as −Ae = −Ẇed + Aesd , interpreted as follows. In the total
˙e
decrease of free energy −A,
a part of it, −Ẇed , produces useful work (done on
˙
the environment) and the rest, Aesd , is irreversibly dissipated. The irreversibly
dissipated free energy can no longer be used (i.e., is not available) to produce
useful work. Thus the spontaneously dissipated free energy Aesd is identified as
the (ensemble-averaged) ‘dissipative work’ [131, 132], also known as the ‘wasted
work’, ‘lost work’ or ‘lost available work’ in engineering [99, 155]. The free
˙
energy spontaneous dissipation rate Aesd is thus identified as the dissipative power
(i.e., dissipative work per unit time), also called lost power in engineering [155].
89
˙
The notations Wsd and Ẇsd may be used instead of Aesd and Aesd . They simply
˙
reflect two different perspectives to look at the same quantity. Since Aesd ≥ 0,
Eq. (3.54) also means:
− dAe ≥ − d¯Wed .
(3.55)
This is actually the principle of maximum work extended to non-equilibrium
isothermal processes; the amount of free energy decreased is the maximum work
the system can produce in isothermal processes. It is the statement of the second
law of thermodynamics in terms of free energy in isothermal processes. Therefore
Eqs. (3.54) and (3.55) represent another facet of the second law.
˙
The equation, Aesd = T Ṡna , i.e., Ẇsd = T Ṡna , has a close connection with the
Gouy-Stodola theorem known in engineering for more than a century [99, 155,
156]. The difference is that in the Gouy-Stodola theorem Ṡna is simply the entropy production rate, as there was no (need of) distinction of adiabatic entropy
production and nonadiabatic entropy production, since the steady states are usually equilibrium states in the application of engineering. For living organisms,
however, the stable biological structures and functions are sustained by constant
input and output of energy and matter, forming dissipative structures corresponding to non-equilibrium steady states, associated with non-zero adiabatic entropy
production [1]. In engineering, entropy production minimization has been used as
an optimization principle for the design of finite-size and finite-time devices [155].
In the biological context of living organisms, this optimization principle must be
revised. The nonadiabatic entropy production may be minimized. But the adiabatic entropy production associated with maintaining the biological structures and
functions cannot be minimized without limit, as zero adiabatic entropy production would also mean disappearance of the dissipative structure and its biological
function, which literally means death for living organisms. Thus a certain level
of adiabatic entropy production must be maintained as the necessary cost to ensure the required level of stability of the dissipative structure and efficiency of its
biological function.
The non-equilibrium free energy balance equation, Eq. (3.54), also provides
an operational solution to the definition of the non-equilibrium stationary free energy. Consider quasi-static processes in which Pt ≈ Ps all the time. In such
˙
processes Aesd = 0 (also Ṡna = 0). (For perspectives on the constantly generated
housekeeping heat in such processes, see Ref. [100].) Therefore, we can determine the non-equilibrium stationary free energy with the thermodynamic work
performed in quasi-static processes considering ∆Aes = Wed . This is consistent
90
∫∑ i
with giving the equations of state Λis ({λ}) and defining Aes =
i Λs dλi , since
the integration is just the thermodynamic work performed in quasi-static processes
(for constant T ).
3.1.6
Summary and Discussion
To summarize, we have the following set of thermodynamic equations for nonequilibrium isothermal processes (three balance equations, of the non-equilibrium
internal energy, entropy and free energy, and two decomposition equations, of the
entropy production and entropy flow):


Uė = Ẇed − Q̇ex






= Ṡpd − Ṡf l
 Ṡ
˙
˙
(3.56)
Ae = Ẇed − Aesd




Ṡpd = Ṡad + Ṡna




Ṡf l = Ṡad + Ṡex
with the relations Ae = Ue − T S, Ṡf l = Q̇tot /T , Ṡad = Q̇hk /T , Ṡex = Q̇ex /T ,
ėsd /T , Aesd ≡ Wsd and the sign properties Ṡad ≥ 0 (Q̇hk ≥ 0), Ṡna ≥ 0
Ṡna = A
˙e
(Asd ≥ 0), Ṡpd ≥ 0. The sign of heat is positive when flowing from the system to
the environment. The sign of work is positive when the environment does work
on the system. The equation Ṡ = Ṡna − Ṡex (Eq. (3.50)) is implied and thus not
e S and Ae are state functions, while the rest are process functions.
spelled out. U,
The expressions of the thermodynamic quantities are also derived.
Within the five equations in Eq. (3.56), the three entropy equations (the entropy balance equation, entropy flow and entropy production decomposition equations) are a direct reflection of the dynamical decomposition equations on
the thermodynamic level. More specifically, the entropy balance equation Ṡ =
Ṡpd − Ṡf l is a manifestation of the transient dynamical decomposition equation
∇S = D −1 · V⃗t − D −1 · F⃗ ′ (alternatively, F⃗ ′ = −D · ∇S + V⃗t ), where Ṡ corresponds to ∇S, Ṡpd corresponds to D −1 · V⃗t and Ṡf l corresponds to D −1 · F⃗ ′ ,
as seen from their respective expressions in Eqs. (3.34), (3.36) and (3.37). The
entropy flow equation Ṡf l = Ṡad + Ṡex is a mirror of the stationary dynamical decomposition equation D −1 · F⃗ ′ = D −1 · V⃗s − ∇U (or F⃗ ′ = −D · ∇U + V⃗s ), where
Ṡf l corresponds to D −1 · F⃗ ′ , Ṡad corresponds to D −1 · V⃗s and Ṡex corresponds
to −∇U , by examining their respective expressions given in Eqs. (3.37), (3.43)
91
and (3.44). The entropy production decomposition equation Ṡpd = Ṡad + Ṡna is
mapped from the flux decomposition equation D −1 · V⃗t = D −1 · V⃗s + D −1 · V⃗r (or
V⃗t = V⃗s + V⃗r ), where Ṡpd corresponds to D −1 · V⃗t , Ṡad corresponds to D −1 · V⃗s
and Ṡna corresponds to D −1 · V⃗r , comparing with their respective expressions in
Eqs. (3.36), (3.43) and (3.48). To be more accurate,∫‘corresponds to’ in these s⃗ = J⃗t · K
⃗ d⃗q, where K
⃗ is the
tatements is equal to the operation of ⟨V⃗t · K⟩
quantity stated after ‘corresponds to’.
In the non-equilibrium thermodynamics pioneered by Onsager [92–94], termed
‘classical irreversible thermodynamics’ [102], it is a basic result of the local equilibrium assumption (where ‘local’ means local in the physical space) that the local
entropy
rate (also in the physical space) has a flux-force bilinear form,
∑ ⃗ production
⃗
⃗ α are conjugate thermodynamic fluxes and thermoJ
·
K
,
where
J⃗α and K
α
α α
dynamic forces, respectively. (They can also be scalars or tensors [102]; here
we consider vectors.) The total entropy production rate is then given by the spatial integral
∫ ∑of the local entropy production rate over the volume of the system:
∗
⃗ α d⃗x. If there is only one pair of thermodynamic flux and
Ṡpd = V α J⃗α · K
∫
∗
⃗ x. It has the same mathematical form
thermodynamic force, then Ṡpd
= V J⃗ · Kd⃗
∫
as the entropy production rate in Eq. (3.36), Ṡpd = J⃗t · D −1 · V⃗t d⃗q, with J⃗t
⃗ pd ≡ D −1 · V⃗t corresponding to
corresponding to the thermodynamic flux and K
the thermodynamic force. This has been noticed before [34, 109, 118].
However, there is a conceptual distinction, which we believe is important, that
has not been given sufficient attention. The thermodynamic fluxes and forces in
the classical irreversible thermodynamics describe transport processes of matter
and energy in the physical space. In this context, thermodynamic fluxes represent
matter or energy fluxes, which are generated by the corresponding thermodynamic
forces in the physical space. For example, temperature gradient is the thermodynamic force generating the heat flow (energy flux). While in our case, ⃗q represents
a microstate of the system in the microstate space rather than in the physical space.
J⃗t is the probability flux, representing ‘information flow’ in the microstate space.
⃗ pd is the ‘force’ generating information flow in the microstate space. Therefore,
K
⃗ pd describe information transport process in the microstate space, rather
J⃗t and K
than matter or energy transport process in the physical space. We may still call
them thermodynamic flux and thermodynamic force, but their meanings should be
understood in the context of information transport in the microstate space. Care
also need to taken in reading the word ‘local’ as it may have two different meanings, that is, local in the physical space and local in the microstate space. In
particular, local in the microstate space does not imply local in the physical space.
92
This is most clearly seen in spatially inhomogeneous systems (Sec. 3.3), where a
global state of the system in the physical space is represented by a local ‘point’ in
the microstate space (field configuration space). This is crucial for understanding,
(Sec. 3.3), how a global description of spatially inhomogeneous systems in the
physical space can be disguised as a local description in the microstate space, and
how it is that our treatment is not bound by local equilibrium despite its apparent
mathematical similarity with the expressions in classical irreversible thermodynamics that assumes local equilibrium. With the distinction between information
transport and matter or energy transport stressed, we also mention the possible
connection between them. The information transport process in the microstate
space may be a result of the matter and energy transport in the physical space.
Consider the non-equilibrium steady state which is sustained by constant matter
and energy flows generated by constant thermodynamic forces (e.g., temperature
difference or chemical potential difference). On the micro-statistical level, there is
a non-zero stationary probability flux J⃗s on the microstate space, which vanishes
as matter and energy flows also vanish when the system is in thermodynamic equilibrium. The consistency between these two levels of description suggests that
the information flux J⃗s is associated with the matter and energy fluxes and thus
also the thermodynamic forces generating them in the physical space. For a case
study, see Ref. [91].
⃗ pd = D −1 · V⃗t , which we call the thermodynamic force
We have defined K
∫
⃗ pd d⃗q. Since the
generating the total entropy production, given that Ṡpd = J⃗t · K
expressions
of other thermodynamic quantities in Eq. (3.56) also have the same
∫
⃗ d⃗q, we generalize the concept of thermodynamic force (not limited
form J⃗t · K
⃗ corresponding to other thermodynamic
to entropy production) and also call the K
quantities, thermodynamic force. Thus the thermodynamic forces generating the
adiabatic entropy production, nonadiabatic entropy production, total entropy flow,
⃗ ad = D −1 ·
excess entropy flow and transient entropy change are, respectively, K
⃗ na = D −1 · V⃗r = −∇A, K
⃗ f l = D −1 · F⃗ ′ , K
⃗ ex = −∇U and K
⃗ S = ∇S. For
V⃗s , K
internal energy, heat flow, free energy and spontaneously dissipated free energy
(i.e., dissipative work), we can also define their respective thermodynamic forces
accordingly; they only differ from those already introduced above by a factor of
T or T −1 . We stress again that these thermodynamic forces are to be understood
in the context of information transport in the microstate space; they are thus in
nature ‘informational’.
⃗ ex = −∇U , K
⃗ S = ∇S and K
⃗ na = −∇A deserve
These three expressions K
particular attention. Since they express the thermodynamic ‘force’ in a gradient
93
form, we can indeed interpret the potential landscapes U , S and A as ‘potentials’,
in the same sense that the inverse of temperature can be interpreted as the potential of heat flow in classical irreversible thermodynamics [92–94]. Therefore,
the stationary potential (landscape) U is the potential of excess entropy flow; the
transient potential (landscape) S is the potential of transient entropy change; the
relative potential (landscape) A is the potential of nonadiabatic entropy production. These potentials in essence are not energy potentials; they are information (or
entropy) potentials. This has shed light on the physical meanings of these ‘potentials’ introduced in the potential-flux landscape framework in Sec. 3.1.2. Further,
if we interpret the diffusion matrix D as a metric tensor on the microstate space
as in the covariant form of the Fokker-Planck equation [16], then the matrix operation of D or D −1 on a vector simply switches the vector between its covariant
⃗ or D −1 · K
⃗
form and contravariant form. In this sense we can also refer to D · K
⃗
as the thermodynamic force. Thus we can also say the transient flux Vt is the thermodynamic force generating total entropy production. Similarly, V⃗s , V⃗r and F⃗ ′
are, respectively, the thermodynamic forces generating the adiabatic entropy production, nonadiabatic entropy production and total entropy flow. This has clarified
the thermodynamic meanings of these dynamical forces. (Although this is convenient for conceptual simplification, if non-covariant form of the Fokker-Planck
equation is used, the diffusion matrix still need to be tracked.) For some relevant
information-theoretic studies in thermodynamics, see Ref. [157].
The dynamical decomposition equations in Eqs. (3.6)-(3.9) can now be interpreted as thermodynamic
force decomposition equations, which, through the op∫
⃗
⃗
⃗
⃗
eration ⟨Vt · K⟩ = Jt · K d⃗q (also call it the flux-force bilinear form), are mapped
into (three of) the non-equilibrium thermodynamic equations in Eq. (3.56). This
shows explicitly how, in the potential-flux landscape framework, the decomposition equations on the dynamic level are manifested on the thermodynamic level,
thus establishing the connection between the dynamics and thermodynamics of
the system. In particular, the fluxes V⃗t , V⃗s and V⃗r introduced on the dynamic
level directly characterize the non-equilibrium aspects of the system. V⃗s characterizing the non-equilibrium irreversible aspect of the steady state is the thermodynamic force generating adiabatic entropy production, V⃗r characterizing the
non-equilibrium irreversible aspect of the relaxation process is the thermodynamic
force generating nonadiabatic entropy production, and V⃗t capturing the combined
non-equilibrium irreversible effect of both the steady state and the transient relaxation process is the thermodynamic force generating total entropy production.
Thermodynamic equilibrium indicated by zero entropy production Ṡpd = Ṡad =
94
Ṡna = 0 is characterized by the vanishing of all the fluxes: V⃗t = V⃗s = V⃗r = 0. It
is the non-zero fluxes V⃗t , V⃗s and V⃗r that break the reversibility of equilibrium states and create entropy production in non-equilibrium processes. As for the other
two thermodynamic equations (internal energy and free energy balance equations), we mention that their counterparts on the dynamic level are the following two
equations:
e
e
e ∂A
e
DU
∂U
DA
e,
e
=
+ V⃗t · ∇U
=
+ V⃗t · ∇A,
(3.57)
Dt
∂t
Dt
∂t
where D/Dt is the advective derivative defined in Eq. (3.28). Taking ensemble
average, changing the advective derivative to the total derivative when pulling it
out of the ensemble average ⟨·⟩t , and using ⟨∂t S⟩t = 0, we get the internal energy
and free energy balance equations in Eq. (3.56).
3.1.7
Extension to Systems with One General State Transition
Mechanism
The non-equilibrium thermodynamic equations in Eq. (3.56) are derived from
non-equilibrium isothermal processes. We consider a generalization of these equations to non-equilibrium processes with one general (effective) transition mechanism. For such processes we do not introduce temperature into the dynamical equation, such as D ∝ T in isothermal processes. Thus we cannot utilize quantities
based on the introduction of temperature, including the non-equilibrium internal
energy Ue and free energy Ae (see Eqs. (3.16) and (3.22)), to formulate the nonequilibrium thermodynamic equations for such systems. Another perspective is
that temperature introduces an energy scale into the dynamical equation. Without it, we cannot construct, from the dynamical equation, quantities that have the
dimension of energy, such as internal energy, free energy, heat and work.
Fortunately, Ue and Ae have their respective dimensionless entropic analogs
available, namely the cross entropy U and relative entropy A, which are purely
constructed from Ps and Pt , without referring to temperature (see Eqs. (3.17) and
and (3.23)). According to Eqs. (3.24) and (3.25), Ue = T U +Aes and Ae = T A+Aes .
From these relations and the balance equations of Ue and Ae in Eq. (3.56), we can
derive the balance equations of U and A. For example,
(
)
(
)
U̇ = T −1 Uė − Aės = T −1 Ẇed − Q̇ex − Aės
(
)
−1
ė
Ẇed − As − T −1 Q̇ex .
= T
(3.58)
95
The second term in the last equation is the excess entropy flow rate Ṡex = T −1 Q̇ex .
We define the first term in the last equation as the external driving entropic power,
which is the entropic analog of the external driving power Ẇed :
∑
(3.59)
⟨∂λi U ⟩λ̇i ,
Ṡed = T −1 (Ẇed − Aės ) = ⟨∂t U ⟩ =
i
e ⟩ and U
e = T U + Aes . (One difference between
where we have used Ẇed = ⟨∂t U
Ṡed and Ẇed is that in quasi-static processes Wed = ∆Aes , which is not necessarily
zero, while Sed = 0 according to the first equation of Eq. (3.59).) Now Eq. (3.58)
becomes the cross entropy balance equation:
U̇ = Ṡed − Ṡex ,
(3.60)
which states that the system’s cross entropy is increased (decreased) when the environment is driving the system (the system is driving the environment) through
changing external conditions and decreased (increased) when excess entropy flows
into the environment (into the system). Similarly, we can derive the relative entropy balance equation:
Ȧ = Ṡed − Ṡna ,
(3.61)
which states that the system’s relative entropy is increased (decreased) when the
environment is driving the system (the system is driving the environment) and always decreased when there is nonadiabatic entropy production within the system.
Although Eqs. (3.60) and (3.61) can be derived from the internal energy and free
energy balance equations in Eq. (3.56), they can also be derived independently
from the definitions U = ⟨U ⟩t , A = ⟨A⟩t and the dynamical decomposition equations in Eqs. (3.6)-(3.9). The other three thermodynamic equations in Eq. (3.56)
remain unchanged.
Hence, for systems with one general (effective) transition mechanism, we have
the following set of non-equilibrium thermodynamic equations, which are three
balance equations for cross entropy, transient entropy and relative entropy, and
two decomposition equations for entropy production and entropy flow:


U̇
= Ṡed − Ṡex






= Ṡpd − Ṡf l
 Ṡ
(3.62)
Ȧ = Ṡed − Ṡna




Ṡpd = Ṡad + Ṡna




Ṡf l = Ṡad + Ṡex
96
with the definitions U = ⟨− ln Ps ⟩, S = ⟨− ln Pt ⟩, A = ⟨ln(Pt /Ps )⟩ and the sign
properties Ṡad ≥ 0, Ṡna ≥ 0, Ṡpd ≥ 0. U, S and A are state functions while
the rest are process functions. For systems described by Langevin or FokkerPlanck dynamics with one transition mechanism, the thermodynamic quantities
in Eq. (3.62) have explicit expressions, summarized below:
⟨
⟩
⃗
U̇ = ⟨∂t U ⟩ + Vt · ∇U ,
(3.63)
⟨
⟩
Ṡ = V⃗t · ∇S ,
(3.64)
⟨
⟩
Ȧ = ⟨∂t U ⟩ + V⃗t · ∇A ,
(3.65)
∑
Ṡed = ⟨∂t U ⟩ =
⟨∂λi U ⟩λ̇i ,
(3.66)
i
Ṡex
Ṡpd
Ṡf l
Ṡad
Ṡna
= −⟨V⃗t · ∇U ⟩,
= ⟨V⃗t · D −1 · V⃗t ⟩,
= ⟨V⃗t · D −1 · F⃗ ′ ⟩,
(3.67)
= ⟨V⃗t · D −1 · V⃗s ⟩ = ⟨V⃗s · D −1 · V⃗s ⟩,
= ⟨V⃗t · D −1 · V⃗r ⟩ = −⟨V⃗t · ∇A⟩
= ⟨V⃗r · D −1 · V⃗r ⟩ = −⟨V⃗r · ∇A⟩ = ⟨∇A · D · ∇A⟩.
(3.70)
(3.68)
(3.69)
(3.71)
These results for systems with one general transition mechanism also apply, in
particular, to systems in an isothermal environment. By introducing the (constant) temperature of the environment and the quantities defined in relation to temperature (internal energy, free energy, heat and work), we recover the results for
non-equilibrium isothermal processes in Eq. (3.56) and the expressions of the thermodynamic quantities. We leave it as an open question what will happen to the
internal energy and free energy balance equations in Eq. (3.56) if the environment
temperature changes.
The dimensionless quantities U and A are formally called internal energy and
free energy, respectively, in Ref. [125], as mathematical generalizations of the
concepts of internal energy and free energy in equilibrium thermodynamics. In
this work we use the entropy language to describe these dimensionless quantities,
while reserving the energy language for those quantities that truly have the dimension of physical energy. This way we can avoid confusion when dealing with
more general situations, where these dimensionless quantities and those with the
dimension of physical energy may coexist. For instance, for systems in contact
97
with multiple heat baths modeled as a Markov process, we can still define the dimensionless mathematical ‘internal energy’ (the cross entropy) in terms of Ps and
Pt , but we may still be able to define the physical internal energy (if interaction
energy is properly accounted for). Yet these two do not necessarily have a simple
relation as that for non-equilibrium isothermal processes in Eq. (3.24).
Besides, using entropy language to describe these dimensionless quantities
also has validity on its own right. U and A, as with S, are indeed of entropy
(or information) nature, since all three are constructed purely from the ensemble
probability distributions Pt and Ps , with the transient entropy S = ⟨− ln Pt ⟩t , the
cross entropy U = ⟨− ln Ps ⟩t and the relative entropy A = ⟨ln(Pt /Ps )⟩t . We can
regard U, S and A as three distinct yet related aspects of non-equilibrium entropy,
differentiated from a single concept of equilibrium entropy, similar to the differentiation of a single concept of heat in equilibrium thermodynamics into three
aspects of heat in non-equilibrium isothermal processes (i.e., the total heat, the
excess heat and the housekeeping heat). In equilibrium systems where only the
equilibrium ensemble distribution Pe is of concern, these three aspects of entropy
are degenerate. More specifically, in equilibrium states Pt = Ps = Pe , the relative
entropy A vanishes while the transient entropy S and the cross entropy U become
identical with the equilibrium entropy. In non-equilibrium systems, however, both
the stationary distribution Ps and the transient distribution Pt capture certain nonequilibrium aspects of the system and thus both are required in the description of
non-equilibrium systems. This is why the non-equilibrium entropy becomes differentiated. If the microstate vector ⃗q takes on discrete values, then the transient
entropy, cross entropy and relative entropy have direct physical meanings in terms
of the information theory [143]. The transient entropy S measures the average
amount of information needed to identify a microstate of the system when the
system is described by an ensemble distribution Pt . The cross entropy U measures the average amount of information needed to identify a microstate of the
system from the transient ensemble distribution Pt , using a coding scheme based
on the stationary ensemble distribution Ps . The relative entropy A measures the
average amount of extra information needed to identify a microstate of the system
from the transient ensemble distribution Pt , using a coding scheme based on the
stationary ensemble distribution Ps . If the microstate vector ⃗q takes on continuous
values as in the case of Fokker-Planck dynamics, there are difficulties interpreting
these entropies directly, as has been encountered in information theory in the socalled differential entropy or continuous entropy [143]. Yet these interpretational
difficulties can be circumvented as far as entropy difference is concerned, which
means only the relative values of entropy are considered. In particular, the tran98
sient entropy S and the cross entropy U are both defined up to a common additive
constant, while the relative entropy A = U − S does not have this freedom. In
non-equilibrium processes all three aspects of entropy, the transient entropy S,
cross entropy U and relative entropy A, are required to fully characterize nonequilibrium entropy. (The stationary entropy Ss = ⟨− ln Ps ⟩s is a special case of
S and U when Pt = Ps .)
The set of non-equilibrium thermodynamic equations in Eq. (3.62) governing various entropic quantities thus represent the pure information dynamics (or
‘infodynamics’) of the non-equilibrium stochastic dynamical system. The nonequilibrium infodynamics is part of the non-equilibrium thermodynamics (specifically, the second law of thermodynamics), but it does not deal with quantities
with the dimension of physical energy directly (thus the first law of thermodynamics). The formulation of the first law representing energy conservation requires additional thermodynamic input apart from the microscopic internal dynamics [121, 130, 135]. For systems in an isothermal environment, when the environment temperature and the stationary free energy are given, the first law of
thermodynamics can be recovered from the cross entropy balance equation. It
is still an open question whether or not, in more general cases, given certain extra thermodynamic content of the system, the non-equilibrium internal energy and
free energy can still be defined, with which the first law in terms of internal energy
and the second law in terms of free energy can be formulated and recovered from
the infodynamic equations. Without further information of the system’s thermodynamic characterization, the best we can do is extract the infodynamics of the
non-equilibrium thermodynamics from the stochastic dynamics.
3.2
Non-Equilibrium Thermodynamics for Spatially Homogeneous Stochastic Systems with Multiple State Transition Mechanisms
In this section we consider systems described by Langevin and Fokker-Planck
equations with multiple state transition mechanisms (e.g., multiple heat or particle reservoirs) in a multidimensional state space. We also expand the potentialflux landscape framework to accommodate such an extended stochastic dynamics.
We then generalize the non-equilibrium thermodynamics obtained in the previous
section (Sec. 3.1) for one mechanism, specifically, the infodynamic equations and
expressions in (3.62)-(3.71), to systems with multiple mechanisms. Then we clar99
ify under what conditions the results for multiple mechanisms reduce to those for
for one mechanism. We conclude this section with an illustration of the general
formalism using the Ornstein-Uhlenbeck process and a more specific example.
3.2.1
Stochastic Dynamics for Multiple State Transition Mechanisms
We consider the following Langevin equation describing the stochastic dynamics of systems with multiple state transition mechanisms labeled by the index
m, as an extension of the Langevin equation for systems with one state transition
mechanism given by Eq. (3.1):
]
[
∑
∑
∑
⃗ (m) (⃗q, t)dWs (t) .
d⃗q =
d⃗q (m) =
F⃗ (m) (⃗q, t)dt +
G
(3.72)
s
m
m
s
We assume the state space is an n-dimensional Euclidean space Rn . We also
assume for each transition mechanism labeled by the fixed index m, the vectors
⃗ (m)
q , t) (s = 1, 2, ...) all belong to the same vector space R(m)
F⃗ (m) (⃗q, t) and G
s (⃗
that is a linear subspace of Rn . We call R(m) the state transition space of mechanism m. R(m) for different transition mechanisms are not assumed to have the
same dimension and they are not necessarily orthogonal to each other. Mathemat(m)
ically, if ⃗ei (i = 1, 2, ...) is an orthonormal basis of R(m) ⊆ Rn , we have:
∑ (m) (m)
F⃗ (m) (⃗q, t) =
⃗ei Fi (⃗q, t),
(3.73)
i
⃗ (m)
G
q , t) =
s (⃗
∑
(m)
⃗ei
(m)
Gi s (⃗q, t),
(3.74)
i
(m)
Fi (⃗q, t)
(m)
= ⃗ei
⃗ (m)
(m)
(m)
⃗ (m)
where
· F⃗ (m) (⃗q, t) and Gi s (⃗q, t) = ⃗ei · G
q , t) are the
s (⃗
(m)
(m)
⃗
components of F (⃗q, t) and Gs (⃗q, t) in the basis {⃗ei }, respectively. We
introduce a projection operator (or a projection matrix) for each state transition
space R(m) :
∑ (m) (m)
Π(m) =
⃗ei ⃗ei ,
(3.75)
i
which projects a vector α
⃗ ∈ R onto R via the projection operation:
)
(
∑ (m) ( (m) )
∑ (m) (m)
⃗ .
⃗ei · α
·α
⃗=
⃗ei
Π(m) · α
⃗=
⃗ei ⃗ei
n
(m)
i
i
100
Π(m) can also operate on a vector from the right side. When α
⃗ ∈ R(m) , we have
(m)
(m)
Π
·α
⃗ = α
⃗ . The projection operator Π
also has the idempotent property
′
Π(m) · Π(m) = Π(m) . Yet we do not require Π(m) · Π(m ) = 0 for m ̸= m′ , since
′
R(m) and R(m ) do not have to be orthogonal.
The Fokker-Planck equation corresponding to Eq. (3.72) (in Ito’s sense) reads:
(
)
⃗
∂t Pt = −∇ · F Pt − ∇ · (DPt ) ,
(3.76)
where
F⃗ =
∑
F⃗ (m) ,
D=
m
∑
D (m)
m
[
]
∑ 1∑
⃗ (m)
⃗ (m) . (3.77)
=
G
s Gs
2
m
s
The argument (⃗q, t) has been suppressed to simplify notations. The diffusion matrix D (m) of mechanism m and the total diffusion matrix D are all nonnegative
definite symmetric by construction. D (m) is a matrix within the space R(m) ⊆ Rn .
When the dimension of R(m) is smaller than Rn , D (m) is not invertible in Rn . Yet
we require D (m) is invertible within R(m) . Also we require D is invertible within
Rn . Thus D (m) and D are positive definite symmetric, respectively, within R(m)
and Rn . The Fokker-Planck equation (Eq. (3.76)) still has the form of a continuity
equation ∂t Pt = −∇ · J⃗t . The (collective) transient probability flux is given by:
J⃗t = F⃗ ′ Pt − D · ∇Pt ,
(3.78)
where F⃗ ′ = F⃗ − ∇ · D is the effective driving force. We can also define the indi(m)
vidual transient probability flux for each mechanism J⃗t , which is related to the
∑
(m)
collective transient probability flux via J⃗t = m J⃗t . According to Eqs. (3.77)
and (3.78), we have:
(m)
J⃗t
= F⃗
′(m)
Pt − D (m) · ∇Pt ,
(3.79)
where F⃗ ′(m) = F⃗ (m) − ∇ · D (m) is the effective driving force of mechanism m.
The stationary Fokker-Planck equation reads:
(
)
∇ · F⃗ Ps − ∇ · (DPs ) = 0.
(3.80)
It has the form ∇ · J⃗s = 0, where the (collective) stationary probability flux is:
J⃗s = F⃗ ′ Ps − D · ∇Ps .
101
(3.81)
∑ ⃗ (m)
J⃗s can be decomposed into components of each mechanism J⃗s =
,
m Js
where the stationary probability flux of mechanism m is given by:
J⃗s(m) = F⃗
′(m)
Ps − D (m) · ∇Ps .
(3.82)
As with systems having one transition mechanism, we assume that the transient
distribution Pt is always positive and normalized to 1 and that at each instant of
time the stationary distribution Ps is unique, positive and normalized to 1 in the
accessible state space.
3.2.2
Potential-Flux Landscape Framework for Multiple State
Transition Mechanisms
The stationary dynamical decomposition equation for each mechanism, from
Eq. (3.82), reads:
F⃗ ′(m) = −D (m) · ∇U + V⃗s (m) ,
(3.83)
where the stationary potential landscape is U = − ln Ps and the stationary flux
(m)
(m)
= J⃗s /Ps . The transient dynamical decomvelocity for mechanism m is V⃗s
position equation for each mechanism from Eq. (3.79) reads:
F⃗
′(m)
= −D (m) · ∇S + V⃗t
(m)
,
(3.84)
where the transient potential landscape is S = − ln Pt and the transient flux ve(m)
(m)
locity for mechanism m is V⃗t
= J⃗t /Pt . From Eq. (3.83) and (3.84) we also
have the relative dynamical constraint equation for each mechanism:
V⃗r(m) = −D (m) · ∇A,
(3.85)
where the relative potential landscape is A = U − S = ln(Pt /Ps ) and the relative
(m)
(m)
(m)
flux velocity for mechanism m is V⃗r = V⃗t − V⃗s . The flux decomposition
equation for each mechanism is:
(m)
V⃗t = V⃗s(m) + V⃗r(m) .
(3.86)
The collective form of Eqs. (3.83)-(3.86) as those for systems with one mechanism
given in Eqs. (3.6)-(3.9) can be recovered by summing over the mechanism index
m.
We discuss some points specific to systems with multiple mechanisms. First,
considering that detailed balance means each microscopic process is balanced by
102
its own reverse process, for systems with multiple mechanisms each process within a mechanism should be balanced by its own reverse process in that mechanism.
This implies when detailed balance condition holds, the stationary flux for each
(m)
(m)
mechanism should be zero, i.e. J⃗s = 0 (or V⃗s
= 0) for each m. This is a
stronger condition than the total stationary flux being zero, J⃗s = 0 (or V⃗s = 0). In
(m)
principle, there could be such situations in which J⃗s ̸= 0 (for at least two dif∑
(m)
ferent m’s) while J⃗s = m J⃗s = 0. In that case, detailed balance is broken in
the steady state although the total stationary flux is zero. Hence, we expect there
is non-zero adiabatic entropy production and thus also total entropy production in
that scenario, since there is irreversibility due to detailed balance breaking in the
steady state. This clearly indicates that the expressions of the adiabatic and total
entropy production rates for one mechanism in Eqs. (3.68) and (3.70), in general,
do not apply to systems with multiple mechanisms. What we can expect is that the
adiabatic and total entropy productions, at least, have contributions from the irreversibility within each mechanism (though cross effects of different mechanisms
(m)
(m)
may also contribute). Thus each non-zero J⃗s (or V⃗s ) should have a strictly
positive contribution to the adiabatic and total entropy productions.
Second, although we still have ∇ · J⃗s = 0, this is not generally true for each
(m)
̸= 0 in general. Further, since D (m) in Eq. (3.85)
mechanism, that is, ∇ · J⃗s
is not necessarily invertible in Rn , A = 0 (i.e., Pt = Ps ) is not equivalent to
(m)
(m)
(m)
= 0 (i.e., V⃗t
= V⃗s ) for each m. Yet since we have assumed D is
V⃗r
invertible in Rn , A = 0 (i.e., Pt = Ps ) is still equivalent to V⃗r = 0 (i.e., V⃗t =
∑ ⃗ (m)
∑
(m)
). Moreover, since D (m) is invertible in R(m) but
=
V⃗s or m V⃗t
m Vs
not necessarily in Rn , modifications are required when inverting D (m) in (3.83)(3.85). By first projecting these equations onto R(m) using the projection matrix
Π(m) and then inverting the matrix D (m) , we obtain the following three dynamical
decomposition equations for multiple mechanisms (compare with Eq. (3.26) for
one mechanism):

(m) −1 ⃗ ′(m)
(m)
(m) −1 ⃗ (m)


 Π · ∇U = [D ] · Vs − [D ] · F
(m)
(3.87)
Π(m) · ∇S = [D (m) ]−1 · V⃗t − [D (m) ]−1 · F⃗ ′(m)


 (m)
(m)
Π · ∇A = −[D (m) ]−1 · V⃗r
where [D (m) ]−1 means the inverse of the matrix D (m) in the subspace R(m) ⊆ Rn .
In addition, we also have the flux decomposition equation for each mechanism
(m)
(m)
(m)
V⃗t = V⃗s + V⃗r .
103
The dynamical quantities in these dynamical decomposition equations constructed directly from the Fokker-Planck equation also have thermodynamic meanings, which will be revealed shortly in Sec. 3.2.3. Here we mention that the stationary, transient and relative potential landscapes U , S and A are, respectively,
the microscopic stationary, transient and relative entropies. They are also, respectively, the potentials of the thermodynamic forces generating excess entropy
flow, transient entropy change and nonadiabatic entropy production. The tran(m)
(m)
(m)
sient, stationary and relative flux velocities V⃗t , V⃗s and V⃗r are, respectively,
the thermodynamic forces generating total, adiabatic and nonadiabatic entropy
productions from mechanism m. The effective driving force F⃗ ′(m) is the thermodynamic force generating entropy flow from mechanism m. The dynamical
decomposition equations act as a bridge connecting the stochastic dynamics with
the non-equilibrium thermodynamics of systems with multiple transition mechanisms.
3.2.3
Non-Equilibrium Thermodynamics for Multiple State Transition Mechanisms
The microscopic dynamics of the system with multiple mechanisms is governed by the Langevin equation in Eq. (3.72). The ensemble distribution is governed by the corresponding Fokker-Planck equation in Eq. (3.76). The definitions
of the three basic non-equilibrium entropies for systems with multiple mechanisms are the same as those for one mechanism: the cross entropy U = ⟨− ln Ps ⟩t ,
the transient entropy S = ⟨− ln Pt ⟩t and the relative entropy A = U − S =
⟨ln(Pt /Ps )⟩t . Taking the time derivative of these three entropies and using Eq.
(3.87), we can derive the same set of non-equilibrium thermodynamic (infodynamic) equations in Eq. (3.62), namely the cross, transient and relative entropy
balance equations and the entropy production and entropy flow decomposition
equations, which we copy here for completeness:


U̇
= Ṡed − Ṡex






= Ṡpd − Ṡf l
 Ṡ
(3.88)
Ȧ = Ṡed − Ṡna




Ṡpd = Ṡad + Ṡna




Ṡf l = Ṡad + Ṡex
104
with the sign properties Ṡad ≥ 0, Ṡna ≥ 0, Ṡpd ≥ 0. The difference for systems
with multiple transition mechanisms lies in the specific expressions of the various
thermodynamic quantities in Eq. (3.88), which are given below, as an extension of
Eqs. (3.63)-(3.71) for one mechanism:
⟨
⟩
U̇ = ⟨∂t U ⟩ + V⃗t · ∇U ,
(3.89)
⟨
⟩
Ṡ = V⃗t · ∇S ,
(3.90)
⟨
⟩
Ȧ = ⟨∂t U ⟩ + V⃗t · ∇A ,
(3.91)
∑
⟨∂λi U ⟩λ̇i ,
(3.92)
Ṡed = ⟨∂t U ⟩ =
i
Ṡex = −⟨V⃗t · ∇U ⟩ =
Ṡpd =
∑
∑
(m)
−⟨V⃗t
· ∇U ⟩,
(3.93)
m
(m)
(m)
· [D (m) ]−1 · V⃗t ⟩,
(3.94)
(m)
· [D (m) ]−1 · F⃗
(3.95)
(m)
· [D (m) ]−1 · V⃗s(m) ⟩ =
⟨V⃗t
m
Ṡf l =
∑
⟨V⃗t
′(m)
⟩,
m
Ṡad =
∑
⟨V⃗t
m
Ṡna = ⟨V⃗t · D
−1
· V⃗r ⟩ =
= −⟨V⃗t · ∇A⟩ =
∑
m
= ⟨V⃗r · D −1 · V⃗r ⟩ =
= −⟨V⃗r · ∇A⟩ =
∑
m
= ⟨∇A · D · ∇A⟩ =
⟨V⃗s(m) · [D (m) ]−1 · V⃗s(m) ⟩,(3.96)
m
(m)
⟨V⃗t
· [D
(m) −1
]
· V⃗r(m) ⟩
m
(m)
−⟨V⃗t
∑
∑
∑
· ∇A⟩
⟨V⃗r(m) · [D (m) ]−1 · V⃗r(m) ⟩
m
−⟨V⃗r(m) · ∇A⟩
∑
⟨∇A · D (m) · ∇A⟩.
(3.97)
m
We use the transient entropy balance equation Ṡ = Ṡpd − Ṡf l as an example to
illustrate how this equation and the expressions of the thermodynamic quantities
105
involved are obtained. We calculate the rate of change of the transient entropy:
∑ (m)
∑ (m)
Ṡ = ⟨V⃗t · ∇S⟩ =
⟨V⃗t · ∇S⟩ =
⟨V⃗t · Π(m) · ∇S⟩
=
∑
m
(m)
⟨V⃗t
· [D
m
(m) −1
]
·
(m)
V⃗t ⟩
m
∑ (m)
−
⟨V⃗t · [D (m) ]−1 · F⃗
′(m)
⟩,(3.98)
m
(m)
(m)
(m)
where we have used V⃗t
= V⃗t · Π(m) (since V⃗t
is a vector in R(m) , its
(m)
projection into R is itself) and the dynamical decomposition equation for Π(m) ·
∇S in Eq. (3.87). By identifying the first term (nonnegative) in the last equation
as the entropy production rate and the second term as the entropy flow rate, we
obtain the entropy balance equation Ṡ = Ṡpd − Ṡf l , together with the expressions
in Eqs. (3.94) and (3.95). Other equations and expressions can be constructed
similarly. An extension of the results in Eqs. (3.88)-(3.97) as well as the way
to construct them for spatially inhomogeneous systems will be given in Sec. 3.3
using the functional language. Therefore, we shall not further elaborate along this
line in this section in order to reduce redundancy.
The expressions of the total entropy production rate Ṡpd in Eq. (3.94) and that
of the adiabatic entropy production rate Ṡad in Eq. (3.96), as a sum over the mechanism index m, agree with our analysis that they should have contributions from
each individual transition mechanism. According to Eq. (3.96), for each non-zero
(m)
(m)
(m)
V⃗s , there is a strictly positive contribution ⟨V⃗s · [D (m) ]−1 · V⃗s ⟩ to the adiabatic entropy production rate, which is also part of the total entropy production
rate. One may wonder about the cross effects of different mechanisms. It seems that Eqs. (3.94) and (3.96) as well as the rest show no sign of cross effect, as
there is no term involving two different mechanism indexes. However, this is
(m)
(m)
(m)
merely an illusion. In fact, the fluxes V⃗t , V⃗s and V⃗r of different mechanisms in those equations are not independent of each other; they are coordinated
by the global potential landscapes U = − ln Ps , S = − ln Pt and A = (ln Pt /Ps ),
which are determined by the collective inputs of F⃗ (m) and D (m) from individual mechanisms via the Fokker-Planck equation. Put it the other way around.
F⃗ (m) and D (m) from each individual mechanism collectively determine F⃗ and D
in the Fokker-Planck equation and thus collectively determine its transient solution Pt and stationary solution Ps . Pt and Ps then determine the potential land(m)
(m)
scapes S, U and A, which in turn instructs the individual fluxes V⃗t , V⃗s and
(m)
V⃗r of each mechanism through the dynamical decomposition equations in Eqs. (3.83)-(3.85). Therefore, the individual flux of each mechanism emerges as
the collective effect of all the mechanisms, with the potential landscapes play-
106
ing the role of a ‘self-consistent field’ in some sense. Thus the individual flux
of each mechanism has already incorporated the cross effects of different mechanisms. Further, thermodynamic equilibrium indicated by zero entropy production,
Ṡpd = Ṡad = Ṡna = 0, is characterized by all the fluxes of all the individual mech(m)
(m)
(m)
anisms vanishing, V⃗t = V⃗s = V⃗r = 0 for all m, according to Eqs. (3.94),
(3.96) and (3.97). Non-zero fluxes break reversibility of equilibrium states and
create entropy production in non-equilibrium processes. This also justifies their
roles as the thermodynamic forces generating entropy production.
Then we investigate the individual and collective forms of the expressions
of thermodynamics quantities in Eqs. (3.92)-(3.97). From those expressions we
can see that the external driving entropic power Ṡed , the excess entropy flow rate
Ṡex and the nonadiabatic entropy production rate Ṡna have expressions within
which the mechanism index m does not appear. Therefore these three quantities
are not dependent on or not sensitive to the recognition of individual transition
mechanisms. In other words, they can be defined collectively (i.e., expressed
by collective rather than individual quantities). On the other hand, the entropy
production rate Ṡpd , the adiabatic entropy production rate Ṡad and the entropy
flow rate Ṡf l can only be expressed as a sum over the index m, which means they
are sensitive to the correct recognition of individual transition mechanisms.
Equations (3.93)-(3.95) show that Ṡex , Ṡpd and Ṡf l have only one unique expression that is decomposed into each individual mechanism. Hence, they can be
defined uniquely for each mechanism:
(m)
(m)
Ṡex
= −⟨V⃗t · ∇U ⟩,
(m)
(m)
(m)
Ṡ
= ⟨V⃗t · [D (m) ]−1 · V⃗t ⟩,
pd
(m)
Ṡf l
(m)
= ⟨V⃗t
· [D (m) ]−1 · F⃗
′(m)
⟩.
(3.99)
(3.100)
(3.101)
However, there are subtleties in defining the adiabatic and nonadiabatic entropy
production rates Ṡad and Ṡna for each individual mechanism. For Ṡad there are
(m)
(m)
two inequivalent expressions for mechanism m, ⟨V⃗t
· [D (m) ]−1 · V⃗s ⟩ and
(m)
(m)
⟨V⃗s · [D (m) ]−1 · V⃗s ⟩, which generally are not equal to each other. If we
additionally require that the adiabatic entropy production rate for each mechanism
is nonnegative, then we can fix its definition as:
Ṡad = ⟨V⃗s(m) · [D (m) ]−1 · V⃗s(m) ⟩.
(m)
(3.102)
The difference between those two inequivalent expressions of Ṡad for mechanism
107
m can be defined as the mixing entropy production rate for mechanism m:
(m)
Ṡmix = ⟨V⃗s(m) · [D (m) ]−1 · V⃗r(m) ⟩ = −⟨V⃗s(m) · ∇A⟩.
(3.103)
(m)
The word ‘mixing’ in its name is motivated by the observation that ⟨V⃗s ·[D (m) ]−1 ·
(m)
(m)
(m)
(m)
V⃗r ⟩ is a ‘mixing’ of V⃗s and V⃗r . The stationary flux V⃗s is an indicator
of detailed balance breaking in the steady state from mechanism m. When it is
non-zero, it has a positive contribution to the adiabatic entropy production rate in
(m)
(m)
(m)
the form of ⟨V⃗s · [D (m) ]−1 · V⃗s ⟩. The relative flux V⃗r is an indicator of nonstationarity from mechanism m. When it is non-zero, it has a positive contribution
(m)
(m)
to the nonadiabatic entropy production rate in the form of ⟨V⃗r ·[D (m) ]−1 · V⃗r ⟩.
(m)
Therefore Ṡmix characterizes the ‘mixing’ or ‘cross effect’ of these two basic aspects of entropy production in the same mechanism m. Although called an en(m)
tropy production rate, Ṡmix does not have a definite sign. In fact, when summed
∑
∑
(m)
(m)
over m, m Ṡmix = − m ⟨V⃗s · ∇A⟩ = −⟨V⃗s · ∇A⟩ = 0. (This goes back to
(m)
̸= 0 in general.) This means the contributhe fact that ∇ · J⃗s = 0, but ∇ · J⃗s
(m)
tions of Ṡmix from different mechanisms cancel each other and thus do not have
an effect on the collective level. This property ensures that the two inequivalent
(m)
expressions of Ṡad , when summed over m, give the same collective expression
(m)
(m)
of Ṡad . A necessary (but not sufficient) condition for Ṡmix ̸= 0 is that V⃗r ̸= 0
(m)
(m)
and V⃗s ̸= 0. In other words, in order to have a non-zero Ṡmix , the mechanism
m must create both the non-stationary condition and detailed balance breaking
condition in the steady state, which has a positive contribution to both the adiabat(m)
ic and nonadiabatic entropy production rates; if either one of them is zero, Ṡmix
would be zero.
The situation for the nonadiabatic entropy production rate Ṡna of each mechanism is similar. Among the five mechanism-wise expressions of Ṡna , two of
(m)
(m)
(m)
them ⟨V⃗t · [D (m) ]−1 · V⃗r ⟩ = −⟨V⃗t · ∇A⟩, which do not have a definite
(m)
(m)
sign property, are not equivalent to the other three: ⟨V⃗r · [D (m) ]−1 · V⃗r ⟩ =
(m)
(m)
−⟨V⃗r · ∇A⟩ = ⟨∇A · D (m) · ∇A⟩ ≥ 0. Their difference is again Ṡmix , which
does not have an effect on the collective level. If we also require that Ṡna for each
mechanism is nonnegative, then we fix its definition as:
(m)
= ⟨V⃗r(m) · [D (m) ]−1 · V⃗r(m) ⟩ = −⟨V⃗r(m) · ∇A⟩ = ⟨∇A · D (m) · ∇A⟩. (3.104)
Ṡna
(m)
Because of the presence of Ṡmix , the decomposition equations of entropy production and entropy flow, Ṡpd = Ṡad + Ṡna and Ṡf l = Ṡad + Ṡex , generally do not
108
hold for each individual mechanism. The amended equations for each individual
mechanism read as follows:
(m)
(m)
(m)
(m)
Ṡpd = Ṡad + Ṡna
+ 2Ṡmix ,
(m)
Due to the property
equation.
3.2.4
(m)
(m)
(m)
Ṡf l = Ṡad + Ṡex
+ Ṡmix .
∑
(m)
m
(3.105)
(3.106)
(m)
Ṡmix = 0, the term Ṡmix does not appear in the collective
Necessary and Sufficient Condition for the Collective Definition Property
We investigate under what conditions the expressions of thermodynamic quantities derived for systems with one state transition mechanism also apply to systems with multiple state transition mechanisms. To be more specific, we study the
conditions under which the thermodynamic quantities calculated using combined
collective quantities from all the mechanisms give the same results as those calculated using individual quantities from each mechanism and then combined together. This can be termed the collective definition property. For example, the nonadiabatic entropy production rate has this collective definition property according
∑
(m)
(m)
to Eq. (3.97): Ṡna = ⟨V⃗r · D −1 · V⃗r ⟩ = m ⟨V⃗r · [D (m) ]−1 · V⃗r ⟩. The result
of Ṡna calculated using the combined collective quantities V⃗r and D from all the
(m)
mechanisms is the same as that calculated using the individual quantities V⃗r and
D (m) from each mechanism and then combined together. This property, however,
is generally not true for the adiabatic or total entropy production rates. In other
∑
(m)
(m)
words, we do not always have Ṡad = m ⟨V⃗s ·[D (m) ]−1 · V⃗s ⟩ = ⟨V⃗s ·D −1 · V⃗s ⟩
∑ ⃗ (m)
(m)
or Ṡpd = m ⟨Vt · [D (m) ]−1 · V⃗t ⟩ = ⟨V⃗t · D −1 · V⃗t ⟩. It is therefore worthwhile to investigate under what conditions they do hold true. An obvious starting
point is to see what makes Ṡna have such a property. By investigating Eq. (3.97)
one will find that this collective definition property originates from the relative
(m)
dynamical constraint equation: V⃗r = −D (m) · ∇A. This observation leads to
the discovery of a necessary and sufficient condition for the collective definition
property, which is proven in Appendix B. We state the result as a theorem in the
following.
∑
Theorem: For given vectors β⃗ (m) ∈ R(m)∑⊆ Rk , β⃗ = m β⃗ (m) ∈ Rk and
invertible matrixes D (m) ∈ M (m)×(m) , D = m D (m) ∈ M k×k , the necessary
109
∑ (m)
and sufficient condition for ∑ m α
⃗
· [D (m) ]−1 · β⃗ (m) = α
⃗ · D −1 · β⃗ to hold for
(m)
(m)
(m)
k
all α
⃗
∈R
with α
⃗ = mα
⃗
∈ R is that there exists ⃗γ ∈ Rk such that
⃗
β⃗ (m) = D (m) · ⃗γ (m = 1, 2, ...). When ⃗γ does exist, it is given by D −1 · β.
The above theorem is stated in its most general form. We need to adapt it
to our particular situation. We first specify Rk to be the entire state space Rn .
Also we notice that the vectors and matrixes involved in the theorem can also be
dependent on the state vector ⃗q and time t, in which case ‘hold’ or ‘exist’ means
hold or exist for all ⃗q and t. Furthermore, the theorem is applicable when averaged
over the ensemble distribution. With these specifications applied, the theorem
shows that the collective definition property of Ṡna is a result of the dynamical
(m)
constraint equation V⃗r = −D (m) · ∇A, where −∇A plays the role of ⃗γ in the
theorem, which exists by definition. The reason why Ṡad and Ṡpd in general do
not have the collective definition property is due to the structure of the stationary
(m)
= D (m) · ∇U + F⃗ ′(m)
and transient dynamical decomposition equations, V⃗s
(m)
and V⃗t
= D (m) · ∇S + F⃗ ′(m) , both of which have an additional term F⃗ ′(m)
that cannot be absorbed into the first term without changing the structure of the
equation.
We further investigate why ⃗γ ∈ Rk satisfying β⃗ (m) = D (m) · ⃗γ (m = 1, 2, ...)
does not always exist and under what conditions it does exist. The condition
β⃗ (m) = D (m) · ⃗γ is equivalent to Π(m) · ⃗γ = [D (m) ]−1 · β⃗ (m) , which specifies the
⃗ (m) . Because the spaces R(m) (m = 1, 2, ...)
projection of ⃗γ onto each space R
could have non-trivial overlaps (the collection of the base vectors of these spaces are linearly dependent), specification of the projection of a vector onto
these spaces can be contradictory to each other, resulting in the vector ⃗γ nonexistent. If the collection of the base vectors of R(m) (m = 1, 2, ...) are linearly
⃗ In that case
independent, the vector ⃗γ always exists and is given by D −1 · β.
∑
(m)
(m) −1 ⃗ (m)
−1 ⃗
⃗
· [D ] · β
=α
⃗ · D · β according to the theorem.
mα
Placing this analysis into the context of transition mechanisms, it means the
thermodynamic expressions for one transition mechanism also apply to multiple
transition mechanisms under the conditions specified in the following. If there
is a subset of transition subspaces R(m) , with the collection of their base vectors linearly independent, then the thermodynamic expressions for one transition
mechanism also apply to the collection of that particular subset of transition mechanisms. Recognizing or not the individual transition mechanisms within that particular subset does not influence the result. In particular, if the collection of the
base vectors of all the transition subspaces R(m) (m = 1, 2, ...) form a complete
(not necessarily orthogonal) basis of the entire state space Rn , then the expres110
sions of thermodynamic quantities are insensitive to the (non-)recognition of the
individual transition mechanisms of the entire system. To put it another way,
for a system with multiple transition mechanisms, whenever there are transition
mechanisms with the collection of the base vectors of the corresponding transition
spaces linearly independent, they can be grouped together as one effective transition mechanism, in terms of calculating the thermodynamic quantities using the
formula for one transition mechanism given in Eqs. (3.63)-(3.71).
The above understanding of the collective definition property is mainly from
the abstract mathematical point of view. There is a more intuitive approach to
understand this property by considering the paths on the state space. The various
thermodynamic quantities introduced so far, such as those in Eq. (3.88), can also
be defined on the trajectory level on the microstate space [128–130]. The average
over the paths on the microstate space then gives the macroscopic thermodynamic
quantities. A trajectory on the microstate space is a depiction of the evolution
of the system’s microstate (a process the system goes through) that is composed
of successive state transitions. When there are multiple physically different state
transition mechanisms, a phenomenon emerges that does not exist for systems
with only one state transition mechanism, which is configuration degeneracy of
physically different paths. That means physically different paths that are realized via different transition mechanisms (or different combinations of successive
transition mechanisms) can share the same path configuration on the microstate
space, resulting in degeneracy of the configuration of physically different paths.
The correct expressions of those various quantities, defined via paths in the state
space, are dependent on taking into account the path degeneracy phenomenon.
If physically different paths realized via different (combinations of) mechanisms
that share the same path configuration are not identified correctly, they would be
treated as if they are a single path without degeneracy, as in systems with only one
state transition mechanism. This has an influence on the derived expressions of
the various quantities defined via paths. However, there are also situations where
the paths on the state space are not degenerate, even if there are multiple state
transition mechanisms. In that case, the expressions for one transition mechanism
also apply to those of multiple transition mechanisms. The condition we have derived for the entire state space, that the collection of the base vectors within R(m)
form a complete basis of Rn , can be understood as a statement of no path degeneracy. This is because each elementary transition vector d⃗q along a specific path
configuration can be uniquely decomposed, in that basis, into components in each
state transition subspace R(m) . Therefore the contribution from each transition
mechanism to each path element d⃗q along the path configuration is specific and
111
has no ambiguity. A path on the state space therefore can be uniquely identified
by its configuration. In other words, there are no physically different paths sharing
the same path configuration leading to the path degeneracy phenomenon. This is
the reason why the expressions for systems with one transition mechanism also
apply to systems with multiple transition mechanisms in such situations.
3.2.5
Ornstein-Uhlenbeck Processes of Spatially Homogeneous
Systems
We apply the formalism developed so far to Ornstein-Uhlenbeck processes
(OU process for short), processes with a linear force and additive (state-independent)
noise, which can be studied analytically [15, 16]. We first consider the general n
dimensional OU process with one effective state transition mechanism. Then we
consider a more specific example of a two dimensional OU process with three state transition mechanisms, which cannot be reduced to one effective mechanism.
General OU processes with one state transition mechanism
The Langevin equation of the OU process has the following Gaussian whitenoise form:
⃗
⃗q˙ = −γ · ⃗q + ξ(t),
(3.107)
⃗ is Gaussian white noise with the following statiswhere the fluctuating force ξ(t)
tical property:
⃗
⃗ ξ(t
⃗ ′ )⟩ = 2Dδ(t − t′ ).
⟨ξ(t)⟩
= 0,
⟨ξ(t)
(3.108)
The matrices γ and D do not depend on the state vector ⃗q. Yet, for generality,
we allow them to be time-dependent (e.g., via dependence on external control
parameters). We also assume that γ is invertible and D is positive definite. The
Fokker-Planck equation corresponding to the above Langevin equation is:
∂t Pt = ∇ · (γ · ⃗q Pt + ∇ · (DPt )) .
(3.109)
Due to the particular form of the drift vector (linear in ⃗q) and the diffusion matrix
(independent of ⃗q), the transient distribution Pt (⃗q) is a Gaussian distribution all
the time as long as the initial condition is a Gaussian distribution [15, 16]. This
is still true even when γ and D are time-dependent. We focus on such Gaussian
solutions:
}
{
1
1
−1
(3.110)
Pt (⃗q) = √
exp − (⃗q − ⃗µ) · σ · (⃗q − ⃗µ) ,
2
det(2πσ)
112
where the mean vector ⃗µ and the covariance matrix σ are time-dependent and
determined by the following equations [16]:
⃗µ˙ = −γ · ⃗µ
σ̇ = −γ · σ − σ · γ | + 2D,
(3.111)
(3.112)
where γ | is the transpose of the matrix γ. With these two equations one can verify
that the Gaussian distribution in Eq. (3.110) is indeed the solution of Eq. (3.109).
When γ and D are time-dependent, we can introduce the instantaneous stationary
solution by setting the right side of Eq. (3.109) as zero. It turns out that this instantaneous stationary solution is still a Gaussian distribution, whose mean vector
and covariance matrix are the instantaneous stationary solutions of Eqs. (3.111)
and (3.112), respectively, by setting the right side of these two equations as zero.
Since γ is invertible by assumption, the instantaneous stationary mean vector vanishes according to Eq. (3.111). Therefore the instantaneous stationary distribution
is given by:
{
}
1
1
−1
e · ⃗q ,
Ps (⃗q) = √
exp − ⃗q · σ
(3.113)
2
e)
det(2π σ
e is the instantaneous stationary covariance matrix, determined by, accordwhere σ
ing to Eq. (3.112):
e +σ
e · γ | = 2D.
γ·σ
(3.114)
Ps (⃗q) in Eq. (3.113) can be verified to be the instantaneous stationary distribution
by plugging it into the right side of Eq. (3.109) and proving it to be zero using
Eq. (3.114).
From the transient and stationary distributions given by Eqs. (3.110) and (3.113),
we are able to derive the expressions of the stationary, transient and relative potential landscapes of OU processes, using the definitions U = − ln Ps , S = − ln Pt
and A = U − S:
1
1
e −1 · ⃗q + tr (ln(2π σ
e )) ,
⃗q · σ
(3.115)
2
2
1
1
S =
(⃗q − ⃗µ) · σ −1 · (⃗q − ⃗µ) + tr (ln(2πσ)) ,
(3.116)
2
2
1
1
1
e −1 · ⃗q − (⃗q − ⃗µ) · σ −1 · (⃗q − ⃗µ) + tr (ln σ
e − ln σ)(3.117)
A =
⃗q · σ
2
2
2
U =
where tr(·) represents the trace of the matrix in the bracket and we have used
e and σ. Their
ln(detB) = tr(ln B) which holds for positive-definite matrices σ
113
gradients are given, respectively, by:
e −1 · ⃗q,
∇U = σ
∇S = σ −1 · (⃗q − µ
⃗ ),
−1
e · ⃗q − σ −1 · (⃗q − ⃗µ).
∇A = σ
(3.118)
(3.119)
(3.120)
Then using the dynamical decomposition equations in Eqs. (3.6)-(3.8), we can further derive the expressions of the stationary, transient and relative flux velocities
for the OU process:
e −1 · ⃗q,
V⃗s = −γ · ⃗q + D · σ
V⃗t = −γ · ⃗q + D · σ −1 · (⃗q − ⃗µ),
e −1 · ⃗q.
V⃗r = D · σ −1 · (⃗q − ⃗µ) − D · σ
(3.121)
(3.122)
(3.123)
Therefore for OU processes we have obtained the explicit expressions of each
term in the stationary dynamical decomposition equation, F⃗ ′ = −D · ∇U + V⃗s ,
with the effective driving force (also the deterministic driving force in this case)
e −1 · ⃗q and
F⃗ ′ = −γ · ⃗q, the gradient-like potential term −D · ∇U = −D · σ
e −1 · ⃗q. For the transient dynamical
the curl-like flux term V⃗s = −γ · ⃗q + D · σ
′
decomposition equation F⃗ = −D · ∇S + V⃗t , we have explicitly the effective
driving force F⃗ ′ = −γ · ⃗q, the potential term −D · ∇S = −D · σ −1 · (⃗q − ⃗µ)
and the flux term V⃗t = −γ · ⃗q + D · σ −1 · (⃗q − ⃗µ). And for the relative dynamical
constraint equation V⃗r = −D · ∇A, the expression of both sides of the equation
e −1 · ⃗q. These explicit expressions
is given explicitly by D · σ −1 · (⃗q − ⃗µ) − D · σ
facilitate the study of global stability and dynamics of OU process in the potentialflux landscape framework [9, 10, 13, 14].
So far we have not specified the state transition mechanisms which are necessary physical inputs for the calculation of certain thermodynamic quantities. In
the following we consider systems with only one effective state transition mechanism. This includes situations where the system actually has multiple physically
different transition mechanisms, but the collection of the base vectors of the corresponding state transition subspaces are linearly independent, thus enabling them
to be treated as one effective transition mechanism. The cross entropy U, transient
entropy S and relative entropy A can be calculated explicitly from their definitions, as the averages of U , S, A in Eqs. (3.115)-(3.117) over the transient ensemble
Pt in Eq. (3.110), respectively. Their time derivatives as well as the rest of the
thermodynamic quantities can be calculated using (3.63)-(3.71), supplemented by
114
Eqs. (3.118)-(3.123) specific to OU processes. We list the results in the following
and leave the details of calculation in Appendix C:
(
)
1 −1
1
e (σ + ⃗µ⃗µ) + ln(2π σ
e)
U = tr
σ
(3.124)
2
2
(
)
1
1
S = tr
I + ln(2πσ)
(3.125)
2
2
(
)
] 1
1 [ −1
e (σ + ⃗µµ
e − ln σ)
σ
⃗ ) − I + (ln σ
A = tr
(3.126)
2
2
(
)
1
d −1
−1
−1
e −σ
e Dσ
e (σ + ⃗µµ
e) σ
U̇ = tr
(σ + ⃗µµ
⃗ −σ
⃗ ) + γ (3.127)
2
dt
(
)
Ṡ = tr σ −1 D − γ
(3.128)
(
1
d −1
e) σ
e −σ
e −1 D σ
e −1 (σ + ⃗µµ
Ȧ = tr
(σ + ⃗µµ
⃗ −σ
⃗)
2
dt
)
−σ −1 D + 2γ
(3.129)
(
)
1
d −1
e) σ
e
Ṡed = tr
(σ + ⃗µµ
⃗ −σ
(3.130)
2
dt
( −1
)
e Dσ
e −1 (σ + ⃗µµ
Ṡex = tr σ
⃗) − γ
(3.131)
( | −1
)
Ṡf l = tr γ D γ(σ + ⃗µ⃗µ) − γ
(3.132)
( | −1
)
Ṡpd = tr γ D γ(σ + ⃗µ⃗µ) + σ −1 D − 2γ
(3.133)
( | −1
)
−1
−1
e Dσ
e )(σ + ⃗µµ
Ṡad = tr (γ D γ − σ
⃗)
(3.134)
( −1
)
−1
−1
e Dσ
e (σ + ⃗µµ
Ṡna = tr σ
⃗ ) + σ D − 2γ
(3.135)
where I represents the identity matrix and µ
⃗ ⃗µ is the matrix with entries [⃗µ⃗µ]ij =
µi µj . It is easy to verify that these expressions satisfy the set of non-equilibrium
thermodynamic equations in Eq. (3.88). Therefore, for OU processes with one
effective transition mechanism, the set of non-equilibrium thermodynamic equations is realized explicitly, with the expressions of thermodynamic quantities given
explicitly by (3.124)-(3.135).
When there are multiple transition mechanisms that cannot be treated as one
effective mechanism, some of the results above need to be amended. Yet only the
total entropy production rate Ṡpd , the adiabatic entropy production rate Ṡad and
the total entropy flow rate Ṡf l will be different; the expressions of other thermodynamic quantities in Eqs. (3.124)-(3.135) still apply since they can be defined
collectively without referring to individual mechanisms.
115
A two dimensional OU process with three state transition mechanisms
We consider a more specific example of an OU process, with a two dimensional state space and three different state transition mechanisms. Each mechanism
induces a one dimensional OU process in the state space. Thus the collection of
the base vectors of these three state transition subspaces are necessarily linearly
dependent. They cannot be reduced to one effective mechanism. This example
can model, for example, two Brownian particles in contact with three heat reservoirs. The state vector is represented by ⃗q = (q+ , q− )| . For convenience we call
the subspace corresponding to the component q+ the upper space and that corresponding to the component q− the lower space. We assume that both mechanism
1 and mechanism 2 induce an OU process only in the upper space, while mechanism 3 induces an OU process only in the lower space. To be more specific, for
mechanism 1 the drift vector is F⃗ (1) = (−γ1 q+ , 0)| and the diffusion coefficient
D1 is in the upper space. For mechanism 2, the drift vector is F⃗ (2) = (−γ2 q+ , 0)|
and the diffusion coefficient D2 is in the upper space. For mechanism 3, the drift
vector is F⃗ (3) = (0, −γ3 q− )| and the diffusion coefficient D3 is in the lower space. Therefore, the total linear drift matrix and the total diffusion matrix are both
diagonal, given by:
γ = diag(γ+ , γ− ),
D = diag(D+ , D− ),
(3.136)
where γ+ = γ1 + γ2 , γ− = γ3 , D+ = D1 + D2 , and D− = D3 . The mean vector
⃗µ = (µ+ , µ− )| of the transient distribution is determined by Eq. (3.111), which in
this case becomes:
µ̇+ = −γ+ µ+ ,
µ̇− = −γ− µ− .
(3.137)
The solutions are given by:
µ+ (t) = e
µ− (t) = e
−
−
∫t
t0
∫t
t0
γ+ (t1 )dt1
µ+ (t0 ),
γ− (t1 )dt1
µ− (t0 ).
(3.138)
The covariance matrix σ of the transient distribution is determined by Eq. (3.112),
which in the component form reads:
σ̇+ = −2γ+ σ+ + 2D+ ,
σ̇− = −2γ− σ− + 2D− ,
σ̇o = −(γ+ + γ− )σo ,
116
(3.139)
where σ+ and σ− are diagonal elements of the covariance matrix σ while σo is
the off-diagonal element, which means σ+ = var(q+ ), σ− = var(q− ) and σo =
cov(q+ , q− ) of the transient probability distribution. These are all first-order linear
ODEs. The general solutions are given by:
[∫ t
]
∫
∫t
−2 tt γ+ (t1 )dt1
2 t 2 γ+ (t1 )dt1
0
σ+ (t) = e
2D+ (t2 )e 0
dt2 + σ+ (t0 ) ,
t0
[∫ t
]
∫
∫t
−2 tt γ− (t1 )dt1
2 t 2 γ− (t1 )dt1
0
σ− (t) = e
2D− (t2 )e 0
dt2 + σ− (t0 ) ,
σo (t) = e
−
∫t
t0
t0 [γ+ (t1 )+γ− (t1 )]dt1
σo (t0 ).
(3.140)
They can be simplified when γ± and D± are time-independent. The instantae is the instantaneous stationary solution of
neous stationary covariance matrix σ
Eq. (3.139), solved algebraically by
σ
e+ = D+ /γ+ ,
σ
e− = D− /γ− ,
σ
eo = 0,
(3.141)
where σ
e+ = var(q+ ), σ
e− = var(q− ) and σ
eo = cov(q+ , q− ) of the instantaneous
stationary probability distribution.
The thermodynamic quantities in Eqs. (3.124)-(3.135) can then be evaluated
for this 2D case, except Ṡpd , Ṡad and Ṡf l which need to be treated differently by
considering each individual mechanism. Since only algebraic manipulations of
2D matrices are involved, we simply list the results below.
[
]
1 σ+ + µ2+ σ− + µ2−
1
U =
+
+ ln(e
σ+ σ
e− ) + ln(2π)
(3.142)
2
σ
e+
σ
e−
2
1
S =
ln(σ+ σ− − σ02 ) + ln(2π) + 1
(3.143)
2[
]
(
)
1 σ+ + µ2+ σ− + µ2−
1
σ
e+ σ
e−
A =
+
+ ln
− 1 (3.144)
2
σ
e+
σ
e−
2
σ+ σ− − σ02
117
[
]
1
σ
ė
σ
ė
+
−
U̇ =
(e
σ+ − σ+ − µ2+ ) 2 + (e
σ− − σ− − µ2− ) 2
2
σ
e+
σ
e−
[
]
D+
D−
2
2
(σ+ + µ+ ) + 2 (σ− + µ− ) − (γ+ + γ− )
−
(3.145)
2
σ
e+
σ
e−
σ− D+ + σ+ D−
− (γ+ + γ− )
(3.146)
Ṡ =
σ+ σ− − σo2
] [
[
σ
ė
1
σ
ė
D+
−
+
σ− − σ− − µ2− ) 2 −
(σ+ + µ2+ )
Ȧ =
(e
σ+ − σ+ − µ2+ ) 2 + (e
2
2
σ
e+
σ
e−
σ
e+
]
σ − D+ + σ + D−
D−
2
+ 2 (σ− + µ− ) +
− 2(γ+ + γ− )
(3.147)
σ
e−
σ+ σ− − σo2
Ṡed
[
]
1
ė+
ė−
2 σ
2 σ
=
(e
σ+ − σ+ − µ+ ) 2 + (e
σ− − σ− − µ− ) 2
2
σ
e+
σ
e−
D+
(σ+ + µ2+ ) +
2
σ
e+
D+
=
(σ+ + µ2+ ) +
2
σ
e+
−2(γ+ + γ− )
Ṡex =
Ṡna
D−
(σ− + µ2− ) − (γ+ + γ− )
2
σ
e−
D−
σ− D+ + σ+ D−
(σ− + µ2− ) +
2
σ
e−
σ+ σ− − σo2
(3.148)
(3.149)
(3.150)
For comparison, we also list the results of Ṡf l , Ṡpd and Ṡad calculated using the
formula for one effective mechanism in Eqs. (3.124)-(3.135).
γ+2
γ2
(σ+ + µ2+ ) + − (σ− + µ2− ) − (γ+ + γ− )
(3.151)
D+
D−
γ+2
γ2
σ − D+ + σ + D−
=
(σ+ + µ2+ ) + − (σ− + µ2− ) +
D+
D−
σ+ σ− − σo2
−2(γ+ + γ− )
(3.152)
( 2
)
( 2
)
γ+
γ−
D+
D−
=
− 2 (σ+ + µ2+ ) +
− 2 (σ− + µ2− )
D+
σ
e+
D−
σ
e−
=0
(3.153)
Ṡf∗l =
∗
Ṡpd
∗
Ṡad
The star ∗ indicates that these results are not true for the case we consider here. In
∗
deriving the result Ṡad
= 0 in Eq. (3.153) we have used Eq. (3.141). This means
that if the three individual mechanisms are not identified correctly, there would
118
seem to be no detailed balance breaking in the steady state indicated by nonzero adiabatic entropy production. Results obtained with proper identification of
different mechanisms are different, which will be given below.
First we need the results in Eqs. (3.118) and (3.119), which in the component
form reads:
∂q+ U
∂q− U
∂q+ S
∂q− S
=
=
=
=
γ+ q+ /D+
γ− q− /D−
[σ− (q+ − µ+ ) − σo (q− − µ− )]/[σ+ σ− − σo2 ]
[σ+ (q− − µ− ) − σo (q+ − µ+ )]/[σ+ σ− − σo2 ]
(3.154)
For mechanism 1 we work in the upper space. Since the diffusion coefficients are
independent of ⃗q, the (effective) drift coefficient (recall F⃗ ′ = F⃗ − ∇ · D) is given
by
F ′(1) = F (1) = −γ1 q+ .
(3.155)
The stationary flux velocity of mechanism 1 is calculated using Eq. (3.83), which
reads:
[
]
D1
(1)
Vs
= −γ1 q+ + D1 ∂q+ U = −γ1 +
(γ1 + γ2 ) q+ . (3.156)
D1 + D2
Using Eq. (3.84) the transient flux velocity of mechanism 1 given by:
(1)
Vt
= −γ1 q+ + D1 ∂q+ S
= −γ1 q+ + D1 [σ− (q+ − µ+ ) − σo (q− − µ− )]/[σ+ σ− − σo2 ].(3.157)
Similarly, for mechanism 2, we have
F ′(2) = −γ2 q+
[
(2)
Vs
= −γ2 +
(2)
Vt
D2
D1 + D2
]
(γ1 + γ2 ) q+ .
(3.158)
(3.159)
= −γ2 q+ + D2 [σ− (q+ − µ+ ) − σo (q− − µ− )]/[σ+ σ− − σo2(] 3.160)
For mechanism 3, we work in the lower space and have
F ′(3) = −γ3 q−
Vs(3) = 0.
(3)
Vt
(3.161)
(3.162)
= −γ3 q− + D3 [σ+ (q− − µ− ) − σo (q+ − µ+ )]/[σ+ σ− − σo2(] 3.163)
119
(3)
The stationary flux Vs = 0 means there is no detailed balance breaking in the
steady state from mechanism 3. Detailed balance breaking in the steady state is
possible from mechanism 1 and 2 when D1 /γ1 ̸= D2 /γ2 . If D1 /γ1 = D2 /γ2 ,
(1)
(2)
then Vs = 0 and Vs = 0. Therefore the detailed balance condition for this
particular case is D1 /γ1 = D2 /γ2 . If these mechanisms represent heat reservoirs,
then D1 /γ1 = D2 /γ2 is simply T1 = T2 , i.e., the condition of thermal equilibrium.
Then we can calculate Ṡf l , Ṡpd and Ṡad with different mechanisms identified
properly. The total entropy flow rate is calculated using Eq. (3.95) for systems
with multiple mechanisms. With the effective drift coefficients of the three mechanisms given in Eqs. (3.155), (3.158) and (3.161), we have:
Ṡf l =
3 ⟨
⟩
∑
(i)
Vt Di−1 F ′(i)
i=1
(
)
γ12
γ22
γ32
2
=
+
(σ+ + µ+ ) +
(σ− + µ2− )
D1 D2
D3
−(γ1 + γ2 + γ3 ).
(3.164)
The total entropy production rate is calculated using Eq. (3.94) for systems with
multiple mechanisms. With the transient flux velocities given by (3.157), (3.160)
and (3.163), we have
Ṡpd =
3 ⟨
∑
Vt Di−1 Vt
(i)
(i)
⟩
i=1
(
)
γ12
γ22
γ2
=
+
(σ+ + µ2+ ) + 3 (σ− + µ2− )
D1 D2
D3
σ− (D1 + D2 ) + σ+ D3
+
− 2(γ1 + γ2 + γ3 ).
σ+ σ− − σo2
(3.165)
The adiabatic entropy production rate is calculated using Eq. (3.96) for systems
with multiple mechanisms, with the stationary flux velocities given by (3.156),
(3.159) and (3.162):
( 2
)
3
∑
⟨ (i) −1 (i) ⟩
γ1
γ22
(γ1 + γ2 )2
Ṡad =
Vs Di Vs =
+
−
(σ+ + µ2+ ). (3.166)
D
D
D
+
D
1
2
1
2
i=1
The true adiabatic entropy production rate Ṡad is not identically zero, in contrast
γ22
γ12
∗
+
≥
with Ṡad = 0 in Eq. (3.153). Instead, Ṡad ≥ 0 due to the inequality
D1 D2
120
(γ1 + γ2 )2
. The necessary and sufficient condition for Ṡad = 0 is D1 /γ1 =
D1 + D2
(i)
D2 /γ2 , which is also the detailed balance condition indicated by Vs = 0 (i =
1, 2, 3).
∗
∗
We notice that the offsets of Ṡf∗l , Ṡpd
, and Ṡad
calculated using the formula for
one effective mechanism from their true values for multiple mechanisms Ṡf l , Ṡpd ,
∗
= Ṡad ,
and Ṡad , in this particular case, are such that: Ṡf l − Ṡf∗l = Ṡad , Ṡpd − Ṡpd
∗
Ṡad − Ṡad = Ṡad . Therefore, the balance equation of the transient entropy holds
∗
∗
for both set of quantities: Ṡ = Ṡpd − Ṡf l = Ṡpd
− Ṡf∗l , yet individually Ṡpd
and
∗
Ṡf l are both off from the true values Ṡpd and Ṡf l . In general, it is not necessarily
∗
∗
= 0). But the offset of the adiabatic entropy
= Ṡad (i.e., Ṡad
such that Ṡad − Ṡad
production rate, due to the incorrect identification of different mechanisms, Ṡad −
∗
= ∆Ṡad , will also be the offset of the total entropy production rate and the
Ṡad
∗
total entropy flow rate: Ṡpd − Ṡpd
= ∆Ṡad , Ṡf l − Ṡf∗l = ∆Ṡad . This can be
seen from the two decomposition equations: Ṡpd = Ṡad + Ṡna and Ṡf l = Ṡad +
Ṡex , where Ṡna and Ṡex are not influenced by the (non-)recognition of individual
mechanisms since they can be defined using collective quantities. Therefore, the
offset of Ṡad is also the offset of Ṡpd and Ṡf l , which in turn ensures that Ṡ in the
balance equation Ṡ = Ṡpd − Ṡf l is not influenced by the (non-)recognition of
individual mechanisms.
3.3
Non-Equilibrium Thermodynamics for Spatially Inhomogeneous Stochastic Dynamical Systems
In this section (Sec. 3.3), we extend the major results in Sec. 3.1 and Sec. 3.2
for spatially homogeneous systems to spatially inhomogeneous systems. Spatially inhomogeneous systems are systems with infinite degrees of freedom (infinite
dimensional systems). A certain level of mathematical accuracy is necessary in
dealing with such infinite dimensional systems; yet being caught up in too many
technical details at the initial stage of establishing the formalism would also be
counterproductive. Thus we seek to find a balance between physical heuristic motivations and mathematical rigorous treatments. We first look at the description of
spatially inhomogeneous systems. Then we introduce the functional Langevin and
Fokker-Planck equations. Accordingly, the potential-flux landscape framework is
121
generalized to spatially inhomogeneous systems. With its assistance we formulate
the non-equilibrium thermodynamics for spatially inhomogeneous systems. General OU processes are studied using the established formalism. Finally, the spatial
stochastic neuronal model, which can also describe a reaction diffusion process,
serves as the testing ground and an illustration of the practical application of the
general theory.
3.3.1
Description of Spatially Inhomogeneous Systems
We assume the (micro)state of a spatially inhomogeneous system at each moment is described by a ℓ-component function of the physical space (a vector field):
⃗ x) = {ϕ1 (⃗x), ..., ϕa (⃗x), ..., ϕℓ (⃗x)}. For example, in the context of chemical reϕ(⃗
⃗ x) may represent the local concentrations of chemical species
action systems, ϕ(⃗
⃗ has only one component, the state of
involved at any given moment. If the vector ϕ
the system is then described by a scalar field ϕ(⃗x). In the biological context, ϕ(⃗x)
may represent, for instance, the local electric potential on the neuron membrane
at any given moment.
To specify a state of the spatially inhomogeneous system (a global state in
⃗ x)
the physical space; also called a field configuration), the value of the field ϕ(⃗
need to be given at every location in a domain of the physical space. In practice,
the system is always observed at some finite spatial scale, the degrees of freedom
below which are averaged out or coarse-grained, resulting in an effective description [137, 158]. In accord with this consideration, we can discretize the relevant
region of the physical space into spatial cells with volume ∆V characterizing the
resolution scale and introduce a discrete space representation [15, 50–52]. We la⃗ x) in
bel the spatial cells by a discrete space index λ. The average values of ϕ(⃗
a
each spatial cell, denoted collectively by {ϕλ }, represent the effective degrees of
freedom (i.e., state variables) of the system:
∫
1
a
d⃗x ϕa (⃗x).
(3.167)
ϕλ =
∆V Vλ
Each state of the system is now specified by giving the values of {ϕaλ } for both
the index a and the discrete space index λ. Therefore, the pair of discrete indexes a and λ in ϕaλ together labels the effective degrees of freedom of the spatially
inhomogeneous system. This description is analogous to spatially homogeneous
systems with finite degrees of freedom, labeled by a discrete index i of its state
vector ⃗q = (q1 , ..., qi , ..., qn ). This index correspondence (a, λ) ↔ i allows for
122
a formal extension of spatially homogeneous systems with finite degrees of freedom, to spatially inhomogeneous systems with infinite degrees of freedom in the
discrete space representation [50, 51, 158]. An effective continuous space representation can be obtained by taking the continuum limit ∆V → 0. What this
limit practically means is that we are studying the system on a scale much larger
than the resolution scale ∆V , so that ∆V is, relatively speaking, very small [15].
Mathematically,
∫
1
a
lim ϕλ = lim
d⃗x ′ ϕa (⃗x ′ ) = ϕa (⃗x).
(3.168)
∆V →0
∆V →0 ∆V
Vλ
Equations (3.167) and (3.168) are the prescriptions to switch between the discrete
and continuous space representations of the state of the system. We will use this
approach later to introduce and study OU processes of spatially inhomogeneous
systems.
To allow for appropriate mathematical treatments, we introduce further mathematical structures for the state and state space of spatially inhomogeneous systems. We assume the physical space (space of ⃗x) is modeled as a k-dimensional
⃗ lies in an ℓ-dimensional Euclidean space
Euclidean space Rk , while the vector ϕ
Rℓ . The region of the physical space relevant for the description of the system is
modeled as a suitable domain V in Rk . Therefore, a state of the spatially inho⃗ x) : V → Rℓ . The state
mogeneous system is represented by a vector field ϕ(⃗
⃗ x), is also called the field conspace of the system Ω, as the space of the field ϕ(⃗
figuration space, with each ‘point’ or ‘element’ in this space representing a field
⃗ x) (a global state in the physical space). The state space (the field
configuration ϕ(⃗
configuration space) as a function space is an infinite-dimensional space, which
we assume is a (real) Hilbert space (a linear space with an inner product that is
complete) [42,43]. The real-valued inner product between two states in the Hilbert
space is defined as:
∫
∫
∑
⃗
(φ|ϕ) ≡ d⃗x φ
⃗ (⃗x) · ϕ(⃗x) = d⃗x
φa (⃗x)ϕa (⃗x),
(3.169)
a
⃗ x)
where the integral is taken over the domain V and the dot between φ
⃗ (⃗x) and ϕ(⃗
is the standard Euclidean dot product in Rℓ . We have used a variation of the
Dirac notation borrowed from quantum mechanics [159], where the states φ
⃗ (⃗x)
⃗ x) in the Hilbert space within the inner product are represented abstractly in
and ϕ(⃗
the bra-ket notation as (φ| and |ϕ), respectively, with round brackets replacing angle brackets conventional in quantum mechanics (angle brackets are reserved for
123
ensemble average in this work). The interpretation here, however, is quite different from quantum mechanics. Here an element in the real Hilbert space represents
a definite classical state of the spatially inhomogeneous system (a classical field),
while in quantum mechanics an element in the complex Hilbert space represents a
quantum state associated with probabilistic interpretations (a quantum wave function). The inner product in Eq. (3.169) represents a geometrical relation between
two states, allowing for the introduction of angle and distance on the Hilbert state
space, as a generalization of the Euclidean state space. We note that the flux-force
bilinear form (integrated over space) in classical irreversible thermodynamics [94]
has a similar form as Eq. (3.169).
With the inner product defined, an orthonormal basis {⃗en (⃗x)} of the Hilbert state space can be characterized by the following orthonormality and completeness
conditions:
∫
∑
d⃗x ⃗en (⃗x) · ⃗em (⃗x) = δmn ,
⃗en (⃗x)⃗en (⃗x ′ ) = I δ(⃗x − ⃗x ′ ),
(3.170)
n
where I is the identity matrix in Rℓ . In this basis a state in the Hilbert space can
be expanded as:
∑
⃗ x) =
ϕ(⃗
ϕn⃗en (⃗x),
(3.171)
n
∫
where
ϕn =
⃗ x).
d⃗x ⃗en (⃗x) · ϕ(⃗
(3.172)
The coefficient ϕn can be seen as the component of an infinite dimensional vector
⃗ x) and the infinite dimensional vector {ϕn } are merely two
{ϕn }. The field ϕ(⃗
⃗ x) is the
different representations of the same state |ϕ) in the Hilbert space. ϕ(⃗
state |ϕ) in the space configuration representation, while {ϕn } is the state |ϕ) in
the representation of the orthonormal basis {⃗en (⃗x)}. A linear operator B̂ on the
Hilbert space, mapping a state |ϕ) into another state |φ) , can be represented by a
matrix-valued integral kernel, B(⃗x, ⃗x ′ ), in the space configuration representation,
with the operation of linear mapping given by the integral transform:
∫
⃗ x ′ ).
φ
⃗ (⃗x) = d⃗x ′ B(⃗x, ⃗x ′ ) · ϕ(⃗
(3.173)
In an orthonormal basis {⃗en (⃗x)}, B(⃗x, ⃗x ′ ) can be expanded as:
∑
B(⃗x, ⃗x ′ ) =
Bmn⃗em (⃗x)⃗en (⃗x ′ ),
mn
124
(3.174)
∫
where
Bmn =
∫
d⃗x
d⃗x ′⃗em (⃗x) · B(⃗x, ⃗x ′ ) · ⃗en (⃗x ′ ).
(3.175)
The coefficient Bmn can be regarded as the entry of an infinite dimensional square
matrix [Bmn ]. The integral kernel B(⃗x, ⃗x ′ ) and the infinite dimensional square
matrix [Bmn ] are two different representations of the same linear operator B̂ on the
Hilbert space. In the abstract representation using the Dirac notation Eq. (3.173)
is
∑written as |φ) = B̂|ϕ), while in the representation of {⃗en (⃗x)} it reads φm =
n Bmn ϕn , which can be obtained using Eqs. (3.171)-(3.174).
All these different representations are mathematically equivalent and related
by certain transformation rules. Yet one representation may be more convenient or
suitable than another in a particular situation. Generally, the space configuration
representation is more intuitive as it works with the physical space; the abstract representation has the advantage of being very compact; the representation of
{⃗en (⃗x)} is convenient for practical calculations. These statements are not to be
taken as absolute, though. In the following, we work in the space configuration
representation in the course of establishing the non-equilibrium thermodynamic
formalism for spatially inhomogeneous systems. In Appendix D we give the abstract representation of the major equations that will be developed in the main text
as well as the transformation rules to switch between different representations.
When it comes to practical examples we will transform into the representation of
{⃗en (⃗x)}. Occasionally, we also use the abstract representation in the main text.
3.3.2
Stochastic Dynamics of Spatially Inhomogeneous Systems
The spatial-temporal stochastic dynamics of many spatially inhomogeneous
systems can be described by functional Langevin and Fokker-Planck equations [15, 31, 37, 42–51, 54–60]. We consider stochastic spatially inhomogeneous systems governed by the following functional Langevin equation (a stochastic differential equation on the infinite dimensional Hilbert state space Ω) [15, 31, 42–44,
50, 51]:
∑
⃗ x, t) = F⃗ (⃗x, t)[ϕ]dt +
⃗ s (⃗x, t)[ϕ] dWs (t),
dϕ(⃗
G
(3.176)
s
⃗ s (⃗x, t)[ϕ] are, respectively, the deterministic and stochastic
where F⃗ (⃗x, t)[ϕ] and G
driving force fields. They are maps from Ω into Ω, parameterized by t, allowing
⃗ s (⃗x, t) are elements in Ω. The
them to be time-dependent. For any t, F⃗ (⃗x, t) and G
125
square bracket [ϕ] denotes functional dependence, indicating they are functions
⃗ x, t) ∈ Ω. In most applications, F⃗ (⃗x, t)[ϕ] and G
⃗ s (⃗x, t)[ϕ] can be
of the field ϕ(⃗
expressed as (nonlinear) partial differential operators or integro-differential opera⃗ x, t) [31,42–44]. In the former case, the values of F⃗ (⃗x, t)
tors acting on the field ϕ(⃗
⃗ x, t) at ⃗x
⃗ s (⃗x, t) at location ⃗x depend only on the local value of the field ϕ(⃗
and G
or its immediate neighborhood; in the latter case, which is more general, they de⃗ x, t) as a whole. Ws (t) (s = 1, 2, ...), with s allowed to go to
pend on the field ϕ(⃗
infinity, are independent one-dimensional standard Wiener processes. The form
of the fluctuation term in Eq. (3.176) is a little different from, yet is equivalent to, that usually presented in the mathematical literature in terms of cylindrical
Brownian motions [43]. The index s labels statistically independent fluctuation
sources. Yet these fluctuation sources do not necessarily represent different state
transition mechanisms. Thus we introduce the state transition mechanism index
m and further specify Eq. (3.176) as follows:
⃗ x, t) =
dϕ(⃗
∑[
m
=
∑
](m)
⃗ x, t)
dϕ(⃗
[
F⃗ (m) (⃗x, t)[ϕ]dt +
m
∑
]
⃗ (m) (⃗x, t)[ϕ] dW (m) (t) .(3.177)
G
s
s
s
[
](m)
⃗
For each mechanism m, the state transition dϕ(⃗x, t)
has the form of a func⃗ (m)
x, t)[ϕ] are, retional Langevin equation in Eq. (3.176). F⃗ (m) (⃗x, t)[ϕ] and G
s (⃗
spectively, the deterministic and stochastic driving force fields of mechanism m.
(m)
Ws (t) (m = 1, 2, ...; s = 1, 2, ...) are statistically independent one-dimensional
standard Wiener processes. We assume for each mechanism m the vectors F⃗ (m)
(m)
⃗ (m)
and G
⊆ Rℓ , which means
s (s = 1, 2, ...) are in the same linear subspace R
they have the following component forms:
∑ (m) (m)
F⃗ (m) (⃗x, t)[ϕ] =
⃗ei Fi (⃗x, t)[ϕ],
(3.178)
i
⃗ (m) (⃗x, t)[ϕ] =
G
s
∑
(m)
⃗ei
(m)
Gi s (⃗x, t)[ϕ],
(3.179)
i
(m)
where ⃗ei (i = 1, 2, ...) are the base vectors of R(m) ⊆ Rℓ . The projection operator Π(m) defined in Eq. (3.75) and its properties still apply here. The restriction of
Rℓ to R(m) defines a Hilbert subspace Ω(m) of the entire Hilbert state space Ω. The
126
projection operator Π(m) also induces the projection from Ω to Ω(m) . F⃗ (m) (⃗x, t)[ϕ]
⃗ (m)
and G
x, t)[ϕ] are therefore maps from the Hilbert state space Ω into its subs (⃗
(m)
space Ω
⊆ Ω. Equations (3.177)-(3.179) are extensions of Eqs. (3.72)-(3.74)
to spatially inhomogeneous systems. We mention that boundary conditions may
have important effects on the non-equilibrium dynamics of spatially inhomogeneous systems [160].
The functional Fokker-Planck equation (usually called the forward Kolmogrov
equation in the mathematical literature) on the Hilbert state space Ω, corresponding to the functional Langevin equation (Eq. (3.177); interpreted as an Ito stochastic differential equation), reads [15, 31, 43, 45–47, 50, 51]:
∫
(
)
⃗
⃗
∂t Pt [ϕ] = − d⃗x δϕ(⃗
x, t)[ϕ]Pt [ϕ]
⃗ x) · F (⃗
∫
∫
+
d⃗x ⃗δϕ(⃗
d⃗x ′⃗δϕ(⃗
x, ⃗x ′ , t)[ϕ]Pt [ϕ]) , (3.180)
⃗ x) ·
⃗ x ′ ) · (D(⃗
where ⃗δϕ(⃗
⃗ x) is a short notation for the multi-component functional derivative, with
a
a
the component (⃗δϕ(⃗
x). The drift vector field and the diffusion matrix
⃗ x) ) ≡ δ/δϕ (⃗
field in Eq. (3.180) are given by:
∑
F⃗ (⃗x, t)[ϕ] =
F⃗ (m) (⃗x, t)[ϕ],
(3.181)
m
′
D(⃗x, ⃗x , t)[ϕ] =
∑
D (m) (⃗x, ⃗x ′ , t)[ϕ]
m
[
]
∑ 1∑
⃗ (m) (⃗x, t)[ϕ]G
⃗ (m) (⃗x ′ , t)[ϕ] .
=
G
s
s
2
m
s
(3.182)
D(⃗x, ⃗x ′ , t) and D (m) (⃗x, ⃗x ′ , t) are the counterparts of nonnegative symmetric matrices in the infinite dimensional space (i.e., nonnegative self-adjoint operators on
the Hilbert space). By construction they have the following symmetry property:
Dab (⃗x, ⃗x ′ , t)[ϕ] = Dba (⃗x ′ , ⃗x, t)[ϕ], [D(m) ]ab (⃗x, ⃗x ′ , t)[ϕ] = [D(m) ]ba (⃗x ′ , ⃗x, t)[ϕ].
(3.183)
They also have the following nonnegative property according to Eq. (D.5). For
any φ
⃗ (⃗x) ∈ Ω,
∫∫
d⃗xd⃗x ′ φ
⃗ (⃗x) · D(⃗x, ⃗x ′ , t)[ϕ] · φ
⃗ (⃗x ′ ) ≥ 0,
(3.184)
127
∫∫
d⃗xd⃗x ′ φ
⃗ (⃗x) · D (m) (⃗x, ⃗x ′ , t)[ϕ] · φ
⃗ (⃗x ′ ) ≥ 0.
(3.185)
We impose a stronger condition on D and D (m) , requiring them to be positive
definite (we use this term in the strict sense in contrast to nonnegative definite),
which means that for any φ
⃗ (⃗x) ∈ Ω (for any φ
⃗ (⃗x) ∈ Ω(m) ) the equality in Eq. (3.184) (the equality in Eq. (3.185)) holds only when φ
⃗ (⃗x) ≡ 0. We also assume
that D(⃗x, ⃗x ′ , t)[ϕ] is invertible in the sense that there exists M (⃗x, ⃗x ′ , t)[ϕ] on Ω
which has the symmetry property as in Eq. (3.183) and satisfies the following:
∫
∑
d⃗x ′′
M ac (⃗x, ⃗x ′′ , t)[ϕ]Dcb (⃗x ′′ , ⃗x ′ , t)[ϕ] = δab δ(⃗x − ⃗x ′ ),
(3.186)
c
where the indexes a, b, c are vector indexes of the space Rℓ . Define M (⃗x, ⃗x ′ , t)[ϕ] ≡
[D(⃗x, ⃗x ′ , t)]−1 [ϕ], where the notation [D(⃗x, ⃗x ′ , t)]−1 indicates the inverse is not
only with respect to the matrix indexes of D but also the space indexes ⃗x and
⃗x ′ . Note that [D(⃗x, ⃗x ′ , t)]−1 does not mean 1/D(⃗x, ⃗x ′ , t). Similarly, we assume
D (m) (⃗x, ⃗x ′ , t)[ϕ] is invertible on Ω(m) in the sense that there exists M (m) (⃗x, ⃗x ′ , t)[ϕ]
on Ω(m) , with the symmetry property in Eq. (3.183) and the condition:
∫
∑
d⃗x ′′
[M (m) ]ac (⃗x, ⃗x ′′ , t)[ϕ] [D(m) ]cb (⃗x ′′ , ⃗x ′ , t)[ϕ] = δab δ(⃗x − ⃗x ′ ), (3.187)
c
where the indexes a, b, c are restricted to the vector indexes of the space R(m) ⊆
[
]−1
Rℓ . We also define M (m) (⃗x, ⃗x ′ , t)[ϕ] ≡ D (m) (⃗x, ⃗x ′ , t)
[ϕ]. In practice,
[ (m)
]−1
−1
′
′
[D(⃗x, ⃗x , t)] [ϕ] and D (⃗x, ⃗x , t)
[ϕ] can be constructed by decomposing
them into their eigenvalues and eigenfunctions and then inverting the non-zero
eigenvalues while keeping the eigenfunctions unchanged. A rigorous mathematical treatment of the (pseudo-)inverse of linear operators on the Hilbert space can
be found in Ref. [161].
3.3.3
Potential-Flux Field Landscape Framework for Spatially
Inhomogeneous Systems
The functional Fokker-Planck equation (Eq. (3.180)) can be interpreted as a
continuity equation on the field configuration space (Hilbert state space) [50]:
∫
⃗ x)[ϕ],
∂t Pt [ϕ] = − d⃗x ⃗δϕ(⃗
(3.188)
⃗ x) · Jt (⃗
128
where the transient probability flux field is given by:
∫
′
⃗
⃗
Jt (⃗x)[ϕ] = F (⃗x, t)[ϕ]Pt [ϕ] − d⃗x ′ D(⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) Pt [ϕ],
(3.189)
with the effective drift vector field defined as:
∫
′
⃗
⃗
F (⃗x, t)[ϕ] = F (⃗x, t)[ϕ] − d⃗x ′⃗δϕ(⃗
x, ⃗x ′ , t)[ϕ].
⃗ x ′ ) · D(⃗
(3.190)
Equation (3.189) can be reformulated into the transient field dynamical decomposition equation:
∫
′
⃗
⃗ x)[ϕ],
F (⃗x, t)[ϕ] = − d⃗x ′ D(⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
(3.191)
⃗ x ′ ) S[ϕ] + Vt (⃗
where the transient potential field landscape is S[ϕ] = − ln Pt [ϕ] and the transient
flux velocity field is V⃗t (⃗x)[ϕ] = J⃗t (⃗x)[ϕ]/Pt [ϕ]. The (instantaneous) stationary
probability functional Ps [ϕ] satisfies the stationary functional Fokker-Planck e∫
⃗ x)[ϕ] = 0. We assume Ps [ϕ] at each moment is unique (up
quation: d⃗x ⃗δϕ(⃗
⃗ x) · Js (⃗
to a multiplication constant explained later) and positive. Accordingly, we have
the stationary field dynamical decomposition equation:
∫
′
⃗
⃗ x)[ϕ],
F (⃗x, t)[ϕ] = − d⃗x ′ D(⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
(3.192)
⃗ x ′ ) U [ϕ] + Vs (⃗
where the stationary potential field landscape is U [ϕ] = − ln Ps [ϕ] and the stationary flux velocity field is V⃗s (⃗x)[ϕ] = J⃗s (⃗x)[ϕ]/Ps [ϕ]. From Eqs. (3.191) and
(3.192) we can also derive the relative field dynamical constraint equation:
∫
⃗
Vr (⃗x)[ϕ] = − d⃗x ′ D(⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
(3.193)
⃗ x ′ ) A[ϕ],
where the relative potential field landscape is defined as A[ϕ] = U [ϕ] − S[ϕ] =
ln(Pt [ϕ]/Ps [ϕ]) and the relative flux velocity field is V⃗r (⃗x)[ϕ] = V⃗t (⃗x)[ϕ]−V⃗s (⃗x)[ϕ].
The flux field decomposition equation then reads:
V⃗t (⃗x)[ϕ] = V⃗s (⃗x)[ϕ] + V⃗r (⃗x)[ϕ].
(3.194)
The probability flux field can also be defined for each individual mechanism:
(m)
J⃗t (⃗x)[ϕ] = F⃗ ′(m) (⃗x, t)[ϕ]Pt [ϕ]
∫
−
d⃗x ′ D (m) (⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) Pt [ϕ],
129
(3.195)
where the effective drift vector field of mechanism m is given by:
∫
′(m)
(m)
(m)
⃗
⃗
F
(⃗x, t)[ϕ] = F (⃗x, t)[ϕ] − d⃗x ′⃗δϕ(⃗
(⃗x, ⃗x ′ , t)[ϕ].(3.196)
⃗ x ′) · D
Accordingly, Eqs. (3.191)-(3.193) also have their counterparts for each mechanism:
∫
′(m)
⃗
F
(⃗x, t)[ϕ] = − d⃗x ′ D (m) (⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) U [ϕ]
+V⃗s(m) (⃗x)[ϕ],
∫
′(m)
⃗
F
(⃗x, t)[ϕ] = − d⃗x ′ D (m) (⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) S[ϕ]
(m)
+V⃗t (⃗x)[ϕ],
∫
(m)
V⃗r (⃗x)[ϕ] = − d⃗x ′ D (m) (⃗x, ⃗x ′ , t)[ϕ] · ⃗δϕ(⃗
⃗ x ′ ) A[ϕ].
(3.197)
(3.198)
(3.199)
∑ (m)
The sum over the index m gives the combined collective quantity: m J⃗t = J⃗t ,
∑
∑ ⃗ ′(m)
∑
∑
(m)
(m)
(m)
= F⃗ ′ , m V⃗t = V⃗t , m V⃗s = V⃗s , m V⃗r = V⃗r . The diffusion
mF
matrix field in (3.197)-(3.199) can be inverted using Eq. (3.187) to produce an
alternative form of these equations:
∫
[
]−1
(m) ⃗
Π · δϕ(⃗
d⃗x ′ D (m) (⃗x, ⃗x ′ , t)
[ϕ] · V⃗s(m) (⃗x ′ )[ϕ]
⃗ x) U [ϕ] =
∫
[
]−1
−
d⃗x ′ D (m) (⃗x, ⃗x ′ , t)
[ϕ] · F⃗ ′(m) (⃗x ′ , t)[ϕ],
∫
[
]−1
(m)
(m) ⃗
Π · δϕ(⃗
d⃗x ′ D (m) (⃗x, ⃗x ′ , t)
[ϕ] · V⃗t (⃗x ′ )[ϕ]
⃗ x) S[ϕ] =
∫
[
]−1
−
d⃗x ′ D (m) (⃗x, ⃗x ′ , t)
[ϕ] · F⃗ ′(m) (⃗x ′ , t)[ϕ],
∫
[
]−1
(m) ⃗
Π · δϕ(⃗
d⃗x ′ D (m) (⃗x, ⃗x ′ , t)
[ϕ] · V⃗r(m) (⃗x ′ )[ϕ],(3.200)
⃗ x) A[ϕ] = −
[
]−1
[ϕ]
where Π(m) is the projection operator onto space R(m) and D (m) (⃗x, ⃗x ′ , t)
(m)
′
is the inverse of D (⃗x, ⃗x , t)[ϕ] defined in Eq. (3.187). Besides, we also have
the flux field decomposition equation for each mechanism:
(m)
V⃗t (⃗x)[ϕ] = V⃗s(m) (⃗x)[ϕ] + V⃗r(m) (⃗x)[ϕ].
130
(3.201)
We discuss some particularities of spatially inhomogeneous systems. The
functional Fokker-Planck equation, as a continuity equation in Eq. (3.188), describes information (probability) transport process in the field configuration space, rather than matter or energy transport in the physical space. For spatially
inhomogeneous systems, this distinction has become drastic, as the field configuration space is an infinite dimensional space, while the physical space is finite
dimensional (usually assumed to be three dimensional). The various quantities
introduced, the potential field landscapes, U [ϕ], S[ϕ], A[ϕ], and the flux velocity
(m)
(m)
(m)
⃗ x).
fields, V⃗s (⃗x)[ϕ], V⃗t (⃗x)[ϕ], V⃗r (⃗x)[ϕ], are all functionals of the field ϕ(⃗
They are all defined on the field configuration space, with each ‘element’, ‘point’
or ‘vector’ in this space representing a global state of the system in the physical
space (i.e., a field configuration). The (thermo)dynamical interpretations of these
quantities as ‘potentials’ and ‘forces’ thus should also be understood in the context
of information transport in the infinite dimensional field configuration space. With
this distinction from spatially homogeneous systems taken into account, these various quantities can be interpreted similarly as those introduced in Sec. 3.2. The
stationary, transient and relative potential field landscapes, U [ϕ], S[ϕ], A[ϕ], represent, respectively, the microscopic stationary, transient and relative entropies of
the system; they are also the potentials of the thermodynamic force fields generating, respectively, the excess entropy flow, transient entropy change and nonadiabatic entropy production. The transient, stationary and relative flux velocity
(m)
(m)
(m)
fields, V⃗t (⃗x)[ϕ], V⃗s (⃗x)[ϕ], V⃗r (⃗x)[ϕ], are the thermodynamic force fields of
mechanism m generating the total, adiabatic and nonadiabatic entropy productions, respectively.
3.3.4
Non-Equilibrium Thermodynamics of Spatially Inhomogeneous Systems
In the thermodynamic context, the functional Langevin equation (Eq. (3.177))
is interpreted as the dynamical equation of the microstate of the spatially inhomogeneous system. The functional Fokker-Planck equation (Eq. (3.180)) is interpreted as the dynamical equation governing an ensemble of systems under the same
macroscopic external conditions. These macroscopic external conditions (i.e.,
macrostates) can vary both in time and space, thus represented by a set of spacetime functions (i.e., time-dependent fields) {λ(⃗x, t)} ≡ {λ1 (⃗x, t), ..., λi (⃗x, t), ...}.
For instance, an inhomogeneous environmental temperature distribution also changing with time can be represented by T (⃗x, t), which regulates the microscopic and
131
ensemble dynamics. The time dependence in F⃗ (⃗x, t)[ϕ] and D(⃗x, t)[ϕ] and that
which can be traced back to them can be replaced by the functional dependence on
{λ(⃗x, t)} (e.g., F⃗ (⃗x)[ϕ, {λ}], D(⃗x, ⃗x ′ )[ϕ, {λ}]). To simplify notations, however,
we still use an explicit time dependence t instead of {λ}.
The cross, transient and relative entropies of the spatially inhomogeneous system are defined, respectively, as the averages of the microstate functional U , S
and A over the transient ensemble:
∫
U = ⟨U [ϕ]⟩ = − Pt [ϕ] ln Ps [ϕ]D[ϕ],
(3.202)
∫
S = ⟨S[ϕ]⟩ = − Pt [ϕ] ln Pt [ϕ]D[ϕ],
(3.203)
∫
A = ⟨A[ϕ]⟩ = Pt [ϕ] ln (Pt [ϕ]/Ps [ϕ]) D[ϕ],
(3.204)
where the integrals are functional integrals taken over the field configuration space (Hilbert state space) [45–47, 158]. We stress that the definitions of these
non-equilibrium entropies are not expressed as integrals over the physical space,
in contrast with classical irreversible thermodynamics based on the local equilibrium assumption which requires entropy to be locally defined in the physical
space [94]. Here, in Eqs. (3.202)-(3.204), ‘local’ is expressed in the field configuration space, within which each local point represents a global state in the physical
space. Therefore, we are not using a local description in the physical space; the
global description in the physical space is disguised in the local form in the field
configuration space. It is important to understand this point.
Then we calculate time derivatives of Eqs. (3.202)-(3.204) to derive their respective entropy balance equations and the expressions of thermodynamic quantities. For cross entropy we have:
∫
d
U̇ =
Pt [ϕ]U [ϕ]D[ϕ]
dt
∫
∫
=
Pt [ϕ]∂t U [ϕ]D[ϕ] + (∂t Pt [ϕ]) U [ϕ]D[ϕ]
⟨∫
⟩
⃗
⃗
= ⟨∂t U ⟩ +
d⃗xVt (⃗x) · δϕ(⃗
,
(3.205)
⃗ x) U
132
where we have used
]
∫
∫ [∫
⃗ x)[ϕ] U [ϕ]D[ϕ]
(∂t Pt [ϕ]) U [ϕ]D[ϕ] = −
d⃗x ⃗δϕ(⃗
⃗ x) · Jt (⃗
]
⟨∫
⟩
∫ [∫
⃗
⃗
⃗
⃗
=
d⃗xJt (⃗x)[ϕ] · δϕ(⃗
d⃗xVt (⃗x) · δϕ(⃗
⃗ x) U [ϕ] D[ϕ] =
⃗ x) U (3.206)
which is a result of the continuity equation, integration by parts and the definition
J⃗t = Pt V⃗t . We have also suppressed the functional dependence [ϕ] in the ensemble
average to simplify notations, but it is better to keep in mind it is still there. The
first term in the last equation of Eq. (3.205) is due to the time dependence in the
stationary distribution functional Ps [ϕ], which comes from {λ(⃗x, t)} representing
space-time dependent external conditions. Thus the first term is identified as the
external driving entropic power for spatially inhomogeneous systems:
∑∫
⟨
⟩
d⃗x δλi (⃗x) U [∂t λi (⃗x)],
(3.207)
Ṡed = ⟨∂t U ⟩ =
i
where we have used the chain rule for functional derivatives [50, 158] and δλi (⃗x) U
is the short notation for δU/δλi (⃗x). The second term (with a negative sign) in
Eq. (3.205) is identified as the rate of excess entropy flow from the system to the
environment for spatially inhomogeneous systems:
⟨∫
⟩
⟩
∑ ⟨∫
(m)
d⃗xV⃗t (⃗x) · ⃗δϕ(⃗
d⃗xV⃗t (⃗x) · ⃗δϕ(⃗
Ṡex = −
=−
, (3.208)
⃗ x) U
⃗ x) U
m
where the summand of the last equation with the negative sign before it can be
(m)
defined as Ṡex , the excess entropy flow rate of mechanism m. Thus we have the
following cross entropy balance equation for spatially inhomogeneous systems:
U̇ = Ṡed − Ṡex .
(3.209)
The time derivative of the transient entropy is given by:
⟨∫
⟩
∫
Ṡ = − (∂t Pt [ϕ]) ln Pt [ϕ]D[ϕ] =
d⃗xV⃗t (⃗x) · ⃗δϕ(⃗
⃗ x) S , (3.210)
where we have used results similar to Eq. (3.206). Using Eq. (3.200) for S, we
133
further have:
Ṡ =
∑ ⟨∫
⟩
(m)
d⃗xV⃗t (⃗x)
∑ ⟨∫ ∫
· ⃗δϕ(⃗
⃗ x) S
m
=
∑ ⟨∫ ∫
(m)
d⃗xd⃗x ′ V⃗t (⃗x)
m
−
(m)
d⃗xd⃗x ′ V⃗t (⃗x)
=
∑ ⟨∫
⟩
(m)
d⃗xV⃗t (⃗x)
·Π
(m)
· ⃗δϕ(⃗
⃗ x) S
⟩
]−1
(m)
′
(⃗x, ⃗x , t)
· V⃗t (⃗x )
m
[
(m)
[
(m)
· D
· D
′
]−1
· F⃗
(⃗x, ⃗x , t)
′
′(m)
⟩
(⃗x , t) . (3.211)
′
m
The first term of the last equation is always nonnegative. It is identified as the
entropy production rate for the spatially inhomogeneous system:
⟩
∑ ⟨∫ ∫
[ (m)
]−1
(m)
′ ⃗ (m)
′
′
· V⃗t (⃗x ) ,
Ṡpd =
d⃗xd⃗x Vt (⃗x) · D (⃗x, ⃗x , t)
(3.212)
m
(m)
where the summand is defined as Ṡpd , the entropy production rate of mechanis[
]−1
m m. Since D (m) (⃗x, ⃗x ′ , t)
[ϕ] is positive definite and Pt [ϕ] is assumed to be
(m)
positive, the necessary and sufficient condition for Ṡpd ≡ 0 is V⃗t (⃗x)[ϕ] ≡ 0
(m = 1, 2, ...), that is, the transient flux velocity field of each mechanism is identically 0. The second term is identified as the entropy flow rate (from the system
to the environment) for spatially inhomogeneous systems:
⟩
∑ ⟨∫ ∫
[ (m)
]−1
′ ⃗ (m)
′
′(m)
′
⃗
d⃗xd⃗x Vt (⃗x) · D (⃗x, ⃗x , t)
·F
(⃗x , t) , (3.213)
Ṡf l =
m
(m)
with the summand defined as Ṡf l , the entropy flow rate of mechanism m. Thus
we have the transient entropy balance equation for spatially inhomogeneous systems:
Ṡ = Ṡpd − Ṡf l .
(3.214)
The time derivative of the relative entropy is given by:
⟨∫
⟩
⃗
Ȧ = U̇ − Ṡ = ⟨∂t U ⟩ +
d⃗xV⃗t (⃗x) · δϕ(⃗
⃗ x) A ,
(3.215)
where we have used Eqs. (3.205) and (3.210) and the definition A = U − S. The
first term in the last equation of Eq. (3.215) is the external driving entropic power
Ṡed . The second term with a negative sign is identified as the nonadiabatic entropy
134
production rate, which has multiple alternative expressions, several of which are
explicitly nonnegative:
⟩
⟨∫ ∫
−1 ⃗
′⃗
′
′
Ṡna =
d⃗xd⃗x Vt (⃗x) · [D(⃗x, ⃗x , t)] · Vr (⃗x )
⟩
∑ ⟨∫ ∫
[ (m)
]−1
′
′ ⃗ (m)
(m)
′
⃗
=
d⃗xd⃗x Vt (⃗x) · D (⃗x, ⃗x , t)
· Vr (⃗x )
⟨∫
m
= −
⟨∫ ∫
⟩
⟩
∑ ⟨∫
(m)
⃗
⃗
⃗
⃗
d⃗xVt (⃗x) · δϕ(⃗
=−
d⃗xVt (⃗x) · δϕ(⃗
⃗ x) A
⃗ x) A
⟩
m
′⃗
′
−1
· V⃗r (⃗x ′ )
d⃗xd⃗x Vr (⃗x) · [D(⃗x, ⃗x , t)]
⟩
∑ ⟨∫ ∫
[ (m)
]−1
′ ⃗ (m)
′
(m)
′
⃗
d⃗xd⃗x Vr (⃗x) · D (⃗x, ⃗x , t)
· Vr (⃗x )
=
=
⟨∫
m
= −
⟩
⟩
∑ ⟨∫
(m)
d⃗xV⃗r (⃗x) · ⃗δϕ(⃗
d⃗xV⃗r (⃗x) · ⃗δϕ(⃗
=−
⃗ x) A
⃗ x) A
⟨∫ ∫
m
[
]
[
]⟩
′ ⃗
′
d⃗xd⃗x δϕ(⃗
=
x, ⃗x , t) · ⃗δϕ(⃗
⃗ x) A · D(⃗
⃗ x ′)A
[
]
[
]⟩
∑ ⟨∫ ∫
′ ⃗
(m)
′
d⃗xd⃗x δϕ(⃗
=
(⃗x, ⃗x , t) · ⃗δϕ(⃗
.
⃗ x) A · D
⃗ x ′)A
(3.216)
m
⟨∫
⟩
These expressions can be obtained using the property
d⃗xV⃗s (⃗x) · ⃗δϕ(⃗
⃗ x) A = 0
(due to the stationary functional Fokker-Planck equation) and Eq. (3.200) for A.
The summands in the 6th, 8th and 10th expressions in Eq. (3.216), which are
(m)
equal to each other and non-negative, can be defined as Ṡna , the nonadiabatic
entropy production rate of mechanism m. The necessary and sufficient condition
(m)
for Ṡna ≡ 0 is V⃗r (⃗x)[ϕ] ≡ 0, which is also equivalent to V⃗r (⃗x)[ϕ] ≡ 0 for all
m. Then Eq. (3.215) becomes the relative entropy balance equation for spatially
inhomogeneous systems:
Ȧ = Ṡed − Ṡna .
(3.217)
From Ṡf l in Eq. (3.213), Ṡex in Eq. (3.208) and the entropy flow decomposition
equation:
Ṡf l = Ṡad + Ṡex ,
(3.218)
the adiabatic entropy production rate Ṡad for spatially inhomogeneous systems is
135
given by:
Ṡad =
∑ ⟨∫ ∫
∑ ⟨∫ ∫
(m)
d⃗xd⃗x ′ V⃗t (⃗x)
⟩
[ (m)
]−1
′
(m)
′
· D (⃗x, ⃗x , t)
· V⃗s (⃗x )
d⃗xd⃗x ′ V⃗s(m) (⃗x)
⟩
[ (m)
]−1
′
′
(m)
· D (⃗x, ⃗x , t)
· V⃗s (⃗x ) (3.219)
m
=
m
where in deriving the⟩second line we have used Eq. (3.200) for A and the property
⟨
∫
d⃗xV⃗s (⃗x) · ⃗δ⃗ A = 0. The second expression of Ṡad in Eq. (3.219) is manϕ(⃗
x)
ifestly nonnegative. Its summand, which is also nonnegative, can be defined as
(m)
Ṡad , the adiabatic entropy production rate of mechanism m. The necessary and
(m)
sufficient condition for Ṡad ≡ 0 is V⃗s (⃗x)[ϕ] ≡ 0 for all m.
Then we can also verify that the following entropy production decomposition
equation holds for spatially inhomogeneous systems:
Ṡpd = Ṡad + Ṡna ,
(3.220)
where the adiabatic entropy production rate Ṡad and nonadiabatic entropy production rate Ṡna are individually nonnegative; together they ensure the total entropy
production rate Ṡpd is also nonnegative. The adiabatic entropy production Ṡad
(m)
is generated by non-zero stationary flux velocity field V⃗s (⃗x)[ϕ] (m = 1, 2, ...)
which characterizes detailed balance breaking in the steady state of spatially inhomogeneous systems by mechanism m, while the nonadiabatic entropy production
(m)
Ṡna is generated by non-zero relative flux velocity field V⃗r (⃗x)[ϕ] (m = 1, 2, ...)
which characterizes the irreversible relaxational process from the transient state
to the steady state of spatially inhomogeneous systems contributed by mechanism
m. These two forms of non-equilibrium irreversibility together constitute the total
irreversibility generated by the spatially inhomogeneous system indicated by nonzero total entropy production Ṡpd , which is characterized by non-zero transient
(m)
flux velocity field V⃗t (⃗x)[ϕ] (m = 1, 2, ...).
In accord with the mixing entropy production rate for mechanism m introduced in Eq. (3.103), we can define its counterpart for spatially inhomogeneous
systems:
⟨∫ ∫
⟩
[ (m)
]−1
(m)
′ ⃗ (m)
′
(m)
′
⃗
d⃗xd⃗x Vr (⃗x) · D (⃗x, ⃗x , t)
· Vs (⃗x )
Ṡmix =
⟩
⟨∫
(m)
⃗
⃗
=
d⃗xVs (⃗x) · δϕ(⃗
(3.221)
⃗ x) A ,
136
∑
(m)
which has the property m Ṡmix = 0. Then we also have the decomposition
equations of the entropy production and entropy flow for each mechanism m for
spatially inhomogeneous systems, which has the same form of Eqs. (3.105) and
(3.106) for spatially homogeneous systems:
(m)
(m)
(m)
(m)
+ Ṡmix .
Ṡf l = Ṡad + Ṡex
(m)
(m)
(m)
(m)
Ṡpd = Ṡad + Ṡna
+ 2Ṡmix ,
(3.222)
(3.223)
(m)
When summed over m, Ṡmix does not appear in Eqs. (3.218) and (3.220).
Therefore, for spatially inhomogeneous systems described by functional Langevin
and Fokker-Planck equations with multiple state transition mechanisms, we have
also constructed the non-equilibrium thermodynamic equations and the expressions of various thermodynamic functions in these equations, given in (3.205)(3.223). These equations are generalizations of those for spatially homogeneous
systems given in Eqs. (3.88)-(3.106). For completeness and simplicity, in the following we also list the set of non-equilibrium thermodynamic equations and the
expressions of the thermodynamic quantities for spatially inhomogeneous systems with one component in one dimensional physical space and with only one state
transition mechanism:


U̇
= Ṡed − Ṡex






= Ṡpd − Ṡf l
 Ṡ
(3.224)
Ȧ = Ṡed − Ṡna




Ṡpd = Ṡad + Ṡna




Ṡf l = Ṡad + Ṡex
with the definitions U = ⟨− ln Ps ⟩, S = ⟨− ln Pt ⟩, A = ⟨ln(Pt /Ps )⟩
expressions:
⟩
⟨∫
dxVt (x)δϕ(x) U
U̇ = ⟨∂t U ⟩ +
⟨∫
⟩
Ṡ =
dxVt (x)δϕ(x) S
⟨∫
⟩
Ȧ = ⟨∂t U ⟩ +
dxVt (x)δϕ(x) A
137
and the
(3.225)
(3.226)
(3.227)
Ṡed = ⟨∂t U ⟩ =
⟨∫
Ṡex = −
⟨∫ ∫
Ṡpd =
⟨∫ ∫
Ṡf l =
⟨∫ ∫
Ṡad =
⟨∫ ∫
=
⟨∫ ∫
Ṡna =
⟨∫ ∫
=
⟨∫ ∫
=
∑∫
⟨
⟩
dx δλi (x) U [∂t λi (x)]
(3.228)
⟩
i
dxVt (x)δϕ(x) U
(3.229)
⟩
′
′
−1
′
dxdx Vt (x)[D(x, x , t)] Vt (x )
(3.230)
⟩
(3.231)
dxdx Vt (x)[D(x, x , t)] F (x , t)
⟩
′
′
−1
′
dxdx Vt (x)[D(x, x , t)] Vs (x )
⟩
′
′
−1
′
(3.232)
dxdx Vs (x)[D(x, x , t)] Vs (x )
⟩
⟨∫
⟩
′
′
−1
′
dxdx Vt (x)[D(x, x , t)] Vr (x ) = −
dxVt (x)δϕ(x) A
⟩
⟩
⟨∫
′
′
−1
′
dxVr (x)δϕ(x) A
dxdx Vr (x)[D(x, x , t)] Vr (x ) = −
⟩
[
]
[
]
′
′
dxdx δϕ(x) A D(x, x , t) δϕ(x′ ) A
(3.233)
′
′
−1
′
′
On the one hand, Eqs. (3.224)-(3.233) for spatially inhomogeneous systems with
one component in one dimension with one mechanism are generalizations of Eqs. (3.62)-(3.71) for spatially homogeneous systems with one mechanism. On the
other hand, they are special cases of Eqs. (3.205)-(3.223) for spatially inhomogeneous systems with multiple components in multiple dimensions with multiple
mechanisms. The equations for cases in between (e.g., spatially inhomogeneous
systems with multiple components in multiple dimensions with one mechanism)
can be easily deduced from Eqs. (3.205)-(3.223) or extended from Eqs. (3.224)(3.233).
We discuss some particular points in these results. The ensemble average in
Eqs. (3.225)-(3.233), when spelled out, is a functional integral in the field configuration space. If we assume the order of the functional integral and the spatial
integral can be interchanged, they can be expressed
as spatial integrals. For ex∫∫
dxdx′ Ṡpd (x, x′ , t), with
ample, Eq. (3.230) can be written as Ṡpd =
∫
′
Ṡpd (x, x , t) = D[ϕ]Pt [ϕ]Vt (x)[ϕ][D(x, x′ , t)]−1 [ϕ]Vt (x′ )[ϕ],
(3.234)
138
which may be termed the entropy production (spatial) correlation function. The
total entropy production rate Ṡpd is thus expressed as a double spatial integral of
the entropy production correlation function. In the special case when fluctuations
are spatially uncorrelated, i.e., D(x, x′ , t) ∝ δ(x − x′ ), entropy production is also
spatially uncorrelated according to∫ Eq. (3.234), which has the form Ṡpd (x, x′ , t) =
Ṡpd (x, t)δ(x − x′ ). Thus Ṡpd = dxṠpd (x, t), where Ṡpd (x, t) can be identified
as the local entropy production rate in the physical
space. In]general cases, one
[∫
∫
∫
′
may play the following trick, Ṡpd = dx dx Ṡpd (x, x′ , t) ≡ dxṠpd (x, t),
with Ṡpd (x, t) being the effective local entropy production rate. Yet this is done
at the cost of coarse-graining the detailed information of spatial correlations of
entropy production. Moreover, according to Eq. (3.234), the entropy production
correlation function Ṡpd (x, x′ , t), thus also the effective local entropy production
rate Ṡpd (x, t), is expressed as a functional integral in the field configuration space.
This means they are global quantities in the field configuration space. (Thus ‘local’ in the physical space does not imply ‘local’ in the field configuration space
either.) The treatment in classical irreversible thermodynamics based on the local equilibrium assumption [94] is different, as the total entropy production rate
is generically expressed as a single spatial integral of the local entropy production rate, without considering the spatial correlation of entropy production and its
underlying statistical origin. These distinctions from classical irreversible thermodynamics imply that our treatment and results are not limited by the local equilibrium assumption. Further, we note that from Eqs. (3.225)-(3.233), the total
entropy production rate Ṡpd , the adiabatic entropy production rate Ṡad and the total entropy flow rate Ṡf l have the form of a double spatial integral of correlation
functions, while the rest can be expressed as a single spatial integral of local densities in the physical space, determined by a functional integral (thus global) in
the field configuration space.
The non-equilibrium thermodynamic formalism for spatially inhomogeneous
systems developed so far from Sec. 3.3.2 to Sec. 3.3.4 are presented in the space
configuration representation. In Appendix D we present the major results in these
sections in the abstract representation and explain how to switch between different
representations; we also mention how to recover results of spatially homogeneous
systems from those of spatially inhomogeneous systems.
139
3.3.5
Ornstein-Uhlenbeck Processes of Spatially Inhomogeneous
Systems
We apply the formalism developed so far for general spatially inhomogeneous
systems to more specific systems that are analytically tractable (to a certain extent). For spatially homogeneous systems, we have studied the Ornstein-Uhlenbeck
process (OU process). Here we investigate its counterpart for spatially inhomogeneous systems. To fully utilize the available results of the OU process for spatially
homogeneous systems, we proceed as follows. First we discretize the physical space into spatial cells and work in the discrete space representation, whereby the
methods and results for the OU process of spatially homogeneous systems can be
ported in. Then we take the continuum limit and return to the continuous space
representation. We also present the expressions of the thermodynamic quantities
for one state transition mechanism.
Discrete Space Representation
We study the stochastic dynamics and potential-flux field landscape of OU
processes for spatially inhomogeneous systems in the discrete space representation. The state of the system is described by the set of variables {ϕaλ } defined in
Eq. (3.167), which we assume follows an OU process. In the Langevin dynamics,
this means the deterministic force is linear in the state variables {ϕaλ }, while the
stochastic force is independent of the state variables {ϕaλ }. Thus the Langevin
equation has the following form (see Eq. (3.107) for spatially homogeneous systems):
d a
ab b
ϕ = −γλµ
ϕµ + ξλa (t),
(3.235)
dt λ
where we have used Einstein summation convention with repeated indexes in a
term summed over (e.g., both b and µ are summed over in the above equation).
The Gaussian white noise ξλa (t) has the following statistical property:
⟨ξλa (t)⟩ = 0,
ab
⟨ξλa (t)ξµb (t′ )⟩ = 2Dλµ
δ(t − t′ ).
(3.236)
ab
ab
γλµ
and Dλµ
do not depend on {ϕaλ }, but they can be time-dependent (e.g., through
ab
depending on external control parameters). We also require [γλµ
] to be invertible
ab
and [Dλµ ] to be symmetric and positive definite, when seen as a matrix with a
double row index a λ and a double column index b µ. The corresponding Fokker-
140
Planck equation, as the counterpart of Eq. (3.109), is then:
( b
)
∂2
∂
ab ∂
ab
Pt ({ϕ}) = γλµ
ϕ
P
({ϕ})
+
D
Pt ({ϕ}),
t
λµ
∂t
∂ϕaλ µ
∂ϕaλ ∂ϕbµ
(3.237)
where we have used the short notation {ϕ} for the collection of state variables.
We consider the Gaussian solutions of the above Fokker-Planck equation:
{
}
[ −1 ]ab b
1
1 a
a
b
Pt ({ϕ}) = √
exp − (ϕλ − Kλ ) σ λµ (ϕµ − Kµ ) , (3.238)
2
ab
det([2πσλµ
])
ab
ab
where [2πσλµ
] is the matrix whose entries are given by 2πσλµ
. The means Kλa and
ab
are determined by the following equations (see Eqs. (3.111)
the covariances σλµ
and (3.112)):
ab
K̇λa = −γλµ
Kµb ,
ab
ac cb
bc ca
ab
σ̇λµ
= −γλν
σνµ − γµν
σνλ + 2Dλµ
,
(3.239)
The instantaneous stationary distribution of Eq. (3.237) is a Gaussian distribution
with mean zero:
{
}
1
1 a [ −1 ]ab b
Ps ({ϕ}) = √
exp − ϕλ σ
e λµ ϕµ ,
(3.240)
2
det([2πe
σ ab ])
λµ
where the instantaneous stationary covariance matrix σ
e is the instantaneous stationary solution of Eq. (3.239), satisfying:
ac cb
bc ca
ab
γλν
σ
eνµ + γµν
σ
eνλ = 2Dλµ
.
(3.241)
With the transient and stationary distributions given by Eqs. (3.238) and (3.240),
the expressions of the stationary, transient and relative potential landscapes can
be obtained using the definitions U = − ln Ps , S = − ln Pt and A = U − S (see
Eqs. (3.115)-(3.117)):
)
1 a [ −1 ]ab b 1 (
ab
U =
ϕλ σ
e λµ ϕµ + tr ln[2πe
σλµ
]
(3.242)
2
2
]ab
[
)
1 (
1 a
ab
]
S =
(ϕλ − Kλa ) σ −1 λµ (ϕbµ − Kµb ) + tr ln[2πσλµ
(3.243)
2
2
[
]ab
1 a [ −1 ]ab b 1 a
A =
ϕλ σ
e λµ ϕµ − (ϕλ − Kλa ) σ −1 λµ (ϕbµ − Kµb )
2
2
)
1 (
ab
ab
+ tr ln[e
]
] − ln[σλµ
σλµ
(3.244)
2
141
Their gradients are given by (see Eqs. (3.118)-(3.120)):
[ −1 ]ab b
∂U
=
σ
e λµ ϕµ
∂ϕaλ
[ −1 ]ab b
∂S
=
σ λµ (ϕµ − Kµb )
∂ϕaλ
[ −1 ]ab b [ −1 ]ab b
∂A
=
σ
e λµ ϕµ − σ λµ (ϕµ − Kµb )
∂ϕaλ
(3.245)
(3.246)
(3.247)
We can further derive the stationary, transient and relative flux velocities for the
system (see Eqs. (3.121)-(3.123)) using the dynamical decomposition equations:
[ −1 ]bc c
∂U
ab b
ab
+
D
σ
e µν ϕν
ϕ
=
−γ
(3.248)
λµ
µ
λµ
∂ϕbµ
[ −1 ]bc c
a
ab ∂S
ab b
ab
= F ′ λ + Dλµ
=
−γ
ϕ
+
D
σ µν (ϕν − Kνc ) (3.249)
λµ
µ
λµ
b
∂ϕµ
[ −1 ]bc c
[ −1 ]bc c
ab ∂A
ab
ab
= −Dλµ
= Dλµ
σ µν (ϕν − Kνc ) − Dλµ
σ
e µν ϕν(3.250)
b
∂ϕµ
ab
(Vs )aλ = F ′ λ + Dλµ
a
(Vt )aλ
(Vr )aλ
Continuous Space Representation
We transform the above results in the discrete space representation into the
continuous space representation by taking the continuum limit. There is a direct correspondence between the notations in the discrete space representation and
those in the continuous space representation. A brief dictionary to translate notations in one representation to the other is given below [50, 51, 158]:
ϕaλ ⇐⇒ ϕa (⃗x),
Fλa ({ϕ})
∑
λ
∂
∂ϕaλ
⇐⇒ F (⃗x)[ϕ],
∫ d⃗x
⇐⇒
,
∆V
δ
,
⇐⇒ ∆V
a
δϕ (⃗x)
a
P ({ϕ}) ⇐⇒ P [ϕ],
ab
Dλµ
({ϕ})
⇐⇒ Dab (⃗x, ⃗x ′ )[ϕ],
δλµ ⇐⇒ ∆V δ(⃗x − ⃗x ′ ),
∫
∏∫ a
dϕλ ⇐⇒
D[ϕ].
(3.251)
aλ
We first consider
limit of the deterministic force in Eq. (3.235),
∑ the
∑ continuum
a
ab b
Fλ ({ϕ}) = − µ b γλµ ϕµ , where we have spelled out the sum. In the continu∫
∑
um limit, Fλa ({ϕ}) → F a (⃗x)[ϕ], ϕbµ → ϕb (⃗x ′ ) and µ → d⃗x ′ /∆V . Plugging
∫
∑ ab
them in, we have F a (⃗x)[ϕ] = − d⃗x ′ b (γλµ
/∆V )ϕb (⃗x ′ ). To get rid of the
142
ab
volume element ∆V , we must have γλµ
→ γ ab (⃗x, ⃗x ′ )∆V in the continuum limit. Thus the deterministic force of the OU process for spatially inhomogeneous
systems has the form:
∫
∑
a
F (⃗x)[ϕ] = − d⃗x ′
γ ab (⃗x, ⃗x ′ )ϕb (⃗x ′ ).
(3.252)
b
This is an integral transform of (or an integral operator acting on) the vector
⃗ x) in the Hilbert state space, returning another vector field F⃗ (⃗x) in the
field ϕ(⃗
Hilbert state space, with the ab entry of the matrix-valued integral kernel given by
−γ ab (⃗x, ⃗x ′ ). If the kernel function is allowed to be generalized functions (e.g., the
Dirac delta function δ(⃗x − ⃗x ′ ) and its derivatives), which we assume so, then the
integral operator in Eq. (3.252) can also represent differential operators [15, 162].
In the most general sense, Eq. (3.252) is simply a statement that the deterministic
force field F⃗ (⃗x)[ϕ] is a linear map from the Hilbert state space into the Hilbert state space. Also, in the continuum limit the Gaussian white-noise stochastic force
ξλa (t) in Eq. (3.235) becomes ξ a (⃗x, t).
Thus in the continuum limit Eq. (3.235) becomes a functional Langevin equation, describing the OU process of spatially inhomogeneous systems :
∫
∑
∂ a
γ ab (⃗x, ⃗x ′ )ϕb (⃗x ′ , t) + ξ a (⃗x, t),
(3.253)
ϕ (⃗x, t) = − d⃗x ′
∂t
b
where ξ a (⃗x, t) has the following statistical property as the continuum limit of Eq. (3.236):
⟨ξ a (⃗x, t)⟩ = 0,
⟨ξ a (⃗x, t)ξ b (⃗x ′ , t′ )⟩ = 2Dab (⃗x, ⃗x ′ )δ(t − t′ ).
(3.254)
Equation (3.253) is the white-noise form of the functional Langevin equation; it
can also be reformulated into the more rigorous form in Eq. (3.176) [43]. We
note that this functional Langevin equation has an important subclass, namely linear stochastic partial differential equations with additive noise. γ ab (⃗x, ⃗x ′ )
⃗ x), but they are aland Dab (⃗x, ⃗x ′ ) do not depend on the state of the system ϕ(⃗
lowed to be time-dependent. We require them to be invertible in the sense of
Eq. (3.186) when seen as matrices with the row index a, ⃗x and the column index
b, ⃗x ′ . Dab (⃗x, ⃗x ′ ) is symmetric in the sense that Dab (⃗x, ⃗x ′ ) = Dba (⃗x ′ , ⃗x) and positive definite as defined in Eq. (3.184). The condition for the existence and uniqueness of the solution of Eq. (3.253) and even more general functional Langevin
equations on Hilbert spaces (Eq. (3.176)) has already been studied [43, 163].
143
The corresponding functional Fokker-Planck equation is the continuum limit
of Eq. (3.237):
∫∫
∑
)
δ ( b ′
∂
Pt [ϕ] =
d⃗xd⃗x ′
γ ab (⃗x, ⃗x ′ ) a
ϕ (⃗x )Pt [ϕ]
∂t
δϕ (⃗x)
ab
∫∫
∑
δ2
+
d⃗xd⃗x ′
P [ϕ]. (3.255)
Dab (⃗x, ⃗x ′ ) a
b (⃗
′) t
δϕ
(⃗
x
)δϕ
x
ab
Accordingly, the Gaussian solution in Eq. (3.238) becomes a Gaussian distribution
functional:
{
∫∫
∑
1
1
Pt [ϕ] = √
exp −
d⃗xd⃗x ′
[ϕa (⃗x) − K a (⃗x)]
ab
′
2
det([2πσ (⃗x, ⃗x )])
ab
[ −1 ]ab
[ b ′
]}
′
b
′
(3.256)
× σ
(⃗x, ⃗x ) ϕ (⃗x ) − K (⃗x ) .
[σ −1 ]ab (⃗x, ⃗x ′ ) as the inverse of σ ab (⃗x, ⃗x ′ ) is defined in the sense of Eq. (3.186).
In practice, it can be found by inverting the eigenvalues of σ ab (⃗x, ⃗x ′ ). The determinant in the denominator det([2πσ ab (⃗x, ⃗x ′ )]) is a functional determinant [158],
defined on an infinite dimensional Hilbert space. It can be formally defined as
the product of all the eigenvalues of [2πσ ab (⃗x, ⃗x ′ )]. Yet the problem is that the
functional determinant may be divergent or approach zero, which also makes the
probability distribution functional in Eq. (3.256) not well-defined. This is also a
situation often encountered in quantum field theory; the solution often proposed is
to demand that only the ratio of two functional determinants are meaningful [158].
Here we also adopt this solution. This also means only the ratio of two Gaussian
probability distribution functionals are meaningful; alternatively, these probability distribution functionals are only defined up to a multiplicative constant. A more
rigorous yet also more technically demanding treatment is to resort to the Gaussian
measure theory on an infinite dimensional Hilbert space [42]. The mean K a (⃗x)
and covariance σ ab (⃗x, ⃗x ′ ) of the transient Gaussian distribution in Eq. (3.256) are
determined by the following equations as the continuum limit of Eq. (3.239):
∫
∑
a
K̇ (⃗x) = − d⃗x ′
(3.257)
γ ab (⃗x, ⃗x ′ )K b (⃗x ′ ),
σ̇ ab (⃗x, ⃗x ′ ) = −
∫
b
∑[
]
d⃗x ′′
γ ac (⃗x, ⃗x ′′ )σ cb (⃗x ′′ , ⃗x ′ ) + γ bc (⃗x ′ , ⃗x ′′ )σ ca (⃗x ′′ , ⃗x)
c
+2D (⃗x, ⃗x ′ ).
ab
(3.258)
144
We assume that with the initial conditions of K a (⃗x) and σ ab (⃗x, ⃗x ′ ) given, the solutions of these two equations exist and are unique when γ ab (⃗x, ⃗x ′ ) and Dab (⃗x, ⃗x ′ )
satisfy certain conditions [43, 163]. The instantaneous stationary distribution in
the continuous space representation reads:
{
}
∑ a
1 ∫∫
′
−1 ab
′
b
′
exp −
d⃗xd⃗x
x) [e
σ ] (⃗x, ⃗x )ϕ (⃗x )
ab ϕ (⃗
2
√
Ps [ϕ] =
,
(3.259)
det([2πe
σ ab (⃗x, ⃗x ′ )])
where the instantaneous stationary covariance σ
eab (⃗x, ⃗x ′ ) satisfies the following
equation:
∫
∑[
]
d⃗x ′′
γ ac (⃗x, ⃗x ′′ )e
σ cb (⃗x ′′ , ⃗x ′ ) + γ bc (⃗x ′ , ⃗x ′′ )e
σ ca (⃗x ′′ , ⃗x) = 2Dab (⃗x, ⃗x ′ ).
c
(3.260)
These two equations are the continuous limit of Eqs. (3.240) and (3.241). We
assume the solution σ
ecb (⃗x, ⃗x ′ ) of Eq. (3.260) is unique when γ ab (⃗x, ⃗x ′ ) and
Dab (⃗x, ⃗x ′ ) satisfy appropriate conditions [43, 163].
The potential field landscapes U = − ln Ps , S = − ln Pt and A = ln (Pt /Ps )
are then given by
∫∫
∑
[ −1 ]ab
1
ϕa (⃗x) σ
e
(⃗x, ⃗x ′ )ϕb (⃗x ′ )
U [ϕ] =
d⃗xd⃗x ′
2
ab
)
1 (
+
tr ln[2πe
σ ab (⃗x, ⃗x ′ )]
(3.261)
2 ∫∫
∑
[
]ab
[
]
1
[ϕa (⃗x) − K a (⃗x)] σ −1 (⃗x, ⃗x ′ ) ϕb (⃗x ′ ) − K b (⃗x ′ )
d⃗xd⃗x ′
S[ϕ] =
2
ab
)
1 (
+
tr ln[2πσ ab (⃗x, ⃗x ′ )]
(3.262)
2 ∫∫
∑
[ −1 ]ab
1
d⃗xd⃗x ′
ϕa (⃗x) σ
e
(⃗x, ⃗x ′ )ϕb (⃗x ′ )
A[ϕ] =
2
ab
∫∫
∑
[
]ab
[
]
1
−
d⃗xd⃗x ′
[ϕa (⃗x) − K a (⃗x)] σ −1 (⃗x, ⃗x ′ ) ϕb (⃗x ′ ) − K b (⃗x ′ )
2
ab
)
1 (
tr ln[e
σ ab (⃗x, ⃗x ′ )] − ln[σ ab (⃗x, ⃗x ′ )] .
(3.263)
+
2
U and S both contain a term coming from a functional determinant, given that
ln det(B) = tr(ln B). Hence they may be divergent. Yet since U and S are only
145
defined up to a common additive constant, the divergence may be removed. The
last term in A comes from a ratio between two functional determinants, which may
cancel the divergence. Their functional derivatives, however, are all independent
of the functional determinants:
∫
∑[
]ab
δU
′
=
d⃗
x
σ
e−1 (⃗x, ⃗x ′ )ϕb (⃗x ′ )
(3.264)
a
δϕ (⃗x)
b
∫
∑[
]ab
[
]
δS
(3.265)
=
d⃗x ′
σ −1 (⃗x, ⃗x ′ ) ϕb (⃗x ′ ) − K b (⃗x ′ )
a
δϕ (⃗x)
b
∫
∑[
]ab
δA
=
d⃗x ′
σ
e−1 (⃗x, ⃗x ′ )ϕb (⃗x ′ )
a
δϕ (⃗x)
b
∫
∑[
]ab
[
]
−
d⃗x ′
σ −1 (⃗x, ⃗x ′ ) ϕb (⃗x ′ ) − K b (⃗x ′ )
(3.266)
b
The expressions of the stationary, transient and relative flux velocity fields can be
obtained from the dynamical decomposition equations in Eqs. (3.191)-(3.193):
∫
∑
a
γ ab (⃗x, ⃗x ′ )ϕb (⃗x ′ )
Vs (⃗x)[ϕ] = − d⃗x ′
∫∫
+
b
d⃗x ′ d⃗x ′′
∫
Vta (⃗x)[ϕ]
= −
d⃗x
′
∑
∑
[ −1 ]bc ′ ′′ c ′′
Dab (⃗x, ⃗x ′ ) σ
e
(⃗x , ⃗x )ϕ (⃗x )
bc
ab
′
b
′
γ (⃗x, ⃗x )ϕ (⃗x ) +
∫∫
d⃗x ′ d⃗x ′′
∑
(3.267)
Dab (⃗x, ⃗x ′ )
bc
b
[
]bc
× σ −1 (⃗x ′ , ⃗x ′′ )(ϕc (⃗x ′′ ) − K c (⃗x ′′ ))
(3.268)
∫∫
∑
[
]bc
Vra (⃗x)[ϕ] =
d⃗x ′ d⃗x ′′
Dab (⃗x, ⃗x ′ ) σ −1 (⃗x ′ , ⃗x ′′ )(ϕc (⃗x ′′ ) − K c (⃗x ′′ ))
∫∫
−
bc
d⃗x ′ d⃗x ′′
∑
[ −1 ]bc ′ ′′ c ′′
Dab (⃗x, ⃗x ′ ) σ
e
(⃗x , ⃗x )ϕ (⃗x ).
(3.269)
bc
Therefore, for OU processes of spatially inhomogeneous systems, we have derived the explicit expressions of the various terms in the dynamical decomposition equations (Eqs. (3.191)-(3.193)), with the (effective) force field given by
Eq. (3.252), the potential field landscapes given by Eq. (3.261)-(3.263), the functional gradient of the potential field landscapes given by Eqs. (3.264)-(3.266), and
the flux velocity fields given by Eqs. (3.267)-(3.269). These results can either be
146
obtained by taking the continuum limit of the results in the discrete space representation or by directly working in the continuous space representation using the
functional language from the start. These explicit expressions can be used to study
the global stability and dynamics of spatially inhomogeneous OU processes in the
potential-flux landscape framework using the methods presented in chapter 2.
These equations and expressions in the continuous space representation can
also be written compactly in the abstract representation, which are given in Appendix D. Here we mention a few. Equation (3.262) in the abstract representation reads S(ϕ) = (ϕ − K|σ̂ −1 |ϕ − K)/2 + tr(ln(2πσ̂))/2, Eq. (3.265) reads
|δϕ S(ϕ)) = σ̂ −1 |ϕ − K) and Eq. (3.268) reads |Vt (ϕ)) = −γ̂|ϕ) + D̂σ̂ −1 |ϕ − K).
Non-Equilibrium Thermodynamics for One State Transition Mechanism
We consider OU processes of spatially inhomogeneous system with one effective state transition mechanism. The thermodynamic quantities in the nonequilibrium thermodynamic equations can be calculated using those for one mechanism (extension of Eqs. (3.225)-(3.233) to multiple dimensional physical space),
by plugging in the specific expressions of the potential field landscapes and the
flux velocity fields for OU processes derived in (3.261)-(3.269). The results can
also be obtained by formally generalizing those already worked out for spatially
homogeneous systems in Eqs. (3.124)-(3.135). More specifically, if we replace
e , σ and ⃗µ⃗µ with their counterparts in spatially inhomogethe matrices I, γ, D, σ
ˆ γ̂, D̂, σ,
neous systems (linear operators on the Hilbert space), I,
ê σ̂ and |K)(K|,
then Eqs. (3.124)-(3.135) formally carry over to spatially inhomogeneous systems,
as long as the operations involved of finite-dimensional matrices are also legitimate for these linear operators on the infinite dimensional Hilbert space. This,
however, is not a trivial issue.
We first write down the formal results and then touch upon some technical
issues involved. The thermodynamic expressions in (3.124)-(3.135), when formally generalized to spatially inhomogeneous systems, are given by the following
expressions in the abstract representation:
)
(
1
1 −1
σ
ê (σ̂ + |K)(K|) + ln(2π σ)
ê
U = tr
(3.270)
2
2
(
)
1ˆ 1
S = tr
I + ln(2πσ̂)
(3.271)
2
2
( (
)
) 1
1 −1
ˆ
A = tr
σ
ê (σ̂ + |K)(K|) − I + (ln σ
ê − ln σ̂)
(3.272)
2
2
147
(
1
d −1
(σ̂ + |K)(K| − σ)
ê σ
ê
2
dt
)
−1
−1
−σ
ê D̂σ
ê (σ̂ + |K)(K|) + γ̂
(3.273)
(
)
tr σ̂ −1 D̂ − γ̂
(3.274)
(
−1
−1
d −1
1
ê − σ
ê D̂σ
(σ̂ + |K)(K| − σ)
ê σ
ê (σ̂ + |K)(K|)
tr
2
dt
)
−σ̂ −1 D̂ + 2γ̂
(3.275)
( (
) d −1 )
1
tr
σ̂ + |K)(K| − σ
ê
σ
ê
(3.276)
2
dt
( −1 −1
)
tr σ
ê D̂σ
(3.277)
ê (σ̂ + |K)(K|) − γ̂
(
)
tr γ̂ | D̂−1 γ̂ (σ̂ + |K)(K|) − γ̂
(3.278)
(
)
tr γ̂ | D̂−1 γ̂ (σ̂ + |K)(K|) + σ̂ −1 D̂ − 2γ̂
(3.279)
((
)
)
−1
−1
tr γ̂ | D̂−1 γ̂ − σ
ê D̂σ
ê
(σ̂ + |K)(K|)
(3.280)
( −1 −1
)
tr σ
ê D̂σ
ê (σ̂ + |K)(K|) + σ̂ −1 D̂ − 2γ̂
(3.281)
U̇ = tr
Ṡ =
Ȧ =
Ṡed =
Ṡex =
Ṡf l =
Ṡpd =
Ṡad =
Ṡna =
It is easy to verify that Eqs. (3.273)-(3.281) formally satisfy the set of non-equilibrium
thermodynamic equations (Eq. (3.224)). The expressions of the thermodynamic
quantities in Eqs. (3.273)-(3.281) are also given in the space configuration representation in Appendix E, written in the explicit functional language. But the
space configuration representation is not practical for actual calculations. These
thermodynamic quantities are most conveniently calculated by working in an orthonormal basis of the Hilbert space, such as the eigenfunctions of the operator γ̂
(if its eigenfunctions form an orthonormal basis), within which the linear operators are represented by infinite-dimensional matrices and the calculations can be
much simplified if they are diagonal.
Then we take a look at a technical issue in these expressions. On an infinite dimensional Hilbert space, not all linear operators have a well-defined trace
(e.g., the trace of the identity operator Iˆ diverges); those that do have a trace are
called trace-class operators [162]. Therefore, it is possible that the thermodynamic quantities in Eqs. (3.270)-(3.281) may diverge. At this moment they are
only formal expressions. For these expressions to be physically meaningful, they
have to produce finite results themselves or we have to use some techniques such
148
as renormalization to remove the infinities. Shortly in Sec. 3.3.6 we shall study
a specific example, within which the expressions in Eqs. (3.270)-(3.281) can be
worked out explicitly to be checked whether they are divergent or not. We speculate from that example that, in general, when the operators γ̂ and D̂ (they govern
the dynamics of the system and determine K, σ̂ and σ)
ê satisfy certain conditions [43, 163], the thermodynamic quantities in Eqs. (3.272)-(3.281) produce finite
results themselves, while the cross entropy U in Eq. (3.270) and the transient entropy S in Eq. (3.271) may still be divergent, but the divergence can be removed
by a simple ‘renormalization’ considering they are only defined up to a common
additive constant. We illustrate these ideas in the following example.
3.3.6
Spatial Stochastic Neuronal Model
We study a spatial stochastic neuronal model described by the linear stochastic
cable equation [43, 54, 55], which governs the stochastic evolution of the membrane potential of a spatially extended neuron. In the context of chemical reactions, the same equation can describe the stochastic degradation-diffusion process.
Mathematically, the linear stochastic cable equation is a particular case of the spatially inhomogeneous OU process just studied in Sec. 3.3.5. Thus it can serve
as a testing ground for the non-equilibrium thermodynamic framework we have
formulated. This equation has been solved previously from the perspective of a
stochastic partial differential equation (functional Langevin equation) [43]. Here
we solve it from the perspective of the functional Fokker-Planck equation. More
importantly, we use the solution to calculate the potential-flux field landscapes
and the expressions of the various non-equilibrium thermodynamic quantities for
a specific initial condition. Discussions as well as a picture illustrating the results
are also given.
Stochastic Dynamics
Let ϕ(x, t) represent the electric potential on the membrane of a neuron modeled as a one dimensional infinitely thin cylinder extended from x = 0 to x = π.
The linear stochastic cable equation (a functional Langevin equation) governing
the evolution of ϕ(x, t) reads [43]:
∂2
∂
ϕ(x, t) =
ϕ(x, t) − ϕ(x, t) + ξ(x, t),
∂t
∂x2
149
0 ≤ x ≤ π,
t > 0,
(3.282)
where the space-time Gaussian white noise has the statistical property:
⟨ξ(x, t)⟩ = 0,
⟨ξ(x, t)ξ(x′ , t′ )⟩ = δ(x − x′ )δ(t − t′ ).
(3.283)
On the right side of Eq. (3.282), the first term represents diffusion of the electric
potential over the neuron fiber; the second term represents ions leaking across
the membrane; the third term represents space-time random disturbance to the
membrane potential [43]. We remark that Eq. (3.282) can also be interpreted in
the context of chemical reactions, where ϕ(x, t) represents the local concentration
of a chemical species and the three terms on the right side represent, respectively,
diffusion of the chemical species across space, degradation of the chemical species
and space-time random fluctuations of the local concentration. In the following we
work in the context of the neuron model, but the results also apply to the stochastic
chemical degradation-diffusion process. The boundary condition is assumed to be
Neumann:
∂
∂
ϕ(0, t) =
ϕ(π, t) = 0,
∀t > 0,
(3.284)
∂x
∂x
which means the neuron fiber is insulated at both ends all the time. The initial
condition is:
ϕ(x, 0) = f (x),
0 ≤ x ≤ π.
(3.285)
Equation (3.282) can be written in an operator form:
∂
ϕ(x, t) = −γ̂ϕ(x, t) + ξ(x, t),
∂t
0 ≤ x ≤ π,
t > 0,
(3.286)
where γ̂ is a differential operator given by:
γ̂ = 1 −
∂2
.
∂x2
(3.287)
To establish a connection with our formalism, we notice that this differential operator γ̂ also has the following representation as an integral operator:
[
]
∫
∂2
′
′
′
γ̂ = dx δ(x − x ) − 2 δ(x − x ) ∗,
(3.288)
∂x
with the integral kernel
γ(x, x′ ) = δ(x − x′ ) −
150
∂2
δ(x − x′ ).
∂x2
(3.289)
The action of the integral operator on ϕ(x′ , t) gives the same result as the differential operator 1 − ∂ 2 /∂x2 acting on ϕ(x, t). The eigenvalues and eigenfunctions of
the differential operator γ̂ has been worked out [43], which is also easy to verify.
The eigenfunctions are given by:
√
1
2
en (x) =
e0 (x) = √ ,
cos nx (n ≥ 1),
(3.290)
π
π
with the corresponding eigenvalues
γn = n2 + 1 (n ≥ 0).
(3.291)
These eigenfunctions are orthonormal and complete, satisfying Eq. (3.170) adapted to this particular case of one dimensional space:
∫ π
+∞
∑
dx en (x)em (x) = δmn ,
en (x)en (x ′ ) = δ(x − x ′ ).
(3.292)
0
n=0
Therefore {en (x)} form an orthonormal basis of the Hilbert space Ω = L2 ([0, π])
(square integrable functions on the interval [0, π]). Using Eq. (3.292), the integral
kernel γ(x, x′ ) can be decomposed using the eigenvalues and eigenfunctions:
′
γ(x, x ) =
+∞
∑
γn en (x)en (x′ ).
(3.293)
n=0
Then we consider the functional Fokker-Planck equation corresponding to the
functional Langevin equation (Eq. (3.282)). The statistical property of the fluctuation in Eq. (3.283) means the diffusion coefficient in the functional Fokker-Planck
equation is given by (see Eq. (3.254))
1
D(x, x′ ) = δ(x, x′ ).
(3.294)
2
Therefore, according to Eq. (3.255), the functional Fokker-Planck equation reads:
[(
)
]
(
)2
∫
∫
1
δ
∂
δ
∂2
dx
Pt [ϕ] = dx
ϕ(x) − 2 ϕ(x) Pt [ϕ] +
Pt [ϕ].
∂t
δϕ(x)
∂x
2
δϕ(x)
(3.295)
The transient Gaussian solution has the following form:
{
∫∫
1
1
exp −
Pt [ϕ] = √
dxdx ′ [ϕ(x) − K(x)]
′
2
det([2πσ(x, x )])
}
−1
′
′
′
× σ (x, x ) [ϕ(x ) − K(x )] ,
(3.296)
151
where the mean and covariance are now determined by (see Eqs. (3.257) and
(3.258)):
∫
K̇(x) = − dx ′ γ(x, x ′ )K(x ′ ),
(3.297)
∫
′
σ̇(x, x ) = − dx ′′ [γ(x, x ′′ )σ(x ′′ , x ′ ) + γ(x ′ , x ′′ )σ(x ′′ , x)]
+2D(x, x ′ ).
(3.298)
Yet we are not going to solve these two equations directly in their current forms.
Rather, we work in the orthonormal basis {en (x)}. In this basis, γ(x, x ′ ) is given
by Eq. (3.293). According to Eq. (3.292) the space-time diffusion coefficient in
Eq. (3.294) has the following form:
D(x, x ′ ) =
+∞
∑
1
n=0
2
en (x)en (x′ ),
(3.299)
which means in the basis {en (x)} the diffusion matrix is diagonal, with the matrix
elements given by Dmn = δmn /2 (see Eq. (3.174)). We also expand K(x) and
σ(x, x′ ) in this basis (see Eqs. (3.171)-(3.175)), which are given respectively by:
K(x) =
+∞
∑
Kn en (x),
′
σ(x, x ) =
n=0
+∞
∑
σmn em (x)en (x′ ),
(3.300)
m,n=0
where σmn = σnm due to the symmetry property σ(x, x′ ) = σ(x′ , x). Thus in the
basis {en (x)}, Eqs. (3.297) and (3.298) take on the following simple form:
K̇n = −γn Kn ,
σ̇mn = −(γm + γn )σmn + δmn .
(3.301)
(3.302)
The initial condition Eq. (3.285) is a definite state, which means the mean K(x)|t=0 =
ϕ(x, 0) = f (x) and the covariance σ(x, x′ )|t=0 = 0. In the basis {en (x)}, this
means
Kn (t = 0) = fn ,
σmn (t = 0) = 0,
(3.303)
∑+∞
where fn is the expansion coefficient in f (x) =
n=0 fn en (x), given by (see
Eq. (3.172))
∫
π
fn =
en (x)f (x)dx.
0
152
(3.304)
The solutions of Eqs. (3.301) and (3.302) with the initial condition Eq. (3.303) are
given by:
−γn t
Kn (t) = e
fn ,
1 − e−2γn t
σmn (t) =
δmn .
2γn
(3.305)
Plugging them back into Eq. (3.300), we thus have solved the mean and covariance
of the transient Gaussian distribution functional:
E[ϕ(x)]t = K(x, t) =
+∞
∑
e−γn t fn en (x),
n=0
+∞
∑
Cov[ϕ(x), ϕ(x′ )]t = σ(x, x′ , t) =
n=0
(3.306)
1 − e−2γn t
en (x)en (x′ ). (3.307)
2γn
The stationary solution of Eq. (3.305) can be obtained by taking the limit t →
+∞, giving:
e n = 0,
K
σ
emn =
δmn
.
2γn
(3.308)
Therefore the mean and covariance of the stationary Gaussian distribution functional Ps [ϕ] are, respectively, given by:
e
E[ϕ(x)]s = K(x)
= 0,
Cov[ϕ(x), ϕ(x′ )]s = σ
e(x, x′ ) =
(3.309)
+∞
∑
n=0
1
en (x)en (x′ ).
2γn
(3.310)
These results agree with those obtained in Ref. [43] by working with the functional
Langevin equation. Here we have obtained the same results using the functional
Fokker-Planck equation, without resorting to the technique of stochastic calculus
on the Hilbert space.
Potential-Flux Field landscape
The expressions of the potential field landscapes, their functional gradients and
the flux velocity fields of OU processes in the space configuration representation
are given by Eqs. (3.261)-(3.269). They can be calculated conveniently in the
basis
∫ π in this basis as ϕ(x) =
∑∞ of {en (x)}. The state of the system ϕ(x) is expanded
n=0 ϕn en (x), with the coefficient given by ϕn = 0 en (x)ϕ(x)dx. We leave
153
the details of calculation in Appendix F and list the results in the following. The
potential field landscapes are given by:
∞ [
∑
]
π
1
U [ϕ] =
(n +
+ ln 2
(3.311)
2
n
+
1
n=0
∞ [
)
)2 1 (
∑
(n2 + 1) (
−2(n2 +1)t
−(n2 +1)t
S[ϕ] =
ln
1
−
e
ϕ
−
f
e
+
n
n
2
1 − e−2(n2 +1)t
n=0
]
1
π
+ ln 2
(3.312)
2 n +1
∞ [
)2
∑
(n2 + 1) (
2
2
−(n2 +1)t
A[ϕ] =
(n + 1)ϕn −
ϕ n − fn e
1 − e−2(n2 +1)t
n=0
)]
1 (
−2(n2 +1)t
− ln 1 − e
(3.313)
2
2
1)ϕ2n
Their functional gradients are:
∞
∑
δU
=
2(n2 + 1)ϕn en (x)
δϕ(x)
n=0
(3.314)
∞
∑
δS
2(n2 + 1)
−(n2 +1)t
=
(ϕ
−
f
e
) en (x)
(3.315)
n
n
2
−2(n +1)t
δϕ(x)
1
−
e
n=0
]
∞ [
∑
δA
2(n2 + 1)
2
−(n2 +1)t
=
2(n + 1)ϕn −
(ϕn − fn e
)
δϕ(x)
1 − e−2(n2 +1)t
n=0
×en (x)
(3.316)
The flux velocity fields are calculated to be:
Vs (x)[ϕ] = 0
Vt (x)[ϕ] = Vr (x)[ϕ]
∞ [
∑
=
−(n2 + 1)ϕn +
n=0
(3.317)
]
(n2 + 1)
−(n2 +1)t
(ϕn − fn e
) en (x)(3.318)
1 − e−(n2 +1)t
In the expressions of
∑U∞[ϕ] and S[ϕ]2 in Eqs. (3.311) and (3.312), there is a common
divergent constant n=0 (ln[π/(n + 1)])/2 which can be removed, considering
that U [ϕ] and S[ϕ] are defined only up to a common additive constant. With
154
these expressions given explicitly, we can verify the field dynamical decomposition equations given in Eqs. (3.191)-(3.193). In our current case, the state of the
system ϕ(x) has only one component and the physical space is one dimensional. The (effective) driving force field F ′ (x)[ϕ] = F (x)[ϕ]. The diffusion matrix
field D(x, x′ ) = δ(x − x′ )/2. Therefore, the stationary, transient and relative
field decomposition equations in this particular case take on the following simplified form: F (x)[ϕ] = −δϕ(x) U/2 + Vs (x)[ϕ]; F (x)[ϕ] = −δϕ(x) S/2 + Vt (x)[ϕ];
Vr (x)[ϕ] = ∑
−δϕ(x) A/2. They can be verified easily by using the expression
2
F (x)[ϕ] = ∞
n=0 −(n + 1)ϕn en (x) and the expressions given in Eqs. (3.314)(3.318).
According to Eq. (3.317), the stationary flux velocity field Vs (x)[ϕ] is identically zero. This means there is no non-equilibrium irreversibility from detailed
balance breaking in the steady state. Therefore the adiabatic entropy production
rate Ṡad should be identically zero. The only contribution to entropy production is
the nonadiabatic entropy production Ṡna , indicated by non-zero relative flux velocity field Vr (x)[ϕ], which comes from the non-equilibrium irreversible process
of relaxing from the transient state to the equilibrium steady state, if the system is
initially not already in that state. Eventually, the system will reach the equilibrium steady state and the total entropy production should be zero. This can also be
seen from Eq. (3.318), Vt (x)[ϕ] = Vr (x)[ϕ] = 0 when t → +∞, which indicates
Ṡpd = Ṡna = 0 (t → +∞). These observations are confirmed by calculating the
non-equilibrium thermodynamic quantities directly in the following.
Non-Equilibrium Thermodynamics
Since this particular system is a special case of the spatially inhomogeneous
OU process, it also obeys the set of non-equilibrium thermodynamic equations in
Eq. (3.224). We calculate the expressions of the thermodynamic quantities using Eqs. (3.270)-(3.281), by working in the basis {en (x)}. The linear operators
ˆ γ̂, D̂, σ,
involved, I,
ê σ̂ and |K)(K|, are represented by infinite dimensional matrices in this basis. According to Eqs. (3.292), (3.293), (3.299), (3.306), (3.307)
and (3.310), the matrix elements of these linear operators in the basis {en (x)} are
given, respectively, by:
Imn = δmn ,
γmn = γn δmn ,
σ
emn = δmn /(2γn ),
Dmn = δmn /2,
σmn = δmn (1 − e−2γn t )/(2γn ),
Km Kn = e−(γm +γn )t fm fn ,
155
(3.319)
where γn = n2 + 1 and fn is given by Eq. (3.304) determined by the initial condition. For specificity and simplicity, we consider the particular initial condition
ϕ(x, 0) = f (x) ≡ 0 (thus fn = 0). That means the electric potential on the membrane is initially 0 everywhere; the evolution of the membrane potential is purely
initiated by stochastic fluctuations. With this initial condition fn = 0, the operator
|K)(K| becomes the zero operator as its matrix elements Km Kn are all zero.
First we have calculate the cross, transient and relative entropies. The cross
entropy is calculated using Eq. (3.270):
(
)
1 −1
1
U = tr
σ
ê (σ̂ + |K)(K|) + ln(2π σ)
ê
2
2
]
∞ [
∑
1 σnn + Kn2 1
=
+ ln(2πe
σnn )
2 σ
enn
2
n=0
[
]
∞
∑
1
π
−2(n2 +1)t
=
−e
+ 1 + ln 2
.
(3.320)
2
n
+
1
n=0
Using Eq. (3.271), we have the transient entropy given by:
(
)
1ˆ 1
S = tr
I + ln(2πσ̂)
2
2
[
]
∞
∑ 1
1
=
δnn + ln(2πσnn )
2
2
n=0
[
]
∞
∑1
π
−2(n2 +1)t
=
ln(1 − e
) + 1 + ln 2
.
2
n
+
1
n=0
(3.321)
The relative entropy is then:
A=U −S =
∞
]
∑
1 [ −2(n2 +1)t
2
−e
− ln(1 − e−2(n +1)t ) .
2
n=0
(3.322)
We examine whether these quantities are divergent or not. Define the summand in
the infinite series of the expression of U, S∑
and A in Eqs. (3.320)-(3.322)
as un (t),
∑∞
∞
s
(t)
and
A=
u
(t),
S
=
s∑
(t)
and
a
(t),
respectively.
Thus
U
=
n
n
n=0 n
n=0 n
∞
n=0 an (t). It is not difficult to check that for any t ∈ (0, +∞), un (t) → −∞
(n → +∞), sn (t) → −∞
∑∞ Therefore,
∑∞ (n → +∞) while an (t) → 0 (n → +∞).
the cross entropy U = n=0 un (t) and the transient entropy S = n=0 sn (t) are
both divergent for any t ∈ (0, +∞). The fact that an (t) → 0 (n → +∞) alone
156
∑
does not guarantee that A = ∞
n=0 an (t) is convergent. Yet further information
does. The inequality ln(1−x)+x ≤ 0 (x ≤ 1) shows that an (t) ≥ 0. And one can
2
also prove n2 an (t) = an (t)/(1/n
) → 0 (n → +∞) for
any t ∈ (0, +∞). The
∑∞
∑∞
2
convergence of the series n=1 1/n ensures that A = n=0 an (t) is convergent
and thus finite for any t ∈ (0, +∞). Therefore, in this case, the cross entropy U
and the transient entropy S are divergent, while the relative entropy A is finite.
We notice from Eqs. (3.320)-(3.322) that there is a common term cn = (1 +
ln[π/(n2 + 1)])/2 in un (t) and sn (t) preventing them from approaching zero in
the limit n → +∞, which is canceled in the expression of A. We exploit this
observation and the fact that U and S are only defined up to a common additive
constant, to implement the ‘renormalization’ of the cross entropy and the transient
entropy. We define the following constant in terms of an infinite series:
(
)
∞
∞
∑
∑
1
π
SC =
cn =
1 + ln 2
.
(3.323)
2
n +1
n=0
n=0
Since cn → −∞ (n → +∞), SC is a divergent constant. We subtract this common
divergent constant from U and S and define the ‘renormalized’ U and S. The
renormalized cross entropy is:
U
ren
= U − SC =
∞
∑
1
2
− e−2(n +1)t ,
2
n=0
(3.324)
and the renormalized transient entropy reads:
S
ren
= S − SC =
∞
∑
1
n=0
2
(
)
2
ln 1 − e−2(n +1)t .
(3.325)
The relative entropy remains the same: A = U − S = U ren − S ren . (One can
still add a finite constant to both U ren and S ren , which is simply a change in
the reference point of entropy.) Define the summand in the series expression of
ren
U ren and S ren as uren
n (t) and sn (t), respectively. One can prove that for any
2 ren
t ∈ (0, +∞), we have uren
n (t) ≤ 0 and n un (t) → 0 (n → +∞) as well
2 ren
ren
as sn (t) ≤ 0 and n sn (t) → 0 (n → +∞) . Using the same argument for
the convergence of A, we conclude that the renormalized cross entropy U ren and
transient entropy S ren are both finite and thus well-defined for any t ∈ (0, +∞).
In fact, U ren has an
expression, U ren = −e−2t [ϑ3 (0, e−2t ) + 1]/4, where
∑analytical
+∞ n2
ϑ3 (a, q) = 1 + 2 n=1 q cos(2na) is the third elliptic theta function. Thus we
have successfully removed the divergence of U and S through ‘renormalization’.
157
The rate of change of U, S and A can either be calculated directly from their
expressions (or the renormalized expressions) in Eqs. (3.320)-(3.325) or using
the formula in Eqs. (3.273)-(3.275). We take the former approach to check their
consistency with the rest of the thermodynamic quantities. The rate of change of
the cross entropy calculated using Eq. (3.320) or Eq. (3.324) is given by:
∞
∑
2
U̇ =
(n2 + 1)e−2(n +1)t .
(3.326)
n=0
Denote the summand of the infinite series as u̇n (t). One can prove that u̇n (t) ≥ 0
and n2 u̇n (t) → 0 (n → +∞) for any t ∈ (0, +∞). Therefore, U̇ is finite for any
t ∈ (0, +∞). According to Eq. (3.321) or Eq. (3.325), the rate of change of the
transient entropy is:
∞
∑
e−2(n +1)t
Ṡ =
(n + 1)
.
−2(n2 +1)t
1
−
e
n=0
2
2
(3.327)
Using the same argument for U̇, we can prove Ṡ is finite for any t ∈ (0, +∞).
The rate of change of the relative entropy is given by:
∞
∑
e−4(n +1)t
Ȧ = U̇ − Ṡ = −
(n + 1)
,
−2(n2 +1)t
1
−
e
n=0
2
2
(3.328)
which is also finite for any t ∈ (0, +∞) as U̇ and Ṡ are both finite.
Then we calculate the rest of the thermodynamic quantities. Since we did
not consider external driving in this particular case, the external driving entropic
power is zero:
Ṡed = 0,
(3.329)
which can also be seen by noticing that in Eq. (3.276) dσ/dt
ê
= 0 since σ
ê given by
Eq. (3.310) does not depend on time. Then we calculate the excess entropy flow
rate using Eq. (3.277):
( −1 −1
)
Ṡex = tr σ
ê D̂σ
ê (σ̂ + |K)(K|) − γ̂
]
∞ [
∑
Dnn
2
=
(σnn + Kn ) − γnn
2
σ
enn
n=0
= −
∞
∑
(n2 + 1)e−2(n
n=0
158
2 +1)t
.
(3.330)
We notice that this expression of Ṡex is simply the negative of U̇. This is consistent
with the cross entropy balance equation U̇ = Ṡed − Ṡex , within which Ṡed = 0
in this particular case. The total entropy flow rate calculated using Eq. (3.278) is
given by
(
)
Ṡf l = tr γ̂ | D̂−1 γ̂ (σ̂ + |K)(K|) − γ̂
]
∞ [ 2
∑
γnn
2
(σnn + Kn ) − γnn
=
Dnn
n=0
= −
∞
∑
(n2 + 1)e−2(n
2 +1)t
.
(3.331)
n=0
However, this is exactly Ṡex . In other words, in this particular case Ṡf l = Ṡex . The
expression in Eq. (3.331) shows that the entropy flow rate Ṡf l is negative for any
t ∈ (0, +∞) and goes to 0 as t → ∞. This means that during the entire process
of relaxing back to the equilibrium state, there is an entropy flow from the environment into the system (e.g., by absorbing heat from the environment), which
disappears eventually as the system reaches equilibrium. The fact that Ṡf l = Ṡex
also implies that the adiabatic entropy production rate Ṡad is identically zero, according to the entropy flow decomposition equation Ṡf l = Ṡad + Ṡex . Indeed, this
can be proven directly, using the expression in Eq. (3.280):
((
)
)
−1
−1
Ṡad = tr γ̂ | D̂−1 γ̂ − σ
ê D̂σ
ê
(σ̂ + |K)(K|)
)
∞ ( 2
∑
)
γnn
Dnn (
=
− 2
σnn + Kn2
Dnn
σ
enn
n=0
=
∞
∑
(
)
2γn2 − 2γn2 )(σnn + Kn2 = 0.
(3.332)
n=0
This agrees with our analysis based on the observation that Vs (x)[ϕ] = 0. This
means there is no detailed balance breaking in the steady state in this particular
case. The only source of irreversibility comes from the relaxation process due to
deviation from the steady state, which is characterized by nonadiabatic entropy
159
production rate. According to Eq. (3.281), it is given by:
( −1 −1
)
Ṡna = tr σ
ê D̂σ
ê (σ̂ + |K)(K|) + σ̂ −1 D̂ − 2γ̂
]
∞ [
∑
Dnn
Dnn
2
− 2γnn
=
(σnn + Kn ) +
2
σ
e
σ
nn
nn
n=0
∞
∑
e−4(n +1)t
=
(n + 1)
.
1 − e−2(n2 +1)t
n=0
2
2
(3.333)
According to this expression Ṡna is positive for any t ∈ (0, +∞) and goes to 0
as t → +∞ in the process of relaxing back to equilibrium. This also agrees with
our analysis that Vr (x)[ϕ] → 0 (t → +∞). Notice that the expression of Ṡna is
also the negative of Ȧ in Eq. (3.328). This is consistent with the relative entropy
balance equation Ȧ = Ṡed − Ṡna and the fact that Ṡed = 0 in this case. Since
Ṡad = 0, we expect that the total entropy production rate Ṡpd = Ṡna , according to
the entropy production decomposition equation Ṡpd = Ṡad + Ṡna . This is indeed
true:
(
)
Ṡpd = tr γ̂ | D̂−1 γ̂ (σ̂ + |K)(K|) + σ̂ −1 D̂ − 2γ̂
]
∞ [ 2
∑
γnn
Dnn
2
(σnn + Kn ) +
− 2γnn
=
D
σ
nn
nn
n=0
∞
∑
e−4(n +1)t
=
(n + 1)
.
1 − e−2(n2 +1)t
n=0
2
2
(3.334)
As with Ṡna , the total entropy production Ṡpd is also positive for any t ∈ (0, +∞)
and goes to zero as t → +∞. That means eventually the system reaches equilibrium where there is no irreversibility contributing to non-zero entropy production.
This also agrees with the fact that Vt (x)[ϕ] → 0 (t → +∞). We can also verify
the transient entropy balance equation Ṡ = Ṡpd − Ṡf l from the expressions in
Eqs. (3.327), (3.331) and (3.334).
Discussion
We have proven in this particular case that the thermodynamic expressions in
Eqs. (3.326)-(3.334) and the relative entropy A in Eq. (3.322), calculated from Eqs. (3.272)-(3.281), are all finite for any t ∈ (0, +∞). Although the cross entropy
160
U and the the transient entropy S calculated directly from Eqs. (3.270) and (3.271)
are divergent, they can be simply ‘renormalized’ by subtracting a common divergent constant that has no physical significance. This shows the thermodynamic
expressions in Eqs. (3.270)-(3.281) for spatially inhomogeneous OU processes are
not merely formal results. When applied to concrete systems, they can produce
sensible, quantitative, physical results.
This particular example is somewhat special in that Ṡed = Ṡad = 0, that is,
there is neither external driving nor detailed balance breaking in the steady state.
In this case the set of non-equilibrium thermodynamic equations (Eq. (3.224)) reduces to:
Ṡ = Ṡpd − Ṡf l
Ṡpd = Ṡna = −Ȧ
Ṡf l = Ṡex = −U̇
(3.335)
(3.336)
(3.337)
The first equation is the usual (transient) entropy balance equation, stating that
the system entropy is increased by entropy production within the system and decreased by entropy flow into the environment. The other two equations are specific to the situation Ṡed = Ṡad = 0. Equation (3.336) indicates that the nonequilibrium nature of this particular case is merely due to the non-equilibrium
relaxation process from the transient state to the equilibrium steady state (no adiabatic entropy production generated by detailed balance breaking in the steady
state) induced by the initial state preparation (no external-driving induced nonequilibrium relaxation). Equation (3.337) states that the total entropy flow is just
the excess entropy flow (no entropy flow associated with detailed balance breaking in the steady state) into the environment, which is also equal to the decrease
of cross entropy (no change of cross entropy caused by external driving).
We have also shown that throughout the entire relaxation process there is a
non-zero entropy production within the system (Ṡpd > 0) and an entropy flow
from the environment into the system (Ṡf l < 0). Thus the entropy of the system
increases during the entire relaxation process (Ṡ > 0). Eventually when the system has reached equilibrium, both the entropy production and entropy flow stop
and the system entropy remains constant. In fact, the entropy change, entropy
production and entropy flow in this process between any two finite time points
t = t1 and t = t2 (t2 > t1 ) can be calculated quantitatively. The change of
entropy S(t2 ) − S(t1 ) can be calculated using S(t) in Eq. (3.321) or its renormalized form in Eq. (3.325). According to Eq. (3.336), the entropy production
Spd (t1 → t2 ) = A(t1 ) − A(t2 ), with A(t) given by Eq. (3.322). According to
161
Eq. (3.337), the entropy flow Sf l (t1 → t2 ) = U(t1 ) − U(t2 ), with U(t) given by
Eq. (3.320) or its renormalized form in Eq. (3.324). This means in this process the
amount of entropy produced in the system, Spd (t1 → t2 ), is equal to the amount
of relative entropy decreased, −[A(t2 ) − A(t1 )]; the amount of entropy transferred into the system, −Sf l (t1 → t2 ), is equal to the amount of cross entropy
increased, U(t2 ) − U(t1 ); the system entropy change is a result of both factors:
S(t2 )−S(t1 ) = Spd (t1 → t2 )−Sf l (t1 → t2 ) = −[A(t2 )−A(t1 )]+[U(t2 )−U(t1 )].
This is clearly seen in Fig. 3.1. The relative entropy A decreases with time, inNon-Equilibrium Thermodynamic Functions
SHtL
0.6
UHtL
0.4
AHtL
0.2
0.5
1.0
1.5
t
- 0.2
- 0.4
- 0.6
Figure 3.1: Temporal profile of the system’s (renormalized) transient entropy S,
cross entropy U and relative entropy A in the process of relaxing to the equilibrium state in the spatial stochastic neuronal model.
dicating entropy production in the system. The cross entropy U increases with
time, indicating entropy flow into the system. The system entropy S therefore
increases with time, until the equilibrium state is reached in the limit t → +∞,
where both entropy production and entropy flow stop as A and U become constant. We note that if the temperature of the environment is constant, this process is
a non-equilibrium isothermal process. The cross entropy and the relative entropy
are then related, respectively, to the non-equilibrium internal energy Ue and free
energy Ae as in Eqs. (3.24) and (3.25). As mentioned before, in a non-equilibrium
isothermal process without external driving, the change of cross entropy and relative entropy are, respectively, proportional to the change of internal energy and
162
e The increase of the system
free energy: ∆U = T −1 ∆Ue and ∆A = T −1 ∆A.
entropy can thus also be understood in terms of the free energy dissipation generating entropy production in the system [115] and the increase of internal energy
due to heat flow into the system that also increases the system entropy. Another
perspective to look at the increase of system entropy is as follows. The system
was initially prepared in a definite (micro)state ϕ(x, 0) = 0 (this, however, is an
idealization; in reality, the prepared state still has some, though maybe small, uncertainty; this is also why the exact point t = 0 seems unphysical in Fig. 3.1 and
other analytical expressions). The final equilibrium state is a Gaussian probability
distribution of (micro)states (see Eqs. (3.309) and (3.310)). Therefore the system
state becomes more uncertain and random as it evolves to equilibrium, increasing
the system entropy.
The successful application of the non-equilibrium thermodynamic formalism
for spatially inhomogeneous systems to the spatial neuron model gives us confidence in applying the formalism to other similar or even more interesting spatially
inhomogeneous systems, which may also include effects of spatially correlated
fluctuations, detailed balance breaking in the steady state, time-dependent external driving and nonlinearity that were not present in the particular example studied. The results in Sec. 3.3.5 for general spatially inhomogeneous OU processes
are applicable to systems described by those processes. In particular, stochastic
chemical reaction diffusion processes [15, 50–52], with first-order (linear) chemical reactions and general diffusions [51], under the influence of external additive
space-time fluctuations [15,50], can be described by spatially inhomogeneous OU
processes; thus these systems can be studied in a similar fashion illustrated in the
spatial neuron model. For general non-linear stochastic systems (even for spatially homogeneous systems), analytical solutions of the (functional) Fokker-Planck
equation are generally not available; thus approximation techniques and numerical simulations are usually required in studying these systems [15, 16, 52, 53].
The general formalism of the potential-flux landscape and non-equilibrium thermodynamic equations, however, still apply to these general non-linear stochastic
systems, even though analytical expressions cannot be obtained. The formalism established in this work has potential applications in the study of the spatialtemporal non-equilibrium (thermo)dynamics of various spatially inhomogeneous
stochastic systems displaying self-organization and pattern formation behaviors,
such as convection flow in fluids, Turing pattern, cell differentiation, as well as
population dynamics in ecological systems [1, 35–41]. It may also be applied to
study some particular (types of) spatially inhomogeneous stochastic systems, described by, for instance, stochastic Ginzburg-Landau equation in superconductivi163
ty [56], Kardar-Parisi-Zhang equation studying surface growth [58] and stochastic
spatial Hodgkin-Huxley model in neurobiology [54].
3.4
Summary
In this chapter, utilizing the potential-flux landscape framework as a bridge
to connect stochastic dynamics with non-equilibrium thermodynamics, we have
established a general non-equilibrium thermodynamic formalism consistently applicable to both spatially homogeneous and inhomogeneous systems, governed
by the Langevin and Fokker-Planck stochastic dynamics. We first constructed
the non-equilibrium thermodynamics within the (extended) potential-flux landscape framework for spatially homogeneous systems in an isothermal environment and extended the results to systems with one general state transition mechanism. We further expanded the potential-flux landscape framework and nonequilibrium thermodynamics to accommodate spatially homogeneous systems with
multiple state transition mechanisms and then spatially inhomogeneous systems. General Ornstein-Uhlenbeck processes were worked out systematically in the
context of the non-equilibrium thermodynamic formalism established. The spatial stochastic neuronal model was studied in detail as both an application and a
validation of the general formalism.
A conceptual distinction is made between the information transport process in
the state space and the matter or energy transport process in the physical space,
with their possible connections discussed as well. The potential(s) and flux(es) in
the potential-flux landscape framework are in nature ‘informational’, as they characterize the information transport process in the state space, associated with the
matter or energy transport process in the physical space. The construction of the
non-equilibrium thermodynamics within the potential-flux landscape framework
from the stochastic dynamics is facilitated by identifying the double role that the
potential landscape and the flux velocity play. On the one hand, they are dynamical quantities constructed directly from the probability distribution and probability
flux of the stochastic dynamics. On the other hand, they also have thermodynamic meanings connected directly to the non-equilibrium thermodynamics. The
potential landscape is the microscopic entropy, which also acts as the potential
of certain thermodynamic force. The flux velocity is the thermodynamic force
generating entropy production. The flux velocity plays a central role in characterizing the non-equilibrium nature of the system’s (thermo)dynamics. It represents
a force breaking detailed balance on the dynamic level, entailing the dynamical
164
decomposition equations; it also represents a force creating entropy production
on the thermodynamic level, manifested in the non-equilibrium thermodynamic
equations.
It is recognized that there are two fundamental aspects of non-equilibrium
processes. One aspect is detailed balance breaking in the steady state through
constant system-environment interaction and exchange, characterized by the stationary flux velocity which is also the thermodynamic force generating the adiabatic entropy production. The other aspect is relaxation from the transient state to
the steady state due to the non-stationary condition, characterized by the relative
flux velocity which is also the thermodynamic force generating the nonadiabatic
entropy production. (The second aspect can be further identified as containing two
facets, namely initial preparation of the transient state and time-dependent external driving of the steady state.) The combined non-equilibrium effects of the two
fundamental aspects are characterized by the transient flux velocity, which is also
the thermodynamic force generating the total entropy production. The decomposition of the non-equilibrium process into these two fundamental aspects is represented by the flux decomposition equation on the dynamic level and mapped into
the entropy production decomposition equation on the thermodynamic level. As a
result of this fundamental decomposition, the potential landscape is differentiated
into three distinct yet related aspects, that is, the stationary potential landscape as
the microscopic stationary entropy, the relative potential landscape as the microscopic relative entropy, and the transient potential landscape as the microscopic
transient entropy. This conceptual differentiation is further manifested as a structural differentiation in the (thermo)dynamic equations. On the dynamic level we
have the stationary, transient and relative dynamical decomposition equations in
addition to the flux decomposition equation, forming a set of dynamical decomposition equations. On the thermodynamic level we have the cross, transient and
relative entropy balance equations as well as the entropy flow and entropy production decomposition equations, forming a set of non-equilibrium thermodynamic
equations. The set of non-equilibrium thermodynamic equations is a reflection
and manifestation of the set of dynamical decomposition equations, with the flux
velocities playing a crucial role on both levels.
165
Chapter 4
Conclusion
In this dissertation we have established a potential and flux field landscape
theory, quantifying the global stability and dynamics as well as facilitating the establishment of the non-equilibrium thermodynamics of spatially inhomogeneous
non-equilibrium systems governed by Langevin and Fokker-Planck dynamics.1
In chapter 2 we studied the global stability and dynamics of non-equilibrium
systems (spatially inhomogeneous systems in particular) using the potential-flux
landscape theory. We developed a general method to construct Lyapunov functionals that quantify the global stability and robustness of deterministic and stochastic spatially inhomogeneous non-equilibrium systems. We found the intrinsic potential field landscape is the Lyanpunov functional of the deterministic
spatially inhomogeneous system quantifying its global stability. The topography
of the intrinsic potential field landscape can be characterized by the basins of attractions and barrier heights that are directly related to the global stability of the
system. The relative entropy functional is found to be a Lyapunov functional
quantifying the global stability of the stochastic spatially inhomogeneous nonequilibrium system. We discovered that the global dynamics of spatially inhomogeneous non-equilibrium systems is determined by both the functional gradient of
the potential field landscape and the curl probability flux field. Vanishing probability flux field indicates the spatially inhomogeneous system is in detailed balance everywhere in the physical space. In such cases, the potential field landscape
is usually known a priori, given by the interaction potential or energy function1
Most of the material in this chapter was originally co-authored with Jin Wang. Reprinted
with permission from W. Wu and J. Wang, The Journal of Chemical Physics, 139, 121920 (2013).
Copyright 2013, AIP Publishing LLC. Reprinted with permission from W. Wu and J. Wang, The
Journal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIP Publishing LLC.
166
al of the system. Therefore, in spatially inhomogeneous equilibrium systems the
global dynamics is determined by the potential field landscape alone. For nonequilibrium spatially inhomogeneous dynamical systems, in general, an energy
functional governing the equilibrium gradient dynamics cannot be found. Thus
both the potential field landscape and the curl probability flux field have to be considered. The curl probability flux field breaks detailed balance and characterizes
how far the system is away from equilibrium. The potential field landscape and
the probability flux field form a dual pair that gives a complete characterization
of the global stability and dynamics of spatially inhomogeneous non-equilibrium
systems. We applied our general framework to reaction diffusion systems and the
Brusselator reaction diffusion model in particular to illustrate the general theory.
In chapter 3 we established a general non-equilibrium thermodynamic formalism consistently applicable to both spatially homogeneous and, more importantly,
spatially inhomogeneous systems, governed by Langevin and Fokker-Planck dynamics with multiple state transition mechanisms, using the potential-flux landscape framework as a bridge to connect stochastic dynamics with non-equilibrium
thermodynamics. A set of non-equilibrium thermodynamic equations, quantifying the relations of the non-equilibrium entropy, entropy flow, entropy production,
and other thermodynamic quantities, is constructed from a set of dynamical decomposition equations associated with the potential-flux landscape framework.
The flux velocity plays a pivotal role on both the dynamic and thermodynamic
levels. On the dynamic level, it represents a dynamic force breaking detailed balance, entailing the dynamical decomposition equations. On the thermodynamic
level, it represents a thermodynamic force generating entropy production, manifested in the non-equilibrium thermodynamic equations. The non-equilibrium
thermodynamic equations are reflections and manifestations of the dynamical decomposition equations, with the link between them given quantitatively by the flux
velocity. The Ornstein-Uhlenbeck process and, in particular, the spatial stochastic
neuronal model are studied to test and illustrate the general theory. The nonequilibrium thermodynamic formalism established in this work is not limited to
the linear regime or bound by the local equilibrium assumption. It can be applied
to study the spatial-temporal non-equilibrium dynamics and thermodynamics of a
variety of physical, chemical and biological systems in nature.
This work is part of a series of works we are preparing on the non-equilibrium
theory of spatially inhomogeneous stochastic dynamical systems. In future studies
we will explore the generalized fluctuation dissipation theorem, the generalized
fluctuation theorem, the gauge field representation, and the kinetic paths of nonequilibrium spatially inhomogeneous systems.
167
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Appendix A
Proof of Several Equations
This appendix is supplementary to Sec. 3.1 of the main text, proving equations
mentioned there without proof.
e ⟩s and ∂T Aes = ⟨∂T U
e ⟩s − Ss
Proof of ∂λi Aes = ⟨∂λi U
These two equations are in Sec. 3.1.4 of the main text. We list the following
two equations from the main text that will be used in the proof:
Aes ({λ}) = −T ln Z({λ}),
(A.1)
[ (
)]
1 e
e
Ps (⃗q, {λ}) = exp
As ({λ}) − U (⃗q, {λ}) .
(A.2)
T
[
]
∫
e
Also, notice the definitions Z({λ}) = exp −U (⃗q, {λ})/T d⃗q and Ss = ⟨− ln Ps ⟩s .
e ⟩s , where λi is not the temperature
We first prove the first equation ∂λi Aes = ⟨∂λi U
T.
∂λi Aes = −T Z −1 ({λ})∂λi Z({λ})
]
[
∫
1e
−1
= −T Z ({λ}) ∂λi exp − U (⃗q, {λ}) d⃗q
T
[
]
∫ [
]
1e
−1
e
= Z ({λ})
∂λi U (⃗q, {λ}) exp − U (⃗q, {λ}) d⃗q
T
∫ [
]
e (⃗q, {λ}) Ps (⃗q, {λ})d⃗q
=
∂ λi U
e ⟩s .
= ⟨∂λi U
(A.3)
179
e ⟩s − S s .
Then we prove the second equation ∂T Aes = ⟨∂T U
∂T Aes = − ln Z({λ}) − T Z −1 ({λ})∂T Z({λ})
[
]
∫
1e
Aes
−1
=
− T Z ({λ}) ∂T exp − U (⃗q, {λ}) d⃗q
T
T
)
(
∫
e (⃗q, {λ})
e (⃗q, {λ})
Aes
1
∂
U
U
=
− T Z −1 ({λ})
−
T
T2
T
∂T
[
]
1e
× exp − U
(⃗q, {λ}) d⃗q
T
⟩
⟩
⟩
⟨
⟨
⟨
e ⟩s
e
e
e
⟨Aes ⟩s ⟨U
∂U
Aes − U
∂U
=
−
+
=
+
T
T
∂T
T
∂T
s
s
⟨
⟩
⟨ s ⟩
e = ∂T U
e − Ss .
= ⟨ln Ps ⟩s + ∂T U
(A.4)
s
s
e −1 · V⃗s⟩ = ⟨V⃗s · D
e −1 · V⃗s⟩ and ⟨V⃗s ·∇A⟩
e =0
Proof of ⟨V⃗t · D
Using the definition V⃗r = V⃗t − V⃗s and the dynamical decomposition equation
e −1 · V⃗s ⟩ − ⟨V⃗s · D
e −1 · V⃗s ⟩ = ⟨V⃗r · D
e −1 · V⃗s ⟩ =
e we have ⟨V⃗t · D
D · V⃗r = −∇A,
e −1 ·V⃗s ⟩ = ⟨V⃗s ·D
e −1 ·V⃗s ⟩ is equivalent to ⟨V⃗s ·∇A⟩
e Therefore ⟨V⃗t ·D
e = 0.
−⟨V⃗s ·∇A⟩.
e
e
e
e
The relation A = T A + As shows ∇A = T ∇A, since As as a macrostate function
e = T ⟨V⃗s ·∇A⟩. Therefore,
is independent of the microstate vector ⃗q. Thus ⟨V⃗s ·∇A⟩
e −1 · V⃗s ⟩ = ⟨V⃗s · D
e −1 · V⃗s ⟩ or its equivalence ⟨V⃗s · ∇A⟩
e = 0,
in order to prove ⟨V⃗t · D
we only need to prove ⟨V⃗s · ∇A⟩ = 0, which is shown as follows:
e −1
(
)
P
t
⟨V⃗s · ∇A⟩ =
Pt V⃗s · ∇ ln
d⃗q
Ps
( )
∫
Pt
P
s
=
Pt V⃗s · ∇
d⃗q
Pt
Ps
( )
∫
Pt
=
J⃗s · ∇
d⃗q
Ps
∫ (
)P
t
⃗
=
∇ · Js
d⃗q = 0,
(A.5)
Ps
where we have used the definition A = ln(Pt /Ps ), integration by parts with vanishing boundary terms for appropriate boundary conditions, and ∇ · J⃗s = 0.
∫
180
Appendix B
Proof of the Necessary and
Sufficient Condition for the
Collective Definition Property
This appendix is supplementary to Sec. 3.2.4 of the main text. We give the
proof of the necessary and sufficient condition for the collective definition property.
∑
Theorem 1: For given vectors β⃗ (m) ∈ R(m)∑
⊆ Rk , β⃗ = m β⃗ (m) ∈ Rk and
(m)
invertible matrixes D (m) ∈ M (m)×(m) , D =
∈ M k×k , a sufficient
mD
∑ (m)
condition for m α
⃗
· D (m) −1 · β⃗ (m) = α
⃗ · D −1 · β⃗ to hold for all α
⃗ (m) ∈ R(m)
∑ (m)
with α
⃗ = mα
⃗
∈ Rk is that there exists ⃗γ ∈ Rk such that β⃗ (m) = D (m) · ⃗γ
(m = 1, 2, ...).
Proof: On the one hand, β⃗ (m) = D (m) · ⃗γ ∑
= D (m) · Π(m)
Π(m) ·
∑· ⃗γ . Therefore
(m)
(m) −1 ⃗ (m)
(m)
⃗
⃗
⃗γ = D
· β . On the other hand, β = m β
= m D · ⃗γ = D · ⃗γ .
∑ (m)
∑ (m)
(m) −1 ⃗ (m)
−1 ⃗
⃗
· Π(m) · ⃗γ =
⃗
·D
·β
= mα
Therefore ⃗γ = D · β. Then m α
∑ (m)
⃗ ⃗
· ⃗γ = α
⃗ · ⃗γ = α
⃗ · D −1 · β.
mα
∑ ⃗ (m)
k ⃗
Theorem 2: For given vectors β⃗ (m) ∈ R(m)
⊆
R
,
β
=
∈ Rk and
mβ
∑
∑
invertible matrixes D (m) ∈ M (m)×(m) , D = m D (m) ∈ M k×k , if m α
⃗ (m) ·
⃗ (m)
D (m) −1
⃗ · D −1 · β⃗ holds for all α
⃗ (m) ∈ R(m) (m = 1, 2, ...) with
∑ · β (m) = α
k
α
⃗ =
⃗
∈ R , then there exists ⃗γ ∈ Rk such that β⃗ (m) = D (m) · ⃗γ
mα
(m = 1, 2, ...).
∑ (m)
∑ (m)
Proof: m α
⃗
· D (m) −1 · β⃗ (m) − α
⃗ · D −1 · β⃗ = m α
⃗
· D (m) −1 · β⃗ (m) −
∑ (m)
∑
⃗ =0
⃗
· Π(m) · D −1 · β⃗ = m α
⃗ (m) · (D (m) −1 · β⃗ (m) − Π(m) · D −1 · β)
mα
(m)
(m)
(m′ )
′
for all α
⃗
∈R
(m = 1, 2, ...). For any fixed m, take α
⃗
= 0 (m ̸= m)
181
(m)
(m)
and α
⃗ (m) = ⃗ei (i = 1, 2, ...), where {⃗ei (i = 1, 2, ...)} are the base vectors in
(m)
⃗ = 0 for all allowed m
R(m) . Therefore ⃗ei · (D (m) −1 · β⃗ (m) − Π(m) · D −1 · β)
(m) −1 ⃗ (m)
and i, which means
− Π(m) · D −1 · β⃗ = 0 (m = 1, 2, ...). Thus
( D ) ·β
β⃗ (m) = D (m) · D −1 · β⃗ = D (m) · ⃗γ (m = 1, 2, ...), where ⃗γ = D −1 · β⃗ exists
by construction. Combining
1 and theorem 2 we have the necessary and sufficient
∑theorem
(m)
⃗
· D (m) −1 · β⃗ (m) = α
⃗ · D −1 · β⃗ to hold for all α
⃗ (m) ∈ R(m) ,
condition for m α
which is the existence of ⃗γ ∈ Rk such that β⃗ (m) = D (m) · ⃗γ (m = 1, 2, ...). And
⃗
when ⃗γ does exist, it is given by D −1 · β.
182
Appendix C
Ornstein-Uhlenbeck Processes for
Spatially Homogeneous Systems
This appendix is supplementary to Sec. 3.2.5 of the main text. We calculate
the explicit expressions of the various thermodynamic quantities of the OrnsteinUhlenbeck process for spatially homogeneous systems with one effective state
transition mechanism. For convenience, we list the results that have been obtained
in the main text which will be used here.
U =
S =
A =
∇U
∇S
∇A
V⃗s
V⃗t
=
=
=
1
1
e −1 · ⃗q + tr (ln(2π σ
e )) ,
(C.1)
⃗q · σ
2
2
1
1
(C.2)
(⃗q − ⃗µ) · σ −1 · (⃗q − ⃗µ) + tr (ln(2πσ)) ,
2
2
1
1
1
e −1 · ⃗q − (⃗q − ⃗µ) · σ −1 · (⃗q − ⃗µ) + tr (ln σ
e − ln σ) ,(C.3)
⃗q · σ
2
2
2
e −1 · ⃗q,
σ
(C.4)
−1
σ · (⃗q − ⃗µ),
(C.5)
−1
−1
e · ⃗q − σ · (⃗q − ⃗µ).
σ
(C.6)
e −1 · ⃗q,
= −γ · ⃗q + D · σ
= −γ · ⃗q + D · σ −1 · (⃗q − ⃗µ),
(C.7)
(C.8)
e −1 · ⃗q.
V⃗r = D · σ −1 · (⃗q − ⃗µ) − D · σ
(C.9)
183
e in the above equations are determined by the
The three quantities ⃗µ, σ and σ
following equations:
⃗µ˙ = −γ · ⃗µ
σ̇ = −γ · σ − σ · γ | + 2D
e +σ
e · γ | = 2D
γ·σ
(C.10)
(C.11)
(C.12)
Two useful equations can be further derived from Eq. (C.12), which will be used
e −1 from both the left and the right, we
later. By multiplying Eq. (C.12) by σ
derive:
e −1 γ + γ | σ
e −1 = 2e
e −1 .
σ
σ −1 D σ
(C.13)
e −1 from the right and then taking the trace, we
By multiplying Eq. (C.12) by σ
derive:
e −1 ) = tr(γ),
tr(D σ
(C.14)
where we have used the properties of the trace tr(AB) = tr(BA) and tr(A| ) =
e −1 ) = tr (e
e · γ | ) = tr (γ | ) = tr (γ).
tr(A), which gives tr (e
σ · γ| · σ
σ −1 · σ
Then we calculate the expressions of the thermodynamic quantities. The cross
entropy U is calculated using Eq. (C.1), given by
U = ⟨U ⟩
⟨
⟩
1
1
−1
e · ⃗q + tr (ln(2π σ
e ))
=
⃗q · σ
2
2
) 1
1 ( −1
e ⟨⃗q⃗q⟩ + tr (ln(2π σ
e ))
=
tr σ
2
2
) 1
1 ( −1
e (σ + ⃗µ⃗µ) + tr (ln(2π σ
e ))
=
tr σ
2(
2
)
1 −1
1
e (σ + µ
e) ,
= tr
σ
⃗ ⃗µ) + ln(2π σ
2
2
(C.15)
where we have used the identity ⃗a · B · ⃗c = tr(B[⃗c⃗a]), where the dyadic ⃗c⃗a is
interpreted as a matrix with the ij entry given by [⃗c⃗a]ij = ci aj . We have also
used ⟨⃗q⃗q⟩ = σ + ⃗µ⃗µ, which comes from the definition of the covariance matrix of
Gaussian distributions σ = ⟨(⃗q − ⃗µ)(⃗q − ⃗µ)⟩ = ⟨⃗q⃗q⟩ − ⃗µµ
⃗ . Then we calculate the
184
transient entropy S, using Eq. (C.2):
S = ⟨S⟩
⟩
⟨
1
1
−1
(⃗q − ⃗µ) · σ · (⃗q − ⃗µ) + tr (ln(2πσ))
=
2
2
) 1
1 ( −1
=
tr σ ⟨(⃗q − ⃗µ)(⃗q − ⃗µ)⟩ + tr (ln(2πσ))
2
2
1 ( −1 ) 1
=
tr σ σ + tr (ln(2πσ))
2(
2
)
1
1
I + ln(2πσ) .
= tr
2
2
(C.16)
The relative entropy A is then given by
A = U − S = ⟨A⟩
(
)
] 1
1 [ −1
e (σ + µ
e − ln σ) .
= tr
σ
⃗ ⃗µ) − I + (ln σ
2
2
(C.17)
Next we calculate their time derivatives U̇, Ṡ and Ȧ. We first calculate the rate
e can also be time dependent due
of change of the cross entropy U̇. Notice that σ
e [Eq. (C.12)].
to the time dependence of γ and D in the equation determining σ
Therefore, according to Eq. (C.15), we have
(
)
d
1 −1
1
e (σ + ⃗µµ
e)
U̇ =
tr
σ
⃗ ) + ln(2π σ
dt
2
2
(
)
1
d
1
−1
−1
e )
=
tr
(σ + ⃗µµ
⃗ )e
σ − ln(2π σ
dt
2
2
(
)
1
1
d
1
d
−1
−1
−1
e − σ
e σ
e
= tr
(σ̇ + ⃗µ˙ µ
⃗ + ⃗µµ
⃗˙ )e
σ + (σ + ⃗µµ
⃗) σ
2
2
dt
2 dt
(
)
1
1
d −1
|
|
−1
e) σ
e
= tr
(−γσ − σγ + 2D − γ⃗µµ
⃗ − ⃗µµ
⃗ γ )e
σ + (σ + ⃗µµ
⃗ −σ
2
2
dt
(
)
1 −1
1
d −1
| −1
−1
e )(σ + ⃗µ⃗µ) + D σ
e + (σ + ⃗µ⃗µ − σ
e) σ
e
= tr − (e
σ γ+γ σ
2
2
dt
(
)
d −1
1
−1
−1
e) σ
e
e (σ + ⃗µ⃗µ) + γ + (σ + ⃗µµ
⃗ −σ
= tr −e
σ Dσ
2
dt
(
)
1
d −1
−1
−1
e) σ
e −σ
e Dσ
e (σ + ⃗µµ
= tr
(σ + ⃗µµ
⃗ −σ
⃗) + γ ,
(C.18)
2
dt
185
where we have used Eqs. (C.10), (C.11), (C.13), (C.14) and the properties of the
trace tr(AB) = tr(BA) and tr(A| ) = tr(A). We have also implicitly utilized
the so-called Jacobi’s formula of differentiating determinants, in the following
form:
d
d
d
1
det(A) =
ln det(A) = tr(ln A)
det(A) dt
dt
dt
(
)
(
)
d
d
= tr A−1 A = tr −A A−1 .
dt
dt
e is realized via∑
If the explicit time-dependence in γ and D and thus σ
a set of timee −1 .
dependent external control parameters {λi (t)}, then de
σ −1 /dt =
i λ̇i ∂λi σ
The rate of change of the transient entropy, according to Eq. (C.16), is calculated
as follows:
(
)
d
1
1
Ṡ =
tr
I + ln(2πσ)
dt
2
2
(
)
1 −1 d
= tr
σ
σ
2
dt
(
)
1 −1
|
= tr
σ (−γσ − σγ + 2D)
2
(
)
1
−1
−1
|
−1
= tr
(−γσσ − σ σγ ) + σ D
2
(
)
1
|
−1
= tr − (γ + γ ) + σ D
2
−1
= tr(σ D − γ),
(C.19)
where we have used Eq. (C.11), tr(AB) = tr(BA) and tr(A| ) = tr(A). The
rate of change of the relative entropy is then given by
Ȧ = U̇ (
− Ṡ
1
d −1
e) σ
e −σ
e −1 D σ
e −1 (σ + ⃗µµ
= tr
(σ + ⃗µµ
⃗ −σ
⃗)
2
dt
)
−σ −1 D + 2γ .
(C.20)
Then we calculate the rest of the thermodynamic quantities. The external
186
driving entropic power Ṡed is calculated using Eq. (C.1):
Ṡed = ⟨∂t U ⟩
) ⟩
⟨
(
1
d −1
1d
e
e ))
=
⃗q ·
σ
· ⃗q +
tr (ln(2π σ
2
dt
2 dt
(
)
)
1
d −1
1d (
e
e −1 )
= tr
⟨⃗q⃗q⟩ σ
−
tr ln(2π σ
2
dt
2 dt
(
)
(
)
d −1
1
d −1
1
e
e σ
e
= tr
(σ + ⃗µ⃗µ) σ
− tr σ
2
dt
2
dt
(
)
1
d −1
e) σ
e
= tr
(σ + ⃗µ⃗µ − σ
,
2
dt
(C.21)
where we have used the equation ⟨⃗q⃗q⟩ = σ + ⃗µµ
⃗ . The excess entropy flow rate is
calculated as follows from Eqs. (C.4) and (C.8):
Ṡex = −⟨V⃗t · ∇U ⟩
⟨(
) ( −1 )⟩
e · ⃗q
=
⃗q · γ | − (⃗q − ⃗µ) · σ −1 · D · σ
⟨
⟩
| −1
−1
e · ⃗q − (⃗q − ⃗µ) · σ D σ
e −1 · ⃗q
= ⃗q · γ σ
1
e −1 + σ
e −1 γ) · ⃗q − (⃗q − ⃗µ) · σ −1 D σ
e −1 · ⃗q⟩
= ⟨⃗q · (γ | σ
2
e −1 D σ
e −1 · ⃗q − (⃗q − ⃗µ) · σ −1 D σ
e −1 · ⃗q⟩
= ⟨⃗q · σ
( −1
)
e Dσ
e −1 ⟨⃗q⃗q⟩ − σ −1 D σ
e −1 ⟨⃗q(⃗q − ⃗µ)⟩
= tr σ
( −1
)
e Dσ
e −1 (σ + ⃗µµ
e −1 σ
= tr σ
⃗ ) − σ −1 D σ
( −1
)
e Dσ
e −1 (σ + ⃗µµ
e −1
= tr σ
⃗ ) − σσ −1 D σ
( −1
)
e Dσ
e −1 (σ + ⃗µµ
= tr σ
⃗) − γ ,
(C.22)
where we have used ⟨⃗q(⃗q − ⃗µ)⟩ = ⟨⃗q⃗q⟩ − ⟨⃗q⟩⃗µ = σ and Eqs. (C.13) and (C.14).
The (total) entropy production rate is calculated using Eq. (C.8):
Ṡpd = ⟨V⃗t · D −1 · V⃗t ⟩
⟨(
)
(
)⟩
=
−⃗q · γ | + (⃗q − ⃗µ) · σ −1 · D · D −1 · −γ · ⃗q + D · σ −1 · (⃗q − ⃗µ)
⟨
= ⃗q · γ | D −1 γ · ⃗q + (⃗q − ⃗µ) · σ −1 DD −1 Dσ −1 · (⃗q − ⃗µ)
⟩
−2⃗q · γ | D −1 Dσ −1 · (⃗q − ⃗µ)
(
)
= tr γ | D −1 γ⟨⃗q⃗q⟩ + σ −1 Dσ −1 ⟨(⃗q − ⃗µ)(⃗q − ⃗µ)⟩ − 2γ | σ −1 ⟨(⃗q − ⃗µ)⃗q⟩
(
)
= tr γ | D −1 γ(σ + ⃗µµ
⃗ ) + σ −1 Dσ −1 σ − 2γ | σ −1 σ
(
)
= tr γ | D −1 γ(σ + ⃗µµ
⃗ ) + σ −1 D − 2γ .
(C.23)
187
The (total) entropy flow rate is given by:
Ṡf l =
=
=
=
=
=
⟨V⃗t · D −1 · F⃗ ′ ⟩
⟨(
)
⟩
−⃗q · γ | + (⃗q − ⃗µ) · σ −1 · D · D −1 · (−γ · ⃗q)
⟨
⟩
⃗q · γ | D −1 γ · ⃗q − (⃗q − ⃗µ) · σ −1 γ · ⃗q
(
)
tr γ | D −1 γ⟨⃗q⃗q⟩ − σ −1 γ⟨⃗q(⃗q − ⃗µ)⟩
(
)
tr γ | D −1 γ(σ + ⃗µµ
⃗ ) − σ −1 γσ
(
)
tr γ | D −1 γ(σ + ⃗µµ
⃗) − γ .
(C.24)
The adiabatic entropy production rate is calculated using Eq. (C.7):
Ṡad =
=
=
=
=
=
⟨V⃗ · D −1 · V⃗s ⟩
⟨(s
)
(
)⟩
e −1 · D · D −1 · −γ · ⃗q + D · σ
e −1 · ⃗q
−⃗q · γ | + ⃗q · σ
(
)
e −1 D)D −1 (γ − D σ
e −1 )⟨⃗q⃗q⟩
tr (γ | − σ
(
)
e −1 − σ
e −1 γ + σ
e −1 D σ
e −1 )⟨⃗q⃗q⟩
tr (γ | D −1 γ − γ | σ
(
)
e −1 + σ
e −1 D σ
e −1 )⟨⃗q⃗q⟩
tr (γ | D −1 γ − 2e
σ −1 D σ
(
)
e −1 D σ
e −1 )(σ + ⃗µµ
tr (γ | D −1 γ − σ
⃗) ,
(C.25)
where we have used Eq. (C.13). The non-adiabatic entropy production rate is
calculated using Eq. (C.9):
Ṡna = ⟨V⃗r · D −1 · V⃗r ⟩
⟨(
)
(
e −1 · D + (⃗q − ⃗µ)σ −1 · D · D −1 · −D · σ
e −1 · ⃗q
=
−⃗q · σ
)⟩
+D · σ −1 · (⃗q − ⃗µ)
⟨
e −1 D σ
e −1 · ⃗q − 2⃗q · σ
e −1 Dσ −1 · (⃗q − ⃗µ)
= ⃗q · σ
⟩
+(⃗q − ⃗µ) · σ −1 Dσ −1 · (⃗q − ⃗µ)
( −1
e Dσ
e −1 ⟨⃗q⃗q⟩ + σ −1 Dσ −1 ⟨(⃗q − ⃗µ)(⃗q − µ
= tr σ
⃗ )⟩
)
−1
−1
−2e
σ Dσ ⟨(⃗q − ⃗µ)⃗q⟩
( −1
)
e Dσ
e −1 (σ + ⃗µµ
= tr σ
⃗ ) + σ −1 Dσ −1 σ − 2e
σ −1 Dσ −1 σ
( −1
)
e Dσ
e −1 (σ + ⃗µµ
= tr σ
⃗ ) + σ −1 D − 2γ ,
(C.26)
where we have used Eqs. (C.13) and (C.14).
Thus we have derived all the explicit expressions of the thermodynamic quantities in the set of thermodynamic (infodynamic) equations for OU processes of spatially homogeneous systems with one mechanism, given by Eqs. (C.15)-(C.26).
It is easy to verify that these expressions satisfy the set of non-equilibrium thermodynamic equations.
188
Appendix D
Abstract Representation and
Representation Transformation
This appendix is supplementary to Sec. 3.3.2 - Sec. 3.3.4 of the main text. We
first give the abstract representation using Dirac notation of the major equations in
the non-equilibrium thermodynamic formalism for spatially inhomogeneous systems developed in Sec. 3.3.2 - Sec. 3.3.4 in the main text presented in the space
configuration representation. Then we discuss how to transform the abstract representation into other representations and how to relate different representations.
D.1 Abstract Representation
There is a correspondence between results in the space configuration representation and those in the abstract representation in the Dirac bra-ket notation.
⃗ x) representing a state of the spatially inhomogeneous sysA space function ϕ(⃗
tem in the space configuration representation is represented by a ket |ϕ) or a bra
(ϕ| in the abstract representation, depending on its relation to other objects in the
same term. An integral kernel B(⃗x, ⃗x ′ ) in the space configuration representation
is represented by a linear operator B̂ in the abstract representation. The function⃗ x) is represented by a ket |δϕ ) or a bra (δϕ |, which
al derivative ⃗δϕ(⃗
⃗ x) ≡ δ/δ ϕ(⃗
does not only have an algebraic character but also a differential character. The
integral over space and the sum over the vector index in the space configuration
representation can be absorbed into the operation of linear operators acting on a
ket, or they can be absorbed into the inner product (bra-ket form), so that they not
appear explicitly in the abstract representation. Using these prescriptions, we are
189
able to transform results in the space configuration representation into the abstract
representation.
Functional Langevin and Fokker-Planck Dynamics of Spatially
Inhomogeneous Systems
The functional Langevin equation, as a stochastic differential equation on the
Hilbert space, has the following form in the abstract representation:
∑
d|ϕ) = |F (ϕ)) +
|Gs (ϕ))dWs (t),
(D.1)
s
where |F (ϕ)) is the deterministic driving force, |Gs (ϕ)) is the stochastic driving force component from source s, and Ws (t) (s = 1, 2, ...) are independent
one-dimensional standard Wiener processes. If there are multiple state transition
mechanisms indexed by m, Eq. (D.1) can be written more specifically as:
[
]
∑
∑
(m)
d|ϕ) =
|F (m) (ϕ)) +
|G(m)
(t) .
(D.2)
s (ϕ))dWs
m
s
(m)
We assume that |F (m) (ϕ)) and |Gs (ϕ)) from the same mechanism m lie in the
same subspace Ω(m) of the entire Hilbert state space Ω. The projection from Ω into
Ω(m) is done through the projection operator Π̂(m) . The corresponding functional
Fokker-Planck equation is
[
]
(
[
])
∂
Pt (ϕ) = −(δϕ | |F (ϕ))Pt (ϕ) + tr |δϕ )(δϕ | D̂(ϕ)Pt (ϕ) .
(D.3)
∂t
We have a look at each of the two terms on the right side of the equation individually. In the first term, the algebraic character of (δϕ | acts on |F (ϕ)) forming
a bra-ket, producing an algebraically scalar quantity. Yet its differential character acts not only on |F (ϕ)) but also on Pt (ϕ). That is why |F (ϕ)) and Pt (ϕ)
are grouped together. In the second term we can regard |δϕ )(δϕ | as one object constructed from |δϕ ) and (δϕ |. As with |δϕ ) or (δϕ |, it has both an algebraic
character and a differential character. Algebraically, |δϕ )(δϕ | is a linear operator
in the Hilbert space as D̂(ϕ) is. The product of two linear operators |δϕ )(δϕ | and
D̂(ϕ) is again a linear operator. The notation tr() in this case is taking the trace
of the product operator |δϕ )(δϕ |D̂(ϕ). Yet the differential character of |δϕ )(δϕ |
does not only act on D̂(ϕ) but also on Pt (ϕ). Therefore D̂(ϕ) and Pt (ϕ) are also
190
grouped together. The drift vector (deterministic driving force) and the diffusion
operator in Eq. (D.3) are given respectively by contributions from each individual
mechanism:
∑
)
F (m) (ϕ) ,
|F (ϕ)) =
(D.4)
m
D̂(ϕ) =
∑
m
]
[
∑ 1 ∑
)
(
G(m)
.
G(m)
D̂(m) (ϕ) =
s (ϕ)
s (ϕ)
2
s
m
(D.5)
By construction, D̂(ϕ) and D̂(m) (ϕ) are nonnegative real-valued self-adjoint operators in the Hilbert space Ω. They have the property that for any |φ) ∈ Ω,
(φ|D̂(ϕ)|φ) ≥ 0 and (φ|D̂(m) (ϕ)|φ) ≥ 0. We require a stronger condition that for
any |φ) ∈ Ω, (φ|D̂(ϕ)|φ) = 0 only when |φ) = 0. And for any |φ(m) ) ∈ Ω(m) ,
(φ(m) |D̂(m) (ϕ)|φ(m) ) = 0 only when |φ(m) ) = 0. The (left) inverse of D̂(ϕ) deˆ The (left) inverse of D̂(m) (ϕ) in
noted by D̂−1 (ϕ) satisfies: D̂−1 (ϕ)D̂(ϕ) = I.
the space Ω(m) satisfies: [D̂(m) (ϕ)]−1 D̂(m) (ϕ) = Iˆ(m) , where Iˆ(m) is the identity
operator in Ω(m) .
Potential-Flux Field Landscape of Spatially Inhomogeneous Systems
The functional Fokker-Planck equation [Eq. (D.3)] can be written as a continuity equation:
∂t Pt (ϕ) = −(δϕ |Jt (ϕ)),
(D.6)
where the transient flux is given by
|Jt (ϕ)) = |F ′ (ϕ))Pt (ϕ) − D̂(ϕ)|δϕ Pt (ϕ)),
with the effective drift vector defined as
′
[
]|
|F (ϕ)) = |F (ϕ)) − (δϕ |D̂(ϕ) .
(D.7)
(D.8)
The notation | represents the operation of transpose, which is necessary to ensure
that this term is a ket as the other two terms in this equation. Note that we do
not need to use the operation of conjugate transpose since we have assumed the
Hilbert space to be real. Equation (D.7) can be reformulated into the transient
dynamical decomposition equation:
|F ′ (ϕ)) = −D̂(ϕ)|δϕ S(ϕ)) + |Vt (ϕ)),
191
(D.9)
where S(ϕ) = − ln Pt (ϕ) and |Vt (ϕ)) = |Jt (ϕ))/Pt (ϕ). The (instantaneous) stationary probability distribution Ps (ϕ) satisfies the stationary functional FokkerPlanck equation: (δϕ |Js (ϕ)) = 0. The stationary dynamical decomposition equation is:
|F ′ (ϕ)) = −D̂(ϕ)|δϕ U (ϕ)) + |Vs (ϕ)),
(D.10)
where U (ϕ) = − ln Ps (ϕ) and |Vs (ϕ)) = |Js (ϕ))/Ps (ϕ). From Eqs. (D.9) and
(D.10) we also have the relative dynamical decomposition equation:
|Vr (ϕ)) = −D̂(ϕ)|δϕ A(ϕ)),
(D.11)
where A(ϕ) = U (ϕ) − S(ϕ) = ln(Pt (ϕ)/Ps (ϕ)) and |Vr (ϕ)) = |Vt (ϕ)) − |Vs (ϕ)).
For each individual mechanism m, we have:
(m)
|Jt
(ϕ)) = |F ′(m) (ϕ))Pt (ϕ) − D̂(m) (ϕ)|δϕ Pt (ϕ)),
where the effective drift vector of mechanism m is
[
]|
|F ′(m) (ϕ)) = |F (m) (ϕ)) − (δϕ |D̂(m) (ϕ) .
(D.12)
(D.13)
Correspondingly, Eqs. (D.9)-(D.11) also have their counterparts for each individual mechanism:
|F ′(m) (ϕ)) = −D̂(m) (ϕ)|δϕ U (ϕ)) + |Vs(m) (ϕ)),
(D.14)
|F ′(m) (ϕ)) = −D̂(m) (ϕ)|δϕ S(ϕ)) + |Vt
(D.15)
(m)
|Vr(m) (ϕ)) = −D̂(m) (ϕ)|δϕ A(ϕ)).
(ϕ)),
(D.16)
∑
(m)
The sum over the index m gives the combined collective quantity: m |Jt (ϕ)) =
∑
∑
∑
(m)
(m)
|Jt (ϕ)), m |F ′(m) (ϕ)) = |F ′ (ϕ)), m |Vt (ϕ)) = |Vt (ϕ)), m |Vs (ϕ)) =
∑
(m)
|Vs (ϕ)), m |Vr (ϕ)) = |Vr (ϕ)). D̂(m) (ϕ) in Eqs. (D.14)-(D.16) can be inverted
to give:
Π̂(m) |δϕ U (ϕ)) = [D̂(m) (ϕ)]−1 |Vs(m) (ϕ)) − [D̂(m) (ϕ)]−1 |F ′(m) (ϕ)),
Π̂(m) |δϕ S(ϕ)) = [D̂(m) (ϕ)]−1 |Vt
(m)
(ϕ)) − [D̂(m) (ϕ)]−1 |F ′(m) (ϕ)),
Π̂(m) |δϕ A(ϕ)) = −[D̂(m) (ϕ)]−1 |Vr(m) (ϕ)).
192
(D.17)
Non-equilibrium Thermodynamics of Spatially Inhomogeneous
Systems
The set of non-equilibrium thermodynamic equations is:


U̇
= Ṡed − Ṡex





 Ṡ
= Ṡpd − Ṡf l

Ȧ = Ṡed − Ṡna




Ṡpd = Ṡad + Ṡna




Ṡf l = Ṡad + Ṡex
(D.18)
with the sign properties Ṡad ≥ 0, Ṡna ≥ 0, Ṡpd ≥ 0 and the definitions U =
⟨− ln Ps ⟩, S = ⟨− ln Pt ⟩, A = ⟨ln[Pt /Ps ]⟩. For systems with one effective state
transition mechanism, the expressions of these thermodynamic quantities in the
abstract representation are given by:
U̇
Ṡ
Ȧ
Ṡed
Ṡex
=
=
=
=
=
Ṡpd =
Ṡf l =
Ṡad =
Ṡna =
=
⟨∂t U ⟩ + ⟨(Vt |δϕ U )⟩
⟨(Vt |δϕ S)⟩
⟨∂t U ⟩ + ⟨(Vt |δϕ A)⟩
⟨∂t U ⟩
− ⟨(Vt |δϕ U )⟩
⟨
⟩
(Vt |D̂−1 |Vt )
⟨
⟩
−1
′
(Vt |D̂ |F )
⟨
⟩ ⟨
⟩
(Vt |D̂−1 |Vs ) = (Vs |D̂−1 |Vs )
⟨
⟩
−1
(Vt |D̂ |Vr ) = − ⟨(Vt |δϕ A)⟩
⟨
⟩
⟨
⟩
(Vr |D̂−1 |Vr ) = − ⟨(Vr |δϕ A)⟩ = (δϕ A|D̂|δϕ A)
(D.19)
(D.20)
(D.21)
(D.22)
(D.23)
(D.24)
(D.25)
(D.26)
(D.27)
For systems with multiple state transition mechanisms, the expressions of the thermodynamic quantities in the abstract representation are given by:
⟩
∑ ⟨ (m)
U̇ = ⟨∂t U ⟩ + ⟨(Vt |δϕ U )⟩ = ⟨∂t U ⟩ +
(Vt |δϕ U )
(D.28)
Ṡ = ⟨(Vt |δϕ S)⟩ =
∑⟨
(m)
(Vt |δϕ S)
m
193
⟩m
(D.29)
Ȧ = ⟨∂t U ⟩ + ⟨(Vt |δϕ A)⟩ = ⟨∂t U ⟩ +
∑⟨
(m)
(Vt
|δϕ A)
⟩
(D.30)
m
Ṡed = ⟨∂t U ⟩
Ṡex
⟩
∑ ⟨ (m)
= − ⟨(Vt |δϕ U )⟩ = −
(Vt |δϕ U )
(D.31)
(D.32)
⟩
∑ ⟨ (m)
(m)
(Vt |[D̂(m) ]−1 |Vt )
(D.33)
Ṡf l
m
⟩
∑ ⟨ (m)
=
(Vt |[D̂(m) ]−1 |F ′(m) )
(D.34)
Ṡad
m
⟩ ⟨
⟩
∑ ⟨ (m)
(m) −1
(m)
(m)
(m) −1
(m)
=
(Vt |[D̂ ] |Vs ) = (Vs |[D̂ ] |Vs ) (D.35)
Ṡna
⟨m
⟩ ∑⟨
⟩
(m)
−1
(m) −1
(m)
= (Vt |D̂ |Vr ) =
(Vt |[D̂ ] |Vr )
m
Ṡpd =
∑⟨
m
= − ⟨(Vt |δϕ A)⟩ = −
(m)
(Vt
|δϕ A)
⟩
m
⟨
⟩ ∑
⟨
⟩
= (Vr |D̂−1 |Vr ) =
(Vr(m) |[D̂(m) ]−1 |Vr(m) )
m
= − ⟨(Vr |δϕ A)⟩ = −
∑⟨
⟩
(Vr(m) |δϕ A)
m
⟨
⟩ ∑
⟨
⟩
= (δϕ A|D̂|δϕ A) =
(δϕ A|D̂(m) |δϕ A)
(D.36)
m
D.2 Representation Transformation
We discuss how to transform the abstract representation into the space configuration representation and into the orthonormal basis representation {|en )} as well
as how to relate these two different representations. The mathematical formalism
to be presented in the following has a close connection with the representation
transformation theory in quantum mechanics; we refer readers to textbooks of
quantum mechanics for more perspectives [159]. We will use examples from the
above subsections as illustrations.
194
From the abstract representation to the space configuration representation
A concrete representation is characterized by a basis of the Hilbert space. The
space configuration representation is characterized by the space configuration basis, which is written in the abstract representation as {|a, ⃗x)}, with a discrete vector index a and a continuous space index ⃗x. If the state of the system in the space
⃗ x) has only one component (i.e., the vector ϕ
⃗ is one
configuration representation ϕ(⃗
dimensional), then the discrete index a will not be there and the space configuration basis is simply {|⃗x)}. Strictly speaking, |a, ⃗x) is not in the Hilbert space since
its norm is divergent due to ⃗x being a continuous index. But such states can be
accommodated in an extended space of the Hilbert space. We shall not go into the
technical details of this issue here. The orthonormality condition (normalized to
Dirac delta function) and the completeness condition for the space configuration
basis {|a, ⃗x)} are, respectively,
∫
∑
′
′
ˆ
(a, ⃗x|b, ⃗x ) = δab δ(⃗x − ⃗x );
d⃗x
|a, ⃗x)(a, ⃗x| = I.
(D.37)
a
The completeness condition is also called resolution of identity when viewed
backward, where the identity operator Iˆ is resolved into a sum and/or integral
in the basis:
∫
∑
ˆ
I = d⃗x
|a, ⃗x)(a, ⃗x|,
(D.38)
a
which is a very useful equation in facilitating representation transformations.
⃗ x) is obA state of the system in the space configuration representation ϕ(⃗
tained by projecting its abstract representation |ϕ) onto the space configuration
base vector (a, ⃗x|:
ϕa (⃗x) = (a, ⃗x|ϕ).
(D.39)
The space configuration representation of a linear operator as an integral kernel
B(⃗x, ⃗x ′ ) is obtained from its abstract representation B̂ by projecting it onto the
space configuration base vectors on both sides:
B ab (⃗x, ⃗x ′ ) = (a, ⃗x|B̂|b, ⃗x ′ ).
(D.40)
The space configuration representation of other expressions and equations in the
abstract representation can be obtained by projecting onto the basis {|a, ⃗x)} and
inserting resolution of identity in appropriate places. For example, the space configuration representation of the equation |φ) = B̂|ϕ) is derived by projecting the
195
equation on (a, ⃗x| and inserting resolution of identity [Eq. (D.38)] between B̂ and
|ϕ):
φa (⃗x) = (a, ⃗x|φ) = (a, ⃗x|B̂|ϕ)
∫
∑
=
d⃗x ′
(a, ⃗x|B̂|b, ⃗x ′ )(b, ⃗x ′ |ϕ)
∫
=
b
d⃗x ′
∑
B ab (⃗x, ⃗x ′ )ϕb (⃗x ′ ),
(D.41)
b
∫
∑
which gives φa (⃗x) = d⃗x ′ b B ab (⃗x, ⃗x ′ )ϕb (⃗x ′ ) that has been given in the main
text.
Next we have a look at the space configuration representation of the functional
Fokker-Planck equation given in Eq. (D.3). Inserting resolution of identity [Eq.
(D.38)] and using the formula tr(|ϕ)(φ|B̂) = (φ|B̂|ϕ), we have:
∂
Pt (ϕ)
∂t [
]
(
[
])
= −(δϕ | |F (ϕ))Pt (ϕ) + tr |δϕ )(δϕ | D̂(ϕ)Pt (ϕ)
∫
[
]
∑
= −(δϕ | d⃗x
|a, ⃗x)(a, ⃗x| |F (ϕ))Pt (ϕ)
(∫
a
+tr
d⃗x
∫
∑
= −
d⃗x
∫∫
+
d⃗x
∫∫
+
|a, ⃗x)(a, ⃗x|δϕ )(δϕ |
d⃗x ′
∑
∑
[
]
|b, ⃗x ′ )(b, ⃗x ′ | D̂(ϕ)Pt (ϕ)
)
b
[
]
(δϕ |a, ⃗x) (a, ⃗x|F (ϕ))Pt (ϕ)
d⃗xd⃗x ′
∑
∫
a
a
∫
= −
∑
(
[
])
tr |a, ⃗x)(a, ⃗x|δϕ )(δϕ |b, ⃗x ′ ) (b, ⃗x ′ |D̂(ϕ)Pt (ϕ)
ab
[
]
(δϕ |a, ⃗x) (a, ⃗x|F (ϕ))Pt (ϕ)
a
d⃗xd⃗x
′
∑
′
[
′
]
(a, ⃗x|δϕ )(δϕ |b, ⃗x ) (b, ⃗x |D̂(ϕ)|a, ⃗x)Pt (ϕ) .
ab
Replacing Pt (ϕ) with its functional notation Pt [ϕ] and identifying (δϕ |a, ⃗x) =
(a, ⃗x|δϕ ) = δ/δϕa (⃗x), (a, ⃗x|F (ϕ)) = F a (⃗x)[ϕ] and (b, ⃗x ′ |D̂(ϕ)|a, ⃗x) = Dba (⃗x ′ , ⃗x)[ϕ],
we recover the functional Fokker-Planck equation in the space configuration rep196
resentation given in the main text:
∫
)
∑ δ (
∂
a
Pt [ϕ] = − d⃗x
F (⃗x)[ϕ]Pt [ϕ]
∂t
δϕa (⃗x)
a
∫∫
∑
( ab
)
δ2
′
+
d⃗xd⃗x ′
D
(⃗
x
,
⃗
x
)[ϕ]P
[ϕ]
.(D.42)
t
δϕa (⃗x)δϕb (⃗x ′ )
ab
The space configuration representation of other expressions and equations given in
the main text can be obtained from the abstract representation in a similar fashion.
From the abstract representation to the orthonormal basis representation
Then we consider an orthonormal basis {|en )}, with the index n discrete and
going into infinity. One aspect in which this basis differs from the space configuration basis is that there is no continuous index. The orthonormality (normalized
to the Kronecker delta function) and completeness conditions of the basis {|en )}
are
∑
ˆ
(en |em ) = δnm ;
|en )(en | = I.
(D.43)
n
The latter can be regarded as resolution of identity in the basis {|en )}:
∑
Iˆ =
|en )(en |.
(D.44)
n
A state |ϕ) in the basis {|en )} is represented by an infinite dimensional vector
{ϕn }, with the component given by
ϕn = (en |ϕ).
(D.45)
A linear operator B̂ in the basis {|en )} is represented by an infinite dimensional
square matrix [Bmn ], with the matrix element given by
Bmn = (em |B̂|en ).
(D.46)
The expressions and equations in the representation {|en )} can be obtained from
the abstract representation by projecting onto the basis {|en )} and inserting resolution of identity Eq. (D.44) where appropriate. The equation |φ) = B̂|ϕ) in the
basis of {|en )} is derived as follows:
∑
∑
φm = (em |φ) = (em |B̂|ϕ) =
(em |B̂|en )(en |ϕ) =
Bmn ϕn . (D.47)
n
n
197
The functional Fokker-Planck equation [Eq. (D.3)] in the representation {|en )}
is obtained as follows:
∂
Pt (ϕ)
∂t [
]
(
[
])
= −(δϕ | |F (ϕ))Pt (ϕ) + tr |δϕ )(δϕ | D̂(ϕ)Pt (ϕ)
[
]
∑
= −(δϕ |
|en )(en | |F (ϕ))Pt (ϕ)
+tr
( n
∑
|en )(en |δϕ )(δϕ |
∑
)
]
|em )(em | D̂(ϕ)Pt (ϕ)
[
n
m
[
]
∑
= −
(δϕ |en ) (en |F (ϕ))Pt (ϕ)
n
+
∑
(
[
])
tr |en )(en |δϕ )(δϕ |em ) (em |D̂(ϕ)Pt (ϕ)
nm
[
] ∑
[
]
∑
= −
(δϕ |en ) (en |F (ϕ))Pt (ϕ) +
(en |δϕ )(δϕ |em ) (em |D̂(ϕ)|en )Pt (ϕ) .
n
nm
Since a state ϕ in the basis {|en )} is represented by {ϕk }, we replace the state dependence (ϕ) with ({ϕk }). Identifying (δϕ |en ) = (en |δϕ ) = ∂/∂ϕn ,
(en |F (ϕ)) = Fn ({ϕk }) and (em |D̂(ϕ)|en ) = Dmn ({ϕk }), we thus have the functional Fokker-Planck equation in the representation {|en )}:
]
∑ ∂ [
∂
Pt ({ϕk }) = −
Fn ({ϕk })Pt ({ϕk })
∂t
∂ϕn
n
]
∑
∂2 [
+
Dmn ({ϕk })Pt ({ϕk }) ,
∂ϕm ∂ϕn
mn
(D.48)
which is an infinite dimensional Fokker-Planck equation. It is equivalent to the
functional equation in Eq. (D.42), just in a different representation. Other expressions and equations in the representation of {|en )} can be derived similarly.
Transformation between the space configuration representation
and the orthonormal basis representation
The relation of a quantity in one representation with that in another representation can be obtained again by inserting resolution of identity where appropriate.
198
The relation between the force field |F ) in the space configuration representation
and that in the the orthonormal basis representation can be obtained by inserting
the resolution of identity in the basis {|en )}:
∑
∑
F a (⃗x) = (a, ⃗x|F ) =
(a, ⃗x|en )(en |F ) =
Fn ean (⃗x),
(D.49)
n
n
where ean (⃗x) = (a, ⃗x|en ) is the space configuration representation of |en ). Reciprocally, we can also express Fn in terms of F a (⃗x) by inserting the resolution of
identity in the basis {|a, ⃗x)}:
∫
∫
∑
∑
Fn = (en |F ) = d⃗x
(en |a, ⃗x)(a, ⃗x|F ) = d⃗x
ean (⃗x)F a (⃗x). (D.50)
a
a
Another example is the diffusion matrix field:
∑
Dab (⃗x, ⃗x ′ ) = (a, ⃗x|D̂|b, ⃗x ′ ) =
(a, ⃗x|em )(em |D̂|en )(en |b, ⃗x ′ )
=
∑
mn
Dmn eam (⃗x)ebn (⃗x ′ ).
(D.51)
mn
Note that since we have assumed the Hilbert space to be real, there is no complex
conjugate involved when reversing the order of a bra-ket: (φ|ϕ) = (ϕ|φ), so that
(a, ⃗x|em ) = (em |a, ⃗x) = eam (⃗x). Reciprocally, we also have
∫∫
∑
(em |a, ⃗x)(a, ⃗x|D̂|b, ⃗x ′ )(b, ⃗x ′ |en )
Dmn = (em |D̂|en ) =
d⃗xd⃗x ′
∫∫
=
d⃗xd⃗x ′
∑
ab
Dab (⃗x, ⃗x ′ )eam (⃗x)ebn (⃗x ′ ).
(D.52)
ab
The differential operator in these two representations are related as follows:
∑
∑
δ
∂
(a,
⃗
x
|e
)(e
|δ
)
=
ean (⃗x)
.
=
(a,
⃗
x
|δ
)
=
ϕ
n
n
ϕ
a
δϕ (⃗x)
∂ϕn
n
n
(D.53)
Reciprocally,
∂
= (en |δϕ ) =
∂ϕn
∫
∫
∑
∑
d⃗x
(en |a, ⃗x)(a, ⃗x|δϕ ) = d⃗x
ean (⃗x)
a
a
δ
δϕa (⃗x)
.
(D.54)
199
The relation in Eq. (D.53) can be derived in another way. ϕn can be seen as a
⃗ x), given by:
functional of ϕ(⃗
∫
∫
∑
∑
′
′
′
ϕn = (en |ϕ) = d⃗x
(en |b, ⃗x )(b, ⃗x |ϕ) = d⃗x ′
ebn (⃗x ′ )ϕb (⃗x ′ ).
b
b
(D.55)
Using the chain rule of differentiation, we have:
∑ δϕn ∂
δ
=
δϕa (⃗x)
δϕa (⃗x) ∂ϕn
n
[
]
∫
∑
∑
δ
∂
′
b
′
b
′
=
d⃗x
en (⃗x )ϕ (⃗x )
a
δϕ (⃗x)
∂ϕn
n
b
[∫
]
∑
∑
δϕb (⃗x ′ ) ∂
=
d⃗x ′
ebn (⃗x ′ ) a
δϕ (⃗x) ∂ϕn
n
b
]
[∫
∑
∑
∂
b
′
′
′
en (⃗x )δab δ(⃗x − ⃗x )
=
d⃗x
∂ϕn
n
b
∑
∂
=
ean (⃗x)
.
∂ϕn
n
(D.56)
It is the same result as in Eq. (D.53), showing that the definitions (a, ⃗x|δϕ ) =
δ/δϕa (⃗x) and (en |δϕ ) = ∂/∂ϕn are consistent with each other. Plugging Eqs.
(D.49) and (D.51) into Eq. (D.42) and using Eq. (D.54), it is easy to see that the
functional Fokker-Planck equation in the space configuration representation in
Eq. (D.42) is transformed into its representation in the basis {|en )} in Eq. (D.48)
directly, showing their equivalence explicitly. Reciprocally, we can transform the
functional Fokker-Planck equation in the basis {|en )} into its space configuration
representation by plugging Eqs. (D.50) and (D.52) into Eq. (D.48) and using Eq.
(D.53) to obtain Eq. (D.42).
Next we touch upon the integration measure on the Hilbert space in different
representations. Consider integrating a (general)
function f (ϕ) over the Hilbert
∫
space, which is formally represented as D(ϕ)f (ϕ) in the abstract representation. When working in specific representations (e.g., the space configuration representation {|a, ⃗x)} or the representation of {|en )}), the integral takes on different
forms but the result should be the same, i.e.,
∫
∫
[
] [
] ∫
⃗ x) f ϕ(⃗
⃗ x) = D({ϕn })f ({ϕn }).
D(ϕ)f (ϕ) = D ϕ(⃗
(D.57)
200
If the value of the function f (ϕ) is the same in different representations and only
ϕ is replaced by its specific form in the corresponding representation, that is,
[
]
⃗ x) = f ({ϕn }),
f (ϕ) = f ϕ(⃗
(D.58)
then we also have the equality of the integration measures in different representations:
[
]
⃗ x) = D({ϕn }),
D(ϕ) = D ϕ(⃗
(D.59)
⃗ x) and ϕn are related to each other by Eq. (D.55)
where in these equations ϕ, ϕ(⃗
(or Eq. (D.49) with F replaced by ϕ). Practically, the representation {|en )}
seems to be easier to work with as it makes the generalization from finite dimensional spaces to infinite dimensional spaces more direct and explicit. Thus
we can first work things out in the representation {|en )} and then transform into
other representations. For example, if we formally define
the integration mea∫
sure in the representation {|en )} as D({ϕn }) = Πn dϕn , then the integration
measure in the space configuration
representation can be defined indirectly as
∫
⃗
⃗ x)
D[ϕ(⃗x)] = D({ϕn }) = Πn dϕn , where ϕn is regarded as a functional of ϕ(⃗
⃗ x)] in
given by Eq. (D.55). This might avoid some issues involved in defining D[ϕ(⃗
the space configuration representation directly by discretizing the physical space
and then taking the continuum limit.
We mention that the entire formalism developed here in this appendix can
also accommodate the treatment of spatially homogeneous systems with finite
degrees of freedom. In that case, the state space is assumed to be a Euclidean
space, which is a finite-dimensional real Hilbert space. The state of the system
⃗q = (q1 , ..., qi , ..., qn ) can be expressed in the Dirac bra-ket notation as qi = (ei |q),
where |q) is the abstract representation of the state and |ei ) is the base vector of
the state space onto which it projects. The orthonormality and completeness conditions for the basis {|ei )} are formally the same as those in Eq. (D.43), with the
restriction that the index i runs from 1 and n rather than going into infinity, since
the state space is now finite-dimensional. The results for spatially homogeneous
systems can thus be recovered from the abstract representation. Therefore, this
formalism provides a unified language for the treatment of both spatially homogeneous and inhomogeneous systems.
201
Appendix E
Ornstein-Uhlenbeck Processes for
Spatially Inhomogeneous Systems
This appendix is supplementary to Sec. 3.3.5 of the main text. We give the
abstract representation of the dynamical equations and the space configuration
representation of the thermodynamic expressions of OU processes of spatially
inhomogeneous systems.
E.1
Dynamical Equations in the Abstract Representation
We list in the following the compact form of some dynamical equations and
expressions for OU processes of spatially inhomogeneous systems in the abstract
representation using Dirac bra-ket notations. The functional Langevin equation
for OU processes reads as follows in Dirac notation:
d
|ϕ) = −γ̂|ϕ) + |ξ(t)),
(E.1)
dt
where −γ̂|ϕ) is the deterministic driving force |F (ϕ)) and |ξ(t)) is the stochastic
driving force with the following Gaussian white noise statistical properties
⟨|ξ(t))⟩ = 0,
⟨|ξ(t))(ξ(t′ )|⟩ = 2D̂δ(t − t′ ),
(E.2)
where the diffusion operator D̂ does not depend on ϕ. The corresponding functional Fokker-Planck equation is
[
]
d
Pt (ϕ) = (δϕ |γ̂ |ϕ)Pt (ϕ) + (δϕ |D̂|δϕ )Pt (ϕ).
(E.3)
dt
202
Compare with the general equation given in Eq. (D.3). The fact that γ̂ and D̂ do
not depend on ϕ allowed the form of the equation to be simplified a little bit. The
transient probability functional is
]
[
1
1
−1
(E.4)
Pt (ϕ) = √
exp − (ϕ − K|σ̂ |ϕ − K) .
2
det(2πσ̂)
The stationary probability functional is
Ps (ϕ) = √
1
det(2π σ)
ê
[
]
−1
1
exp − (ϕ|σ
ê |ϕ) .
2
(E.5)
The equations determining |K) and σ̂ are:
d
|K) = −γ̂|K)
dt
d
σ̂ = −γ̂ σ̂ − γ̂ | σ̂ + 2D̂.
dt
(E.6)
(E.7)
The equation determining σ
ê is
γ̂ σ
ê + γ̂ | σ
ê = 2D̂.
(E.8)
The potential field landscapes are given by
)
−1
1
1 (
U (ϕ) =
(ϕ|σ
ê |ϕ) + tr ln(2π σ)
ê
(E.9)
2
2
1
1
S(ϕ) =
(ϕ − K|σ̂ −1 |ϕ − K) + tr (ln(2πσ̂))
(E.10)
2
2
)
−1
1
1
1 (
A(ϕ) =
(ϕ|σ
ê |ϕ) − (ϕ − K|σ̂ −1 |ϕ − K) + tr ln σ
ê − ln σ̂ (E.11)
.
2
2
2
Their functional gradients are
−1
|δϕ U (ϕ)) = σ
ê |ϕ)
|δϕ S(ϕ)) = σ̂ −1 |ϕ − K)
(E.12)
(E.13)
−1
|δϕ A(ϕ)) = σ
ê |ϕ) − σ̂ −1 |ϕ − K).
(E.14)
The flux velocity fields are given by
−1
|Vs (ϕ)) = −γ̂|ϕ) + D̂σ
ê |ϕ)
|Vt (ϕ)) = −γ̂|ϕ) + D̂σ̂ −1 |ϕ − K)
−1
|Vr (ϕ)) = D̂σ̂ −1 |ϕ − K) − D̂σ
ê |ϕ).
203
(E.15)
(E.16)
(E.17)
E.2
Thermodynamic Expressions in the Space Configuration Representation
In the following, we give the explicit expressions of some thermodynamic
quantities for OU processes of spatially inhomogeneous systems in the space configuration representation in the functional language. For comparison, we also
write down the compact form of these thermodynamic quantities in the abstract
representation using the Dirac notation. The expression in the space configuration
representation can be obtained by inserting, into appropriate places of the abstract
representation, resolution of identity in the space configuration basis:
∫
∑
ˆ
I = d⃗x
|a, ⃗x)(a, ⃗x|.
(E.18)
a
For example, the rate of change of the transient entropy Ṡ of OU processes is
transformed from the abstract representation into the space configuration representation as follows:
(
)
Ṡ = tr σ̂ −1 D̂ − γ̂
(E.19)
)
((∫
)
(∫
∑
∑
|b, ⃗x ′ )(b, ⃗x ′ | D̂
= tr
d⃗x
|a, ⃗x)(a, ⃗x| σ̂ −1
d⃗x ′
a
(∫
−
d⃗x
∫∫
=
a
d⃗xd⃗x ′
∫
−
d⃗x
∫∫
=
∑
b
|a, ⃗x)(a, ⃗x| γ̂
∑
(
)
tr |a, ⃗x)(a, ⃗x|σ̂ −1 |b, ⃗x ′ )(b, ⃗x ′ |D̂
ab
tr (|a, ⃗x)(a, ⃗x|γ̂)
a
d⃗xd⃗x
∫∫
=
∑
) )
′
∑
−1
′
′
(a, ⃗x|σ̂ |b, ⃗x )(b, ⃗x |D̂|a, ⃗x) −
∫
d⃗x
∑
(a, ⃗x|γ̂|a, ⃗x)
∫
∑
∑[
]
′
ba
′
′
−1 ab
(⃗x, ⃗x )D (⃗x , ⃗x) − d⃗x
γ aa (⃗x, ⃗x), (E.20)
d⃗xd⃗x
σ
a
ab
a
ab
where we have used the formula tr(|ϕ)(φ|B̂) = (φ|B̂|ϕ). The expression of other
thermodynamic quantities in the space configuration representation can be derived
204
similarly by inserting the identity in Eq. (E.18) and using that trace formula. We
just list the results without going through the derivation again.
(
)
−1
−1
1
d −1
U̇ = tr
(σ̂ + |K)(K| − σ)
ê σ
ê − σ
ê D̂σ
ê (σ̂ + |K)(K|) + γ̂
(E.21)
2
dt
∫∫
∑1[
]
=
d⃗xd⃗x ′
σ ab (⃗x, ⃗x ′ ) + K a (⃗x)K b (⃗x ′ ) − σ
eab (⃗x, ⃗x ′ )
2
ab
d [ −1 ]ba ′
×
σ
e
(⃗x , ⃗x)
∫dt∫ ∫ ∫
∑[
]ab
[ −1 ]cd ′′ ′′′
−
d⃗xd⃗x ′ d⃗x ′′ d⃗x ′′′
σ
e−1 (⃗x, ⃗x ′ )Dbc (⃗x ′ , ⃗x ′′ ) σ
e
(⃗x , ⃗x )
×
(
abcd
da
′′′
d
′′′
a
)
σ (⃗x , ⃗x) + K (⃗x )K (⃗x) +
∫
d⃗x
∑
γ aa (⃗x, ⃗x)
(E.22)
a
(
−1
−1
1
d −1
(σ̂ + |K)(K| − σ)
ê σ
ê − σ
ê D̂σ
ê (σ̂ + |K)(K|)
2
dt
)
(E.23)
σ̂ −1 D̂ + 2γ̂
∫∫
∑1[
]
d⃗xd⃗x ′
σ ab (⃗x, ⃗x ′ ) + K a (⃗x)K b (⃗x ′ ) − σ
eab (⃗x, ⃗x ′ )
2
ab
d [ −1 ]ba ′
σ
e
(⃗x , ⃗x)
∫dt∫ ∫ ∫
∑[
]ab
[ −1 ]cd ′′ ′′′
d⃗xd⃗x ′ d⃗x ′′ d⃗x ′′′
σ
e−1 (⃗x, ⃗x ′ )Dbc (⃗x ′ , ⃗x ′′ ) σ
e
(⃗x , ⃗x )
Ȧ = tr
−
=
×
−
×
(
abcd
′′′
′′′
)
σ (⃗x , ⃗x) + K (⃗x )K (⃗x) −
da
× Dba (⃗x ′ , ⃗x) + 2
d
∫
d⃗x
∑
a
∫∫
d⃗xd⃗x ′
∑[
]ab
σ −1 (⃗x, ⃗x ′ )
ab
γ aa (⃗x, ⃗x)
(E.24)
a
Ṡed
( (
) d −1 )
1
σ̂ + |K)(K| − σ
ê
σ
ê
= tr
(E.25)
2
dt
∫∫
∑1[
]
=
d⃗xd⃗x ′
σ ab (⃗x, ⃗x ′ ) + K a (⃗x)K b (⃗x ′ ) − σ
eab (⃗x, ⃗x ′ )
2
ab
d [ −1 ]ba ′
×
(⃗x , ⃗x)
(E.26)
σ
e
dt
205
( −1 −1
)
Ṡex = tr σ
ê D̂σ
ê (σ̂ + |K)(K|) − γ̂
(E.27)
∫∫∫∫
∑[
]ab
=
d⃗xd⃗x ′ d⃗x ′′ d⃗x ′′′
σ
e−1 (⃗x, ⃗x ′ )Dbc (⃗x ′ , ⃗x ′′ )
abcd
[ −1 ]cd ′′ ′′′ ( da ′′′
)
× σ
e
(⃗x , ⃗x ) σ (⃗x , ⃗x) + K d (⃗x ′′′ )K a (⃗x)
∫
∑
−
d⃗x
γ aa (⃗x, ⃗x)
(E.28)
a
(
)
Ṡf l = tr γ̂ | D̂−1 γ̂ (σ̂ + |K)(K|) − γ̂
(E.29)
∫∫∫∫
∑
[
]bc
=
d⃗xd⃗x ′ d⃗x ′′ d⃗x ′′′
γ ba (⃗x ′ , ⃗x) D−1 (⃗x ′ , ⃗x ′′ )γ cd (⃗x ′′ , ⃗x ′′′ )
×
(
abcd
)
σ (⃗x , ⃗x) + K (⃗x )K (⃗x) −
da
′′′
d
′′′
∫
a
d⃗x
∑
γ aa (⃗x, ⃗x)
(E.30)
a
(
)
Ṡpd = tr γ̂ | D̂−1 γ̂ (σ̂ + |K)(K|) + σ̂ −1 D̂ − 2γ̂
(E.31)
∫∫∫∫
∑
[
]bc
γ ba (⃗x ′ , ⃗x) D−1 (⃗x ′ , ⃗x ′′ )γ cd (⃗x ′′ , ⃗x ′′′ )
=
d⃗xd⃗x ′ d⃗x ′′ d⃗x ′′′
×
(
abcd
)
σ (⃗x , ⃗x) + K (⃗x )K (⃗x) +
da
′′′
× Dba (⃗x ′ , ⃗x) − 2
d
∫
d⃗x
′′′
∑
a
∫∫
d⃗xd⃗x ′
∑[
σ −1
]ab
(⃗x, ⃗x ′ )
ab
γ aa (⃗x, ⃗x)
(E.32)
a
((
)
)
−1
−1
Ṡad = tr γ̂ | D̂−1 γ̂ − σ
ê D̂σ
ê
(σ̂ + |K)(K|)
(E.33)
∫∫∫∫
∑{
[
]bc
γ ba (⃗x ′ , ⃗x) D−1 (⃗x ′ , ⃗x ′′ )γ cd (⃗x ′′ , ⃗x ′′′ )
=
d⃗xd⃗x ′ d⃗x ′′ d⃗x ′′′
abcd
[ −1 ]cd ′′ ′′′ }
[ −1 ]ab
′
bc
′
′′
(⃗x , ⃗x )
(⃗x, ⃗x )D (⃗x , ⃗x ) σ
e
− σ
e
( da ′′′
)
× σ (⃗x , ⃗x) + K d (⃗x ′′′ )K a (⃗x)
206
(E.34)
( −1 −1
)
Ṡna = tr σ
ê D̂σ
ê (σ̂ + |K)(K|) + σ̂ −1 D̂ − 2γ̂
(E.35)
∫∫∫∫
∑[
]ab
=
d⃗xd⃗x ′ d⃗x ′′ d⃗x ′′′
σ
e−1 (⃗x, ⃗x ′ )Dbc (⃗x ′ , ⃗x ′′ )
abcd
[ −1 ]cd ′′ ′′′ ( da ′′′
)
× σ
e
(⃗x , ⃗x ) σ (⃗x , ⃗x) + K d (⃗x ′′′ )K a (⃗x)
∫∫
∑[
]ab
+
d⃗xd⃗x ′
σ −1 (⃗x, ⃗x ′ )Dba (⃗x ′ , ⃗x)
∫
− 2
d⃗x
∑
ab
γ aa (⃗x, ⃗x)
a
207
(E.36)
Appendix F
Spatial Stochastic Neuronal Model
This appendix is supplementary to Sec. 3.3.6 of the main text. In the following
we calculate the potential field landscapes, their functional gradients, and the flux
velocity fields of the spatial stochastic neuronal model in the space configuration
representation in the basis of {en (x)}.
Note that the abstract representation of en (x) using the Dirac notation is |en ).
en (x) can be obtained from its abstract representation |en ) by projecting it onto the
space configuration basis |x), i.e., en (x) = (x|en ). Therefore, the basis {en (x)}
can also be abstractly represented as {|en )}. We list the expressions of the vector
component and matrix elements of the involved states and linear operators in the
basis {|en )}, which was derived in the main text and will be used here:
γmn = (em |γ̂|en ) = γn δmn ,
Kn = (en |K) = e−γn t fn ,
σ
emn = (em |σ|e
ê n ) =
δmn
,
2 −2γ t
1−e n
= (em |σ̂|en ) =
δmn ,
2γn
Dmn = (em |D̂|en ) =
σmn
δmn
.
2γn
(F.1)
We start from the expressions of the potential and flux velocity fields for OU
processes in the abstract representation given by Eqs. (E.9)-(E.17). Then we go
into the ∑
representation of {|en )} by inserting resolution of identity in the basis of
ˆ into places where appropriate. The stationary potential
{|en )}: n |en )(en | = I,
208
field landscape is calculated using Eq. (E.9):
)
−1
1
1 (
U [ϕ] =
(ϕ|σ
ê |ϕ) + tr ln(2π σ)
ê
2
2
(∞
)
∞
∑
−1
1∑
1
=
(ϕ|en )(en |σ
ê |ϕ) + tr
|en )(en | ln(2π σ)
ê
2 n=0
2
n=0
1∑
1∑
−1
(en |ϕ) +
(ϕ|en )e
σnn
(en | ln(2π σ)|e
ê n )
=
2 n=0
2 n=0
∞
∞
∞
∞
1 ∑ ϕ2n
1∑
ln(2πe
σnn )
+
2 n=0 σ
enn 2 n=0
]
∞ [
∑
1
π
2
2
=
(n + 1)ϕn + ln 2
,
2 n +1
n=0
=
(F.2)
−1
where we have used the fact that |en ) is also the eigenstate of the operator σ
ê
since this operator is diagonal in the basis of {|en )}; we have also used the property tr(|ϕ)(φ|B̂) = (φ|B̂|ϕ). Similarly, we calculate the transient potential field
landscape using Eq. (E.10):
1
1
(ϕ − K|σ̂ −1 |ϕ − K)) + tr (ln(2πσ̂))
2
2
(∞
)
∞
∑
∑
1
1
=
(ϕ|en )(en |σ̂ −1 |ϕ − K)) + tr
|en )(en | ln(2πσ̂)
2 n=0
2
n=0
S[ϕ] =
∞
∞
1 ∑ (ϕn − Kn )2 1 ∑
=
+
ln(2πσnn )
2 n=0
σnn
2 n=0
∞ [
)2
∑
(n2 + 1) (
−(n2 +1)t
ϕ n − fn e
=
1 − e−2(n2 +1)t
n=0
]
) 1
1 (
π
−2(n2 +1)t
+
ln 1 − e
+ ln 2
.
2
2 n +1
(F.3)
209
The relative potential field landscape is thus given by
A[ϕ] = U [ϕ] − S[ϕ]
∞ [
∑
=
(n2 + 1)ϕ2n −
)2
(n2 + 1) (
−(n2 +1)t
ϕ
−
f
e
n
n
1 − e−2(n2 +1)t
n=0
)]
1 (
−2(n2 +1)t
−
.
ln 1 − e
2
(F.4)
Next we calculate their functional gradients. Starting from the abstract representation of the functional gradient of the stationary potential field landscape
given in Eq. (E.12), we calculate its expression in the space configuration representation, but resolved in the basis of {en (x)}:
δU
= (x|δϕ U (ϕ))
δϕ(x)
−1
= (x|σ
ê |ϕ)
∞
∑
−1
=
(x|en )(en |σ
ê |ϕ)
=
n=0
∞
∑
en (x)
n=0
=
∞
∑
ϕn
σ
enn
2(n2 + 1)ϕn en (x),
(F.5)
n=0
where we have used en (x) = (x|en ). Similarly, using Eq. (E.13) we calculate the
functional gradient of the transient potential field landscape:
δS
= (x|σ̂ −1 |ϕ − K)
δϕ(x)
∞
∑
=
(x|en )(en |σ̂ −1 |ϕ − K)
n=0
=
∞
∑
n=0
=
∞
∑
n=0
en (x)
ϕn − Kn
σnn
2(n2 + 1)
−(n2 +1)t
(ϕ
−
f
e
) en (x).
n
n
2
1 − e−2(n +1)t
210
(F.6)
Thus the functional gradient of the relative potential field landscape is given by
δA
δU
δS
=
−
δϕ(x)
δϕ(x) δϕ(x)
[
]
∞
∑
2(n2 + 1)
2
−(n2 +1)t
=
2(n + 1)ϕn −
(ϕn − fn e
) en (x). (F.7)
−2(n2 +1)t
1
−
e
n=0
Then we calculate the flux velocity fields. Using Eq. (E.15), the stationary
flux velocity field is given by:
Vs (x)[ϕ] = (x|Vs (ϕ))
−1
= −(x|γ̂|ϕ) + (x|D̂σ
ê |ϕ)
∞
∞
∑
∑
−1
= −
(x|en )(en |γ̂|ϕ) +
(x|en )(en |D̂σ
ê |ϕ)
n=0
∞
∑
n=0
[
Dnn
=
en (x) −γnn ϕn +
ϕn
σ
enn
n=0
]
∞
∑
[
]
−(n2 + 1)ϕn + (n2 + 1)ϕn en (x) = 0.
=
(F.8)
n=0
Similarly, using Eq. (E.16), we have the transient flux velocity field given by:
Vt (x)[ϕ] = −(x|γ̂|ϕ) + (x|D̂σ̂ −1 |ϕ − K)
∞
∞
∑
∑
= −
(x|en )(en |γ̂|ϕ) +
(x|en )(en |D̂σ̂ −1 |ϕ − K)
n=0
∞
∑
[
n=0
]
Dnn
=
en (x) −γnn ϕn +
(ϕn − Kn )
σ
nn
n=0
]
∞ [
∑
(n2 + 1)
2
−(n2 +1)t
=
−(n + 1)ϕn +
(ϕn − fn e
) en (x). (F.9)
1 − e−(n2 +1)t
n=0
Therefore, the relative flux velocity field is given by:
Vr (x)[ϕ] = Vt (x)[ϕ] − Vs (x)[ϕ]
]
∞ [
∑
(n2 + 1)
2
−(n2 +1)t
=
−(n + 1)ϕn +
(ϕn − fn e
) en (x). (F.10)
−(n2 +1)t
1
−
e
n=0
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