PHYSICAL REVIEW E 80, 031145 共2009兲
Finite bath fluctuation theorem
Michele Campisi,* Peter Talkner, and Peter Hänggi
Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86153 Augsburg, Germany
共Received 30 April 2009; revised manuscript received 21 July 2009; published 28 September 2009兲
We demonstrate that a finite bath fluctuation theorem of the Crooks type holds for systems that have been
thermalized via weakly coupling them to a bath with energy independent finite specific heat. We show that this
theorem reduces to the known canonical and microcanonical fluctuation theorems in the two respective limiting
cases of infinite and vanishing specific heat of the bath. The result is elucidated by applying it to a twodimensional hard disk colliding elastically with few other hard disks in a rectangular box with perfectly
reflecting walls.
DOI: 10.1103/PhysRevE.80.031145
PACS number共s兲: 05.20.Gg, 05.70.Ln, 05.40.⫺a
I. INTRODUCTION
During the last decade a number of fluctuation theorems
have been reported in the literatures, which have contributed
a good deal to a better understanding of nonequilibrium thermodynamics 关1–8兴. These can be roughly divided in two
categories: steady state fluctuation theorems and transient
fluctuation theorems. The former apply to systems in nonequilibrium steady states and give information on the system
fluctuations in the asymptotic regime of very large times 共see
关9–11兴 for reviews on this topic兲. The latter apply to systems
that are temporarily driven out of equilibrium and give information about the fluctuations of work generated by the driving forces. The most representative example of the latter kind
is the Crooks fluctuation theorem 关5,6兴, that applies to systems that are initially in a canonical state. Although the canonical case is by far the most common case, one may need
to study situations where the system is initially distributed
according to some other statistics, instead. For example the
system might be initially a microcanonical state of well defined energy. In this latter case it has been shown that a
microcanonical fluctuation theorem of the type of Crooks
also exists 关12,13兴. One naturally then wonders whether the
same type of transient fluctuation theorem exists as well for
yet other types of statistics.
In this work we focus on the probability distribution function 共pdf兲 that describes the statistics of a subsystem of a
total classical ergodic system with fixed energy. For the case
where the interaction between the subsystem and the rest of
the total system 共which we will refer to as the bath兲 is weak,
and the specific heat of the bath is independent of the energy
共as for an ideal gas, or for a bath composed of hard spheres兲,
the derivation of the pdf is a standard problem of statistical
mechanics 关14,15兴. We make no assumptions regarding the
size of the bath; in particular we do not assume that it is
much larger or smaller than that of the system of interest as
assumed in the canonical and microcanonical cases respectively. For this reason we refer to this type of bath as to a
finite heat bath, and to the statistics of the subsystem as to
the finite bath statistics 关see Eq. 共6兲 below兴. For this statistics
we show that a fluctuation theorem of the type of Crooks,
*[email protected]
1539-3755/2009/80共3兲/031145共9兲
i.e., a finite bath fluctuation theorem holds. This finite bath
fluctuation theorem includes the Crooks canonical fluctuation theorem and the microcanonical fluctuation theorem, as
the two limiting cases in which the bath specific heat goes to
infinity and zero, respectively.
The present work is organized as follows. In Sec. II we
review the derivation of finite bath statistics and recall some
of its properties. In Sec. III we derive the corresponding
finite bath fluctuation theorem, and show that it reduces to
microcanonical and canonical fluctuation theorems in the
limits of vanishing and infinite baths, respectively. In Sec. IV
we apply the theory to a specific example 关i.e., a twodimensional 共2D兲 hard disk elastically colliding with few
other hard disks in a box兴 and test the validity of the finite
bath fluctuation theorem, both analytically and numerically.
Sec. V contains a discussion of the obtained results. The
conclusions are drawn in Sec. VI.
II. FINITE BATH STATISTICS
Let us consider a finite classical Hamiltonian system of
total Ntot particles and total energy Etot composed of two
weakly interacting subsystems: the “system of interest” 共or
simply the system兲 and the “bath.” Assuming that the total
system is ergodic, the pdf of the system is given in terms of
density of states, ⍀B共E兲, of the bath and density of states of
the total system, ⍀tot共E兲, as 关16兴:
␳共z,␭兲 =
⍀B„Etot − H共z,␭兲…
⍀tot共Etot兲
共1兲
where z = 共p1 , . . . , ps , q1 , . . . , qs兲 stands for the 2s dimensional
phase space point of the system. Here we assume that the
共sub兲system Hamiltonian H共z , ␭兲 may depend on some externally controllable parameter ␭ 共this could be for instance
the volume of a vessel that contains the system, or an applied
magnetic or electric field兲. For example, in the case of a bath
composed of n hard spheres in three dimensions, it is
⍀B共EB兲 ⬀ EB3n/2−1 共see Appendix A兲, and one finds from Eq.
共1兲 关14兴:
␳共z,␭兲 =
冕
关Etot − H共z,␭兲兴+3n/2−1
dz关Etot −
共2兲
H共z,␭兲兴+3n/2−1
which is a known result of classical statistical mechanics
关17兴. The symbol 关x兴+ is defined as 关x兴+ ª x␪共x兲, with ␪共x兲
031145-1
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 031145 共2009兲
CAMPISI, TALKNER, AND HÄNGGI
denoting Heaviside step function. Note that, in this case, the
specific heat of the bath C共EB兲, is energy independent and
equal to 3n / 2 关18兴. More generally one has the following
theorem 关19,20兴:
Theorem 1 The system pdf is given by
␳共z,␭兲 =
冕
关Etot − H共z,␭兲兴+C−1
dz关Etot −
共3兲
H共z,␭兲兴+C−1
if and only if the specific heat of the bath C is energy independent.
Here C is the microcanonical specific heat of the bath, i.e.,
C共EB兲 ª
冉
⳵
TB共EB兲
⳵ EB
冊
−1
共4兲
where EB is the energy, and TB共EB兲 ª ⌽B共EB兲 / ⍀B共EB兲 is the
microcanonical temperature expressed in terms of the phase
space volume ⌽B共EB兲 of the bath, with energy below EB, and
the bath density of states ⍀B共EB兲 = ⳵⌽B共EB兲 / ⳵EB. In the following of this work we restrict ourselves to the case of energy independent, positive specific heat of the bath, abbreviated as C共EB兲 ª C ⬎ 0.
Note that the pdfs in Eq. 共3兲 are parametrized via the total
system energy Etot. It is however convenient to parametrize
the pdfs via a property that pertains to the subsystem only,
e.g., its average energy U. This is accomplished by writing
Etot = U + CT 共here CT represents the average energy of the
bath兲, substituting this expression in Eq. 共3兲, and imposing
that U = 兰dzH共z ; ␭兲␳共z , ␭兲. This leads to solving the following equation for T, given the average energy U and ␭
冕
dzH共z;␭兲†1 − 关H共z;␭兲 − U兴/共CT兲‡+C−1
冕
=U
dz†1 − 关H共z;␭兲 −
共5兲
U兴/共CT兲‡+C−1
We shall denote the value of T that satisfies Eq. 共5兲 for given
U and ␭ as T共U , ␭兲 关in Appendix B we prove that a solution
T共U , ␭兲 always exists兴. With this function at hand we can
parametrize the pdfs in Eq. 共3兲 via the subsystem average
energy U and recast them in the form:
†1 − 关H共z;␭兲 − U兴/关CT共U,␭兲兴‡+C−1
␳C共z;U,␭兲 =
NC共U,␭兲
冕
A. Remark
By expressing the specific heat C as C = 1 / 共1 − q兲 ⬎ 0 one
recognizes that the pdf in Eq. 共6兲 is the Tsallis escort pdf of
index q with q ⬍ 1 关22兴. Note that these do not exhibit heavy
tails but rather have a faster than exponential decay with a
finite cutoff occurring at the energy U + CT = Etot. The physical meaning of this cutoff energy is that the system’s energy
cannot be larger than the total energy.
B. Properties
1. Equipartition
The following equipartition theorem holds for the finite
bath statistics in Eq. 共6兲 关22兴:
冓 冔
pi
⳵H
⳵ pi
共6兲
dz†1 − 关H共z;␭兲 − U兴/关CT共U,␭兲兴‡+C−1 . 共7兲
As discussed in Appendix B it is not always possible to
invert T共U , ␭兲. For sake of simplicity, in this work we shall
assume that T共U , ␭兲 is invertible with respect to the argument U. This means that we could also choose T as an independent parameter and express U as a function of T and ␭.
Thus we are free to choose between two possible parameterizations: a microcanonical-like parameterization 共or U parameterization兲, and a canonical-like parameterization 共or T
parameterization兲 关21兴.
共8兲
= T共U,␭兲
where 具 · 典 denotes average over ␳C in Eq. 共6兲, pi is one of the
momenta and repeated indices are not summed. Equation 共8兲
says that T共U , ␭兲 can be interpreted as the temperature of the
system.
2. Heat theorem
The finite bath statistics provides a mechanical model of
thermodynamics 关23兴, meaning that the temperature T, the
external parameter ␭, its conjugated generalized force f ␭, and
the average energy U are related in such a way as to satisfy
the heat theorem 关24兴:
dU + f ␭d␭
= exact differential
T
共9兲
where f ␭ is defined in the usual way as:
f␭ = −
where NC共U , ␭兲 is the normalization:
NC共U,␭兲 =
We shall refer to the numerator in Eq. 共6兲 as to a “generalized Boltzmann factor.” It is important to stress that a factor of the type 关1 − 共H共z ; ␭兲 − U兲 / 共CT兲兴+C−1 is a generalized
Boltzmann factor only if T = T共U , ␭兲, in agreement with Eq.
共5兲.
冓 冔
⳵H
.
⳵␭
共10兲
This property is an important one because it allows to determine the thermodynamic entropy associated with the finite
bath statistics by finding the integral of the exact differential.
This is given by 关24兴:
SC共U,␭兲 = ln NC共U,␭兲.
共11兲
3. Interpolation
The pdfs in Eq. 共6兲 interpolate between canonical and
microcanonical ensembles. Using the limits of infinite and
null specific heat C, i.e.,
031145-2
冋 册
lim 1 +
C→⬁
x
C
C−1
冋 册
= lim 1 +
+
C→⬁
x
C
C
= e x;
+
共12兲
PHYSICAL REVIEW E 80, 031145 共2009兲
FINITE BATH FLUCTUATION THEOREM
冋 册
冋 册
lim 1 +
C→⬁
x
C
x
lim 1 +
C
C→⬁
C
= ␪共x兲;
共13兲
C−1
= ␦共x兲
C→0
e−H共z;␭兲/T
Z共T,␭兲
共15兲
␦„U − H共z;␭兲…
⍀共U,␭兲
,
共16兲
NC,0共U兲pC,U
t ,t 共W兲 =
f 0
lim NC共T,␭兲 =
C→⬁
冕
dze
lim NC共U,␭兲 =
C→0
=e
U/T
Z共T,␭兲
冉
H f 共z f 兲 − 共U + W兲
CT0共U兲
T0共U兲 + ␦T
T0共U兲
冕
⫻
册
C−1
共22兲
+
冊
C−1
dz f ␦共H0共z0兲 − H f 共z f 兲 + W兲
冋
⫻ 1−
共17兲
dz = ⌽共U,␭兲.
共18兲
H共z;␭兲ⱕU
H f 共z f 兲 − 共U + W − C␦T兲
C共T0共U兲 + ␦T兲
册
C−1
.
+
We now choose ␦T as the solution of the following integral
equation:
冕
The quantity ⌽共U , ␭兲 is the volume of system phase space
with energy below U. The density of states is related to ⌽ via
a partial derivative ⍀ = ⳵⌽ / ⳵U. By taking the logarithm one
recovers canonical and microcanonical entropies; i.e.,
C→⬁
冋
共23兲
冕
lim SC共T,␭兲 =
dz f ␦共H0共z0兲 − H f 共z f 兲 + W兲
where now z0 = z共t0 , t f , z f 兲, is the solution of Hamilton’s
equation with z f as initial condition and time running backward. Note that the second term in the integrand is not a
generalized Boltzmann factor because in general it does not
satisfy Eq. 共5兲. However for any ␦T one can rewrite the
previous equation as:
respectively 关20兴. The microcanonical normalization ⍀共U , ␭兲
is the system density of states. Likewise one has, for the
normalization, the following limits 关20兴:
−共H共z;␭兲−U兲/T
冕
⫻ 1−
共14兲
+
lim ␳C共z;T,␭兲 =
lim ␳C共z;U,␭兲 =
f 0
+
one recovers the canonical and microcanonical pdfs 关25兴:
C→⬁
NC,0共U兲pC,U
t ,t 共W兲 =
U
+ ln Z共T,␭兲
T
lim SC共U,␭兲 = ln ⌽共U,␭兲.
C→0
共19兲
dzH f 共z兲B共z,U,W, ␦T兲
冕
= U + W − C␦T
dzB共z,U,W, ␦T兲
where, for convenience, we use the notation
冋
B共z,U,W, ␦T兲 ª 1 −
共20兲
共24兲
H f 共z兲 − 共U + W − C␦T兲
C共T0共U兲 + ␦T兲
册
C−1
;
+
共25兲
or, equivalently, as a solution of
III. FLUCTUATION THEOREM
Consider an ensemble of systems distributed according to
Eq. 共6兲. Assume the system being decoupled from its bath
and that it is acted upon by an external force that changes the
external parameter ␭ according to some prescribed protocol
␭共t兲 executed between times t0 and t f . The probability density that the external force does a certain work W on the
system in that interval of time reads:
pC,U
t f ,t0 共W兲
ª
N−1
C,0共U兲
冋
⫻ 1−
冕
T0共U兲 + ␦T = T f 共U + W − C␦T兲.
Then, we find
NC,0共U兲pC,U
t ,t 共W兲 =
f 0
册
T f 共U + W − C␦T兲
T0共U兲
⫻
冕
冋
dz0␦共H f 共z f 兲 − H0共z0兲 − W兲
H0共z0兲 − U
CT0共U兲
冋
共21兲
H f 共z f 兲 − 共U + W − C␦T兲
T f 共U + W − C␦T兲
册
C−1
+
共27兲
+
where z f = z共t f , t0 , z0兲 is the solution of Hamilton’s equation
with initial condition z0. For simplicity of notation we drop
the variable ␭ in all quantities that depend on it, and replace
it with a subscript 0 or f, depending on whether the quantity
is taken at values of ␭ equal to ␭共t0兲 or ␭共t f 兲, e.g., H0共z兲
ª H关z , ␭共t0兲兴, T0共U兲 ª T关U , ␭共t0兲兴. By making the change of
variables from z0 → z f with a unitary Jacobian, one obtains
C−1
dz f ␦关H0共z0兲 − H f 共z f 兲 + W兴
⫻ 1−
C−1
册
共26兲
where the second term of the integrand is the Boltzmann
factor of the pdf ␳C(z ; U + W − C␦T , ␭共t f 兲). The integral is the
product of NC,f 共U + W − C␦T兲 and the probability
␦T
共−W兲 that the force performs the work −W when
pC,U+W−C
t0,t f
the protocol is run backward and the system is initially in the
state ␳C(z ; U + W − C␦T , ␭共t f 兲).
Therefore the following fluctuation theorem is obtained:
031145-3
PHYSICAL REVIEW E 80, 031145 共2009兲
CAMPISI, TALKNER, AND HÄNGGI
pC,U
t ,t 共W兲
f 0
f
pC,U
t ,t 共− W兲
=
0 f
冉 冊
Tf
T0
C−1
NC,f 共U f 兲
,
NC,0共U兲
IV. EXAMPLE: A 2D GAS OF HARD DISKS
共28兲
where
U f ª U + W − C␦T
共29兲
T f ª T f 共U f 兲.
共30兲
Using Eqs. 共11兲 and 共28兲 can be rewritten in terms of entropy
as:
pC,U
t ,t 共W兲
f 0
f
pC,U
t ,t 共− W兲
=
0 f
冉 冊
Tf
T0
In this section we illustrate the finite bath fluctuation theorem by applying it to a system composed of n + 1 elastically
colliding hard disks in a two-dimensional box with perfectly
reflecting walls. One disk will be our system of interest,
whereas the remaining n ones will form the bath. We assume
that the disks do not have rotational degrees of freedom. As
shown in the Appendix A, the specific heat is given in this
case by C = dn / 2 where d is the number of translational degrees of freedom of each disk. In this case d = 2, hence C
= n. Note the fact that C does not depend on energy.
C−1
exp关⌬SCf,0共U,W兲兴
共31兲
where ⌬SCf,0共U , W兲 = SC,f 共U f 兲 − SC,0共U兲.
The finite bath fluctuation theorem of Eq. 共31兲 allows to
calculate the ratios of probability of work done on the system
when it is driven arbitrarily away from equilibrium during
the action of the forward and backward protocol, in terms of
equilibrium properties such as entropy and temperature.
A. Probability density function
The energy of the system of interest is simply its kinetic
energy; i.e.,
H共px,py ;M兲 =
冋
␳C共px,py ;U,M兲 = NC−1共U,M兲 1 −
1. Limit of microcanonical ensemble
In the limit C → 0 Eq. 共22兲 becomes 关using the formula
␦共ax兲 = a−1␦共x兲, and Eqs. 共14兲 and 共18兲兴
f 0
冕
dz f ␦„H0共z0兲 − H f 共z f 兲 + W…
⫻␦„H f 共z f 兲 − 共U + W兲….
共32兲
Using the microcanonical equipartition theorem 关16兴
T共U , ␭兲 = ⌽共U , ␭兲 / ⍀共U , ␭兲, one recovers the microcanonical
fluctuation theorem 关12,13兴:
p0,U
t f ,t0共W兲
p0,U+W
t0,t f 共−
W兲
=
⍀ f 共U + W兲
.
⍀0共U兲
共33兲
Alternatively one can take the limit C → 0 of Eq. 共28兲 directly and obtain the expression T0共U兲⌽ f 共U + W兲 / 关T f 共U
+ W兲⌽0共U兲兴, which reduces to the previous one by virtue of
the microcanonical equipartition theorem.
共36兲
which fluctuates permanently due to the collisions with the
bath’s particles. According to Eq. 共6兲, the probability that the
disk has a given momentum 共px , py兲 is given by
Recovering known special cases
⌽0共U兲pC,U
t ,t 共W兲 = T0共U兲
p2x + p2y
,
2M
共p2x + p2y 兲/共2M兲 − U
CT共U,M兲
Likewise, using the T parameterization, it can be seen
that, in the limit C → ⬁ Eq. 共22兲 becomes
W/T
Z0共T兲pC,T
t ,t 共W兲 = e
f 0
冕
共38兲
T共U,M兲 = U,
and
冋 冉
␳C共px,py ;U,M兲 = NC−1共U,M兲 1 −
冊 册
p2x + p2y
− U /共CU兲
2M
f 0
p⬁,T
t0,t f 共−
W兲
=
Z f 共T兲 W/T
e .
Z0共T兲
共35兲
C−1
.
+
共39兲
Using Eq. 共7兲, with Eq. 共39兲 gives
共40兲
where A is the reduced volume 共i.e., area in this twodimensional case兲 of the box 共see the Appendix A for the
definition of reduced volume兲. From Eq. 共39兲, one obtains
the pdf of energy E of the disk:
冋
p共E;U兲 = U−1关1 + C−1兴−C 1 −
共34兲
p⬁,T
t ,t 共W兲
.
+
We consider the mass of the disk M as an external parameter
that can be changed at will in the course of time according to
prespecified protocols. The function T共U , M兲 has to be computed via Eq. 共5兲. In general, the solution of Eq. 共5兲 with a
purely kinetic Hamiltonian with s translational degrees of
freedom gives the usual equipartition of energy 关22兴:
T共U , M兲 = 2U / s. In the specific case of Eq. 共36兲 s = 2, hence
dz f ␦共H0共z0兲 − H f 共z f 兲 + W兲e−H f 共z f 兲/T
One thus obtains the fluctuation theorem for the canonical
ensemble of Crooks 关5,6兴
C−1
共37兲
NC共U,M兲 = 2␲A关1 + C−1兴CMU
2. Limit of canonical ensemble
册
共E − U兲
CU
册
C−1
.
共41兲
+
Interestingly, the energy pdf does not depend on the mass M.
In Fig. 1 we compare Eq. 共41兲, with the result of various
numerical simulations with C = 1 , 2 , 3 , 4. Note that for C = 1
the distribution is flat, for C = 2 it is linear, for C = 3 it is
quadratic etc. In view of theorem 1, the impressive agreement between theory and numerics corroborates the validity
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FINITE BATH FLUCTUATION THEOREM
1.2
pE;U molkJ
1.0
0.8
冉 冊
Tf
T0
0.2
0.0
−1
pC,U
t ,t 共W兲 = NC,0共U兲A
f 0
0.0
冉 冊
NC,f 共U f 兲
Uf
=
NC,0共U兲
U
From Eq. 共21兲 we have:
0.6 0.4
C−1
0.5
1.0
1.5
2.0
2.5
3.0
E kJmol
FIG. 1. 共Color online兲 Energy pdf for a 2D hard disk of radius
r = 1 nm and mass M = 2 amu, in a bath composed of 1共쎲兲, 2共䊏兲,
3共⽧兲, 4共䉱兲 other identical disks. The dots represent histograms of
properly normalized relative frequencies from numerical simulations. All simulations were carried out for the same total energy
Etot = 3.3469 kJ/ mol, which corresponds to measured average energies of the disk of interest U1 = 1.67166 kJ/ mol, U2
= 1.11644 kJ/ mol, U3 = 0.835784 kJ/ mol, U4 = 0.671521 kJ/ mol.
The solid lines represent the pdf predicted by the theory 关Eq. 共41兲兴
for the measured average energies Ui , i = 1 . . . 4.
of the assumed ergodic hypothesis for this model system.
Similar simulations have been reported in 关26兴 for a onedimensional harmonic oscillator coupled to a bath of n onedimensional quartic oscillators. In that case the density of
states of the bath is proportional to E共3n−2兲/4, and accordingly
the specific heat, C = 共3n + 2兲 / 4, is energy independent.
␦T = W/共1 + C兲
共42兲
hence from Eq. 共29兲 we obtain
共43兲
U f = T f = U + W/共1 + C兲.
Using Eq. 共40兲 with Eq. 共43兲 we obtain the normalizations of
the equilibrium pdfs with average energy and external parameters 共U , M 0兲 and 共U f , M f 兲, respectively,
NC,0共U兲 = 2␲A关1 + C−1兴CM 0U
冉
NC,f 共U f 兲 = 2␲A关1 + C−1兴CM f U +
Using Eqs. 共43兲–共45兲 we find
共44兲
冊
W
.
1+C
共45兲
Mf
.
M0
冉
冊 册
共46兲
p2x + p2y p2x + p2y
−
−W
2M f
2M 0
p2x + p2y
− U /共CU兲
2M 0
冊
C−1
共47兲
+
where we use the fact that the momentum 共px , py兲 is a constant of motion. By applying the change of variable E = 共p2x
+ p2y 兲 / 共2M 0兲, and employing Eq. 共44兲 we obtain
−1
−1 −C
pC,U
t ,t 共W兲 = U 关1 + C 兴
f 0
冋 冉
⫻ 1−
Mf
兩M 0 − M f 兩
冊 册
Mf
W − U /共CU兲
M0 − M f
C−1
. 共48兲
+
Similarly one finds the backward pdf of work
−1
−1 −C
f
pC,U
t ,t 共− W兲 = U f 关1 + C 兴
0 f
冋 冉
⫻ 1−
M0
兩M f − M 0兩
冊 册
M0
W − U f /共CU f 兲
M0 − M f
C−1
.
+
共49兲
Taking the ratio of Eq. 共48兲 and Eq. 共49兲 we obtain:
pC,U
t ,t 共W兲
f 0
f
pC,U
t0,t f 共−
B. Analytical test of the finite bath fluctuation theorem
Consider a protocol M共t兲 that changes the mass of the
disk from the value M 0 = M共t0兲 to M f = M共t f 兲. According to
the general assumption of our derivation, the system is decoupled from the bath during the action of the protocol. We
are interested in checking the validity of Eq. 共28兲. To this end
we need to compute the forward pdf of work, pC,U
t f ,t0 共W兲, the
f 共−W兲, and the starting average
backward pdf of work pC,U
t0,t f
energy of the backward protocol U f , given the starting average energy of the forward protocol U. Solving Eq. 共26兲 with
Eq. 共38兲 关note that Eq. 共38兲 does not depend on the value of
M, hence T f 共U兲 = T0共U兲 = U兴 we arrive at:
冋 冉
⫻ 1−
冕
dpxdpy␦
C
W兲
=
冉 冊
Uf
U
C
Mf
.
M0
共50兲
By comparison with Eq. 共46兲 we see that the finite bath fluctuation theorem of Eq. 共28兲 is satisfied.
C. Numerical check of the finite bath fluctuation theorem
In order to check numerically the validity of Eq. 共50兲 we
simulated the forward work pdf pC,U
t f ,t0 共W兲 for a bath of n = C
2D disks, a given value of U and a protocol that changes the
mass of the disk from M 0 to M f = 2M 0. The pdf for the numerical work is calculated as follows. We first run a simulation of the motion of the disk with fixed U and M 0. We then
construct a histogram that counts the number of occurrences
of energy in the intervals In = 关En − ⌬E / 2 , En + ⌬E / 2兲 for a
certain ⌬E 共in our simulations, typically, ⌬E = 0.1 kJ/ mol,
for a total of about 20 intervals and the histogram counts a
total of about 105 events兲. This provides us with the starting
statistics. At this point, we note that, independent of the functional form of M共t兲, acting the protocol on a particle with
energy E gives with probability 1 the work W = E共M 0
− M f 兲 / M f . The reason is that the time dependent system
Hamiltonian 共p2x + p2y 兲 / 关2M共t兲兴 generates the following equation of motion for the momenta: ṗx = ṗy = 0. Hence E共t f 兲
= 共p2x + p2y 兲 / 关2M共t f 兲兴 = E共t0兲M 0 / M f regardless of the details of
the protocol. So we immediately obtain a count of work belonging to the intervals Jn = 关Wn − ⌬W , Wn + ⌬W兲, where Wn
= En共M 0 − M f 兲 / M f and ⌬W = ⌬E共M 0 − M f 兲 / M f . After proper
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PHYSICAL REVIEW E 80, 031145 共2009兲
CAMPISI, TALKNER, AND HÄNGGI
C,U
pt f ,t0
C ,U f
2.0
␦T = W/Ctot
1.5
where Ctot is the total specific heat of the system+ bath compound system: Ctot ª s / 2 + C. This ␦T is therefore the increment of temperature that would result if, after having injected the energy W in the system of interest this is brought
back into contact with the bath and the compound system is
let reach thermal equilibrium. Recall that during the forcing
protocol we assumed that system and bath are decoupled. We
shall refer to this process as to the rethermalization. After
system and bath have rethermalized, the extra energy W,
initially stored in the system, will be shared between system
and bath according to the ratio of the respective specific
heats. In particular the bath gets the energy Q = C␦T, which is
indeed the heat that flows from the system to the bath during
rethermalization. Accordingly the system looses this amount
of energy and its change in energy becomes ⌬U = W − Q, in
agreement with the first law of thermodynamics. This means
that U f represents the average energy of the system after the
rethermalization. To summarize: 共a兲 the system is first in
thermal contact with the bath. Its average energy is Ui and
the temperature is Ti. 共b兲 the system is decoupled from the
bath and the forcing protocol is acted on it. As a result, the
energy W is injected in the system with a certain probability
density pC,U
t f ,t0 共W兲. 共c兲 The system 共carrying the extra energy
W兲, and bath 共still at temperature Ti兲 are now allowed to
rethermalize. During rethermalization the heat C␦T flows in
the bath, the system reaches the average energy U f , and the
new temperature T f is reached in the compound system.
Remarkably, the temperature change ␦T vanishes in the
canonical case: limC→⬁ ␦T = 0. However it is limC→⬁ C␦T
= W, meaning that the whole extra energy W injected in the
system, flows into the bath during rethermalization. However
this does not affect its temperature 共i.e., Ti = T f 兲, the specific
heat being infinite in the canonical case. Therefore the term
T f / T0 does not appear in the canonical fluctuation theorem of
Crooks. In fact the latter gives information about the freeenergy difference of two states with different parameter values, but same temperature. This is a much more fortunate
situation as compared to the finite bath and microcanonical
fluctuation theorems, in the sense that, in the canonical case,
one should not bother to start the backward process from the
“target” temperature T f 共which depends on W兲, but simply
starts it from the same temperature as that of the forward
process.
W
pt0 ,t f W
1.0
0.5
0.0
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
W kJ
kJmol
mol
FIG. 2. 共Color online兲 Comparison between the numerical values 共dots兲 and the theoretical expression in Eq. 共50兲 共continuous
f 共−W兲 for a 2D hard disk of mass M
line兲 of ptC,U
共W兲 / ptC,U
0
f ,t0
0,t f
= 2 amu in a bath composed of three hard disks of the same mass.
The initial energy is U = 0.831447 kJ/ mol and the protocol doubles
the mass of the disk.
normalization, this yields a histogram, labeled
hC,U
t ,t 共n兲
f 0
that provides a numerical estimate for pC,U
t f ,t0 共W兲. Next, for
each n, we simulate the motion of the disk with fixed parameters, M f = 2M 0 and Un = U + Wn / 共C + 1兲, and compute n difn
ferent histograms for the backward probabilities hC,U
t0,t f 共k兲 in
the same way as the forward histogram was computed. By
selecting the k = n value from each of the backward histograms and collecting them to form the new histogram
n
hC,U
t ,t 共n兲
0 f
f
we obtain a numerical estimate for pC,U
t0,t f 共−W兲. Finally, we
C,Un
C,U
compute the ratios ht ,t 共n兲 / ht ,t 共n兲.
f 0
0 f
These ratios are depicted in Fig. 2 along with the theoretical values given by Eq. 共50兲. The figure shows excellent
agreement between analytical theory and numerical experiment. The visible differences are within the statistical errors.
Note that, for the forward protocol, where the mass is increased by a factor 2, the work can only be negative and
vice-versa for the backward protocol. Therefore, the graph
shows only the negative values of nonequilibrium work W.
V. DISCUSSION
共51兲
B. Implications for the second law of thermodynamics
A. Physical meaning of ␦T
The basic quantity that enters the finite bath fluctuation
theorem, and marks a distinction with the canonical fluctuation theorem of Crooks Eq. 共35兲, is the quantity ␦T, defined
formally as the solution of Eq. 共26兲. This quantity enters in
the definition of U f and T f . What is the physical meaning of
these quantities? The hard sphere gas example turns useful in
addressing this question. Calculations analogous to those
leading to Eq. 共42兲 show that for a gas of hard spheres with
a total of s degrees of freedom, in contact with a bath with a
specific heat C, it is
From the canonical fluctuation theorem of Crooks, one
obtains, after proper algebraic manipulations, and integration
over W, the integral form of the fluctuation theorem, namely
the Jarzynski equality 具e−␤W典 = e−␤⌬F 关3兴, which implies the
second law in the form 具W典 ⱖ ⌬F. A similar integral equation
can be obtained for the finite bath fluctuation theorem too. It
reads
S f 共U f 兲
S0共U0兲
N具TC−1
典 = TC−1
0 e
f e
where
031145-6
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PHYSICAL REVIEW E 80, 031145 共2009兲
FINITE BATH FLUCTUATION THEOREM
Nª
冕
f
pU
t ,t 共W兲dW
共53兲
0 f
and 具 · 典 denotes average over the normalized distribution
f
qt0,t f 共W兲 ª pU
t0,t f 共W兲 / N. Equation 共53兲 generalizes both the
canonical Jarzynski equality and the microcanonical entropyfrom-work theorem 关13,27兴. Note that, as for the entropyfrom-work theorem, in general it is N ⫽ 1 because the energy
U f in Eq. 共53兲 is a function of W 共see Eq. 共29兲兲. As pointed
out already in 关27兴, this prevents obtaining the second law
directly from the integral form of the fluctuation theorem.
Nevertheless the validity of the second law of thermodynamics for a driving protocol acting on a system that is initially thermalized with a finite bath, can be proved directly
without invoking the finite bath fluctuation theorem. To this
end it is sufficient to recall the content of two theorems
which have been recently reported in the literature 关28–30兴.
According to these theorems, the second law of thermodynamics, in either the minimal work principle form, or the
entropy increase form of Clausius, is obeyed whenever the
initial phase space pdf ␳共z兲 is a decreasing function of energy, namely ␳共z兲 ⱖ ␳共z⬘兲, for every z , z⬘ such that H共z兲
ⱕ H共z⬘兲. This condition is obeyed by the finite bath statistics,
if the condition C ⱖ 1 is met 共see Eq. 共3兲兲. In this regard we
notice that this condition only is violated in the extremal case
when the bath consists of a single degree of freedom 共in
which case it is C = 1 / 2兲, or if there is no bath at all 共C = 0,
microcanonical case兲.
The Crooks fluctuation theorem Eq. 共35兲 can be seen as a
statement according to which the probability of doing a certain negative work −W during the backward protocol is exponentially suppressed with respect to the probability of doing the positive work W, in the forward protocol. For a cyclic
protocol, this says that it is exponentially more probable to
spend energy, rather harvesting it, in agreement with the
Kelvin postulate 共i.e., no energy extraction from a cyclic
process兲. A similar situation occurs for the finite bath fluctuation theorem, with the exponential suppression being replaced by a power-law suppression. To exemplify this, consider again the gas of N hard spheres in d dimensions.
Imagine the protocol consists of changing the volume of the
box that contains the gas from V0 to V f . Straightforward
calculations lead the following form of the finite bath fluctuation theorem
pC,U
t ,t 共W兲
f 0
f
pC,U
t0,t f 共−
W兲
=
冉 冊冉
Vf
Vi
N/d
1+
W
CtotT0
冊
Ctot−1
共54兲
where it is evident that the power-law term 关1 + W /
共CtotT0兲兴Ctot−1 becomes the exponential term appearing in the
Crooks theorem Eq. 共35兲 for very large C 共Ctot = C + dN / 2
becomes very large for very large C兲.
two ideal cases of absent bath 共microcanonical ensemble兲
and infinite bath 共canonical ensemble兲. The finite bath fluctuation theorem interpolates between microcanonical and canonical fluctuation theorems. It thus generalizes these theorems and reveals a common underlying mathematical
structure.
The validity of the finite bath statistics is illustrated by
means of numerical simulations of a 2D gas of hard disks in
a box with perfectly reflecting walls, see Fig. 1, and the
validity of the finite bath fluctuation theorem is confirmed
both analytically and numerically, cf. Fig. 2, for our system.
Similarity and differences between the finite bath fluctuation theorem and the canonical and microcanonical fluctuation theorems have been discussed, as well as its interrelation
with the second law of thermodynamics. In contrast with the
canonical fluctuation theorem, two temperatures, instead of
one, appear in the finite bath fluctuation theorem. The physical meaning of these two temperatures has been clarified by
considering a rethermalization process.
As shown in Sec. II, the finite bath statistics in Eq. 共6兲 is
a special instance of the general statistical formula according
to which the bath density of states determines the shape of
the system pdf. Based on quasiadiabatic perturbation theory
of chaotic systems, Jarzynski 关31兴 found that a slow particle
coupled to a small bath with fast chaotic degrees of freedom
thermalizes and reaches a stationary pdf whose shape is dictated by the density of states of the bath. Our simulations
provide an example that such behavior of the system pdf
occurs even if there is no time-scale separation between system and bath. In any case, thermalization of the subsystem
toward a pdf of the form in Eq. 共6兲 is expected only if the
total system is ergodic.
An important assumption underlying our main finding is
that we used a specific heat that is energy independent:
whether a finite bath fluctuation theorem exists also in the
case of more realistic energy dependent specific heats remains an open challenge.
ACKNOWLEDGMENTS
Financial support by the DFG via the collaborative research center SFB-486, project A10, via the project no. 1517/
26–2, the German Excellence Initiative via the Nanosystems
Initiative Munich 共NIM兲 and the Volkswagen Foundation
共project I/80424兲 is gratefully acknowledged.
APPENDIX A: SPECIFIC HEAT OF A BATH OF n HARD
SPHERES
Although straightforward, the calculation of the microcanonical specific heat of a gas of hard spheres is not discussed
in statistical mechanics textbooks. We present this calculation below.
The Hamiltonian of a gas of n d-dimensional hard spheres
of radius a reads
VI. CONCLUSIONS
n
We devised a finite bath fluctuation theorem that gives
information about the probability of work on systems that
have been thermalized with a finite heat bath. This corresponds to physical situations which are situated between the
HB共兵p
ជi其,兵qជi其兲 = 兺
i=1
p i2
ជ
+ 兺 V共兩q
q j兩兲,
ជi − ជ
2m i⬍j
共A1兲
where ជ
pi , qជi are the d-dimensional momentum and position
vectors of the ith sphere, and
031145-7
PHYSICAL REVIEW E 80, 031145 共2009兲
CAMPISI, TALKNER, AND HÄNGGI
V共x兲 =
再
xⱖa
0
+⬁ x⬍a
冎
共A2兲
is the hard-core interaction potential. The phase space volume ⌽B with energy below EB becomes
冕兿 冕兿
n
⌽B共EB兲 =
dq
ជi
I␭共U,0兲 =
dp
ជi
i=1
冉
n
⫻ ␪ EB − 兺
i=1
冊
p i2
ជ
− 兺 V共兩q
q j兩兲 ,
ជi − ជ
2m i⬍j
冕兿
n
i=1
冉
n
dp
ជi␪ EB − 兺
i=1
冊
p i2
ជ
,
2m
共A3兲
共A4兲
n
dq
where V⬘n = 兰M兿i=1
ជi, is independent of EB. We shall refer
to V⬘ as to the reduced volume. The integration over the
momenta then yields 关32兴
⌽B共EB兲 = Adn共2m兲dn/2V⬘nEBdn/2
共A5兲
where AN ª ␲N/2 / ⌫共N / 2 + 1兲. By differentiating ⌽B共EB兲 with
respect to EB, one finally obtains the density of states of the
gas of hard spheres
⍀B共EB兲 = Adn共dn/2兲共2m兲dn/2V⬘NEBdn/2−1 .
共A6兲
The only difference with the density of states of an ideal gas
is that the actual volume V is replaced by the reduced volume
V⬘. The temperature TB共EB兲 = ⌽B共EB兲 / ⍀B共EB兲, is given by
the same formula as for the ideal gas, i.e., TB共EB兲
= 2EB / 共dn兲 and so is the specific heat, i.e., C共EB兲 = dn / 2. For
simplicity, in Eq. 共A1兲 we neglected the spheres rotational
degrees of freedom. These however would add to the total
specific heat an energy independent contribution.
APPENDIX B: EXISTENCE AND (NON)UNIQUENESS
OF SOLUTIONS OF Eq. (5)
We prove that, given U and ␭, it is always possible to find
a T such that Eq. 共5兲 is satisfied. For this purpose we define
the function
冕
共B2兲
冕
de⍀␭共e兲共e − U兲共U − e兲C−1
共B3兲
U
0
where each integral in dqជi is restricted to the region V, of
q j兩 smaller than a,
volume V, of the box. For values of 兩q
ជi − ជ
the integrand vanishes, thus reducing the spatial integration
domain to the region M 傺 Vn where 兩q
q j兩 ⬎ a, for each
ជi − ជ
couple i , j. In this region the interaction term is zero and
one obtains
⌽B共EB兲 = V⬘n
I␭共U,T兲 = 0.
For T = 0 it is
n
i=1
I␭共U,T兲 ª
H共z , ␭兲. Equation 共5兲 can be equivalently expressed as:
CT+U
de⍀␭共e兲共e − U兲共CT − e + U兲C−1
0
共B1兲
which is continuous with respect to both U and T. The symbol ⍀␭共e兲 denotes the density of states of the Hamiltonian
Since ⍀␭共e兲 ⱖ 0, and e − U ⱕ 0 in the integration domain, we
have
I␭共U,0兲 ⱕ 0.
共B4兲
On the other hand for T Ⰷ U / C, we find
I␭共U,T兲 ⯝
冕
CT
de⍀␭共e兲共e − U兲共CT − e兲C−1
共B5兲
0
where we neglected the terms U as compared to CT. By
making the change of variable x = CT − e, and neglecting
again the term U as compared to CT, we obtain:
I␭共U,T兲 ⯝
冕
CT
dx⍀␭共CT − x兲共CT − x兲xC−1 .
共B6兲
0
All three terms forming the integrand are non-negative,
hence
I␭共U,T Ⰷ U/C兲 ⱖ 0.
共B7兲
Thus I␭共U , T兲 is nonpositive for T = 0 and non-negative for
very large T. This implies, that there must be at least one
non-negative value of T, for which I␭共U , T兲 = 0. Uniqueness,
however is not guaranteed.
In a similar way it is also possible to prove that
I␭共0,T兲 ⱖ 0,
I␭共U Ⰷ CT,T兲 ⱕ 0
共B8兲
showing that one can also fix T and find a U such that
I␭共U , T兲 = 0. Also in this case only existence is guaranteed
but not uniqueness.
Examples for which two or more different energies correspond to the same temperature were reported in 关33,34兴 for
microcanonical 共C = 0兲 gases with interparticle interaction of
the Lennard-Jones type. These systems undergo a microcanonical phase transition whose signature is the appearance of
oscillations in the function T共U兲, which, therefore, is not
invertible 共i.e, U共T兲 is multivalued兲. These oscillations are
expected to appear also if these Lennard-Jones type systems
are thermalized by means of a finite bath with specific heat
C ⬎ 0. Based on the observation that no oscillation appear in
the canonical treatment 关34兴, one expects that the amplitude
of these oscillations decreases with increasing C.
031145-8
PHYSICAL REVIEW E 80, 031145 共2009兲
FINITE BATH FLUCTUATION THEOREM
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关34兴
031145-9
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