Thermodynamic Formalism for
Inhomogeneous Random Walks
Günter Radons
Theoretical Physics I – Complex Systems and
Nonlinear Dynamics
Content:
• The Thermodynamic Formalism
• A Prototypical Example: Asymmetric
Random Walk with Reflecting Barrier –
Dynamical Phases
• Generalizations
• Discussion and Recent Developments
The Thermodynamic Formalism:
• 1978: Book by D. Ruelle, Thermodynamic
Formalism
• 1980s: Applications to Chaotic Systems
(Generalized Lyapunov Exponents) and Fractals
(“Multifractals”)
• 1990s: Applications to Random Walks
• 2000s: New Developments
Basic idea:
• Consider process in time interval [0,N]
• Define possible histories in [0,N]
• Define dynamical partition function
• Evaluate thermodynamic potentials
• Identify dynamical phases – phase transitions
Thermodynamic Formalism for discrete
random walks:
Process: Evolution equation
Histories: Symbolic representation
Partition function:
Growth rate, “free energy”:
Generalized dynamical entropies:
Thermodynamic potentials, dynamical phase transitions:
Evaluation of the Partition Function:
Evaluation cont’d:
A Prototypical Example: Asymmetric
Random Walk with Reflecting Barrier
0
1
2
3
…
Dynamical phases:
discrete:
band:
free:
Some understanding by mapping to solid state problem:
Localized vs. delocalized states
Mapping:
“Thermodynamic potentials” τ(β) and f(α
α):
p = 0.25
τ(β)
p = 0.6
d
d
f
b
f
d
f
1st
b
b
2nd
2nd
.b
f(α
α)
p = 0.85
f
2nd
f
d
Maxwell
f
Phase diagram:
discrete:
band:
β
free:
p
Discussion of phase diagram:
…
…
…
…
…
Generalizations (methods):
bi-variate thermodynamical formalism
phase f:
generalization of classical case β = 1
confirmation of diffusive nature of phase f
phase d:
all generalized transport coefficients vanish
localized nature of phase d
phase b:
all generalized transport coefficients vanish
delocalized nature of phase b
anomalous diffusion
Generalizations (systems):
symmetric random walk with isolated impurity
…
bi-infinite
…
trap
or
barrier
again, analytically:
• localization – delocalization transition as β varies
• 2nd order phase transitions
Discussion and Recent Developments:
System size L vs. time t infinity
M.H. Ernst et al,, Phys. Rev. Lett. 74, 4416 (1995)
C. Appert et al., Phys. Rev. E 54, R1013 (1996)
G.R., Phys. Rev. Lett. 75, 4719 (1995)
G.R., Phys. Rept. 290, 67 (1997)
Recently:
Extension to continuous time Markov processes
V. Lecomte et al., Phys. Rev. Lett. 95, 010601 (2005)
V. Lecomte et al., J. Stat. Phys. 127, 51 (2007)
Connection to Fluctation Theorems in the form
of Galavotti-Cohen, Lebowitz-Spohn
τ(β) is a large deviation function
Summary:
• Thermodynamic Formalism: Applying equilibrium
concepts to non-equilibrium processes
• Deeper insights into dynamical processes:
deterministic or stochastic, discrete or continuous
in time
• Dynamical phases and phase transitions may
become apparent, mechanisms not well explored
• Generalizations to many other physical
observables are possible, possibly measurable