Chaotic Properties of Systems with Markov Dynamics

Abstract

We present a general approach for computing the dynamic partition function of a continuous-time Markov process. The Ruelle topological pressure is identified with the large deviation function of a physical observable. We construct for the first time a corresponding finite Kolmogorov-Sinai entropy for these processes. Then, as an example, the latter is computed for a symmetric exclusion process. We further present the first exact calculation of the topological pressure for an N-body stochastic interacting system, namely, an infinite-range Ising model endowed with spin-flip dynamics. Expressions for the Kolmogorov-Sinai and the topological entropies follow.

Thermodynamic Formalism for Systems with Markov Dynamics

“…It can also be noted that, if one defines a Lyapunov exponent for the random walk through an equivalent one-dimensional map, as described in [41,19,17], we recover Pesin's theorem (2).…”

“…Given the successes of the Markov approach in understanding the various versions of the fluctuation and work theorems, it seemed natural to turn to the more general dynamical partition function. As briefly sketched in [17], by contrast with the existing treatment of Markov chains [18,19,20,21] there had hitherto been no satisfactory attempt to force the thermodynamic formalism of Ruelle into the framework of systems endowed with continuous-time Markov dynamics. As this was already noticed by Gaspard [22], passing from discrete to continuous time raises specific difficulties.…”

Dynamical First-Order Phase Transition in Kinetically Constrained Models of Glasses

“…Indeed, it has been suggested that the glass transition manifests a firstorder phase transition in space and time between active and inactive phases [3]. Here we apply Ruelle's thermodynamic formalism [4,5] to show that this suggestion is indeed correct, for a specific class of stochastic models. The existence of active and inactive regions of spacetime, separated by sharp interfaces, is dynamic heterogeneity, a central feature of glass forming systems [6].…”

“…The energy of the system is replaced by the dynamical action (the negative of the logarithm of the probability of a given history); the entropy of the system by the Kolmogorov-Sinai entropy [13], and the temperature by an intrinsic field conjugate to the action. This formalism has been exploited recently to describe the chaotic properties of continuous-time Markov processes [5].…”

“…The energy of the system is replaced by the dynamical action (the negative of the logarithm of the probability of a given history); the entropy of the system by the Kolmogorov-Sinai entropy [13], and the temperature by an intrinsic field conjugate to the action. This formalism has been exploited recently to describe the chaotic properties of continuous-time Markov processes [5].In this work, we define the dynamical partition sum [4, where the sum is over histories from time 0 to time t; the probability of a history is Prob(history); andK(history) is the number of configuration changes in that history. K(history) is a direct measure of the activity in a history: an active trajectory has many changes of configuration,…”