Chaotic properties of systems with Markov dynamics

TL;DR

This paper introduces a novel framework for analyzing the chaotic properties of continuous-time Markov systems by computing the topological pressure and entropy, with exact results for specific models.

Contribution

It develops a general method to compute the dynamic partition function and topological pressure for Markov processes, including finite Kolmogorov-Sinai entropy, with new exact calculations for models like the symmetric exclusion process and infinite-range Ising model.

Findings

Computed the dynamic partition function for continuous-time Markov processes.

Derived the first exact topological pressure for an N-body stochastic system.

Established a finite Kolmogorov-Sinai entropy for 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.