Thermodynamic formalism and large deviation functions in continuous time Markov dynamics

TL;DR

This paper extends the thermodynamic formalism to continuous time Markov processes, demonstrating how it can reveal dynamical phase transitions through generating functions of observables, with applications to mean-field models like the Potts model.

Contribution

It reformulates thermodynamic formalism for continuous time Markov processes and explores its implications for dynamical phase transitions in mean-field models.

Findings

Thermodynamic formalism applies to continuous time Markov processes with appropriate interpretation.

The observable K signals dynamical phase transitions.

In the mean-field Potts model, the Ising case differs qualitatively from other cases.

Abstract

The thermodynamic formalism, which was first developed for dynamical systems and then applied to discrete Markov processes, turns out to be well suited for continuous time Markov processes as well, provided the definitions are interpreted in an appropriate way. Besides, it can be reformulated in terms of the generating function of an observable, and then extended to other observables. In particular, the simple observable $K$ giving the number of events occurring over a given time interval turns out to contain already the signature of dynamical phase transitions. For mean-field models in equilibrium, and in the limit of large systems, the formalism is rather simple to apply and shows how thermodynamic phase transitions may modify the dynamical properties of the systems. This is exemplified with the q-state mean-field Potts model, for which the Ising limit q=2 is found to be qualitatively different from the other cases.