Transition records of stationary Markov chains

TL;DR

This paper explores the properties of transition records in stationary Markov chains, establishing their exponential family distribution, deriving related fluctuation theorems, and illustrating applications in entropy and thermodynamics.

Contribution

It introduces a novel approach linking transition records to exponential family distributions, enabling new insights into entropy and thermodynamic relations in Markov processes.

Findings

Transition records in finite state Markov chains belong to the exponential family.

A fluctuation theorem for stationary Markov chains is proved.

A thermodynamic relation for equilibrium systems with Metropolis dynamics is derived.

Abstract

In any Markov chain with finite state space the distribution of transition records always belongs to the exponential family. This observation is used to prove a fluctuation theorem, and to show that the dynamical entropy of a stationary Markov chain is linear in the number of steps. Three applications are discussed. A known result about entropy production is reproduced. A thermodynamic relation is derived for equilibrium systems with Metropolis dynamics. Finally, a link is made with recent results concerning a one-dimensional polymer model.