Cycle counts and affinities in stochastic models of non-equilibrium systems

TL;DR

This paper investigates the statistical properties of cycle counts in non-equilibrium Markov systems, revealing universal formulas dependent on cycle affinity and connecting to fluctuation theorems and large deviation theory.

Contribution

It introduces universal formulas for cycle count distributions in Markov processes, generalizes to cycle families, and links to fluctuation theorems and large deviations.

Findings

Universal formulas depend on cycle affinity

Distribution of forward/backward cycle counts is system-independent

Application of large deviation theory to cycle counting

Abstract

For non-equilibrium systems described by finite Markov processes, we consider the number of times that a system traverses a cyclic sequence of states (a cycle). The joint distribution of the number of forward and backward instances of any given cycle is described by universal formulae which depend on the cycle affinity, but are otherwise independent of system details. We discuss the similarities and differences of this result to fluctuation theorems, and generalize the result to families of cycles, relevant under coarse-graining. Finally, we describe the application of large deviation theory to this cycle counting problem.