Variational principle of counting statistics in master equations

TL;DR

This paper derives a variational principle for counting statistics in stochastic processes using path integral formulation, applicable beyond mesoscopic systems, and clarifies the validity of the saddle point method.

Contribution

It introduces a path integral approach to counting statistics without mesoscopic assumptions and establishes a valid variational principle using replicas and the Euler-Maclaurin formula.

Findings

Path integral formulation applicable to general stochastic systems.

Saddle point method is rigorously justified as exact in this context.

Derivation of a variational principle for counting statistics.

Abstract

We study counting statistics of number of transitions in a stochastic process. For mesoscopic systems, a path integral formulation for the counting statistics has already been derived. We here show that it is also possible to derive the similar path integral formulation without the assumption of mesoscopic systems. It has been clarified that the saddle point method for the path integral is not an approximation, but a valid procedure in the present derivation. Hence, a variational principle in the counting statistics is naturally derived. In order to obtain the variational principle, we employ many independent replicas of the same system. In addition, the Euler-Maclaurin formula is used in order to connect the discrete and continuous properties of the system.