BEST statistics of Markovian fluxes: a tale of Eulerian tours and Fermionic ghosts

TL;DR

This paper derives an exact formula for Markov jump process flux statistics at all times, combining graph theory, Fermionic field theory, and thermodynamics to improve understanding of fluctuations beyond asymptotic limits.

Contribution

It generalizes the BEST theorem to open Eulerian tours and introduces a Fermionic ghost field framework to analyze finite-time flux fluctuations in Markov processes.

Findings

Exact finite-time flux statistics formula derived

Finite-time fluctuations linked to spanning-tree determinants and Fermionic fields

Connections established between stochastic thermodynamics and gauge theories

Abstract

We provide an exact expression for the statistics of the fluxes of Markov jump processes at all times, improving on asymptotic results from large deviation theory. The main ingredient is a generalization of the BEST theorem in enumeratoric graph theory to Eulerian tours with open ends. In the long-time limit we reobtain Sanov's theorem for Markov processes, which expresses the exponential suppression of fluctuations in terms of relative entropy. The finite-time power-law term, increasingly important with the system size, is a spanning-tree determinant that, by introducing Grassmann variables, can be absorbed into the effective Lagrangian of a Fermionic ghost field on a metric space, coupled to a gauge potential. With reference to concepts in nonequilibrium stochastic thermodynamics, the metric is related to the dynamical activity that measures net communication between states, and the connection is made to a previous gauge theory for diffusion processes.