Large deviations and fluctuation theorems for cycle currents defined in the loop-erased and spanning tree manners: a comparative study

TL;DR

This paper compares large deviations and fluctuation theorems for cycle currents defined via loop-erased and spanning tree methods in Markov systems, deriving exact distributions and clarifying their applicability.

Contribution

It provides exact joint distributions, rate functions, and a detailed comparison of fluctuation theorems for LE and ST cycle currents in stochastic thermodynamics.

Findings

LE currents satisfy all fluctuation theorems strongly.

ST currents do not satisfy fluctuation theorems generally.

Analytical results applied to enzyme reaction fluctuations.

Abstract

The cycle current is a crucial quantity in stochastic thermodynamics. The absolute and net cycle currents of a Markovian system can be defined in the loop-erased (LE) or spanning tree (ST) manner. Here we make a comparative study between the large deviations and fluctuation theorems for LE and ST currents, i.e. cycle currents defined in the LE and ST manners. First, we derive the exact joint distribution and large deviation rate function for the LE currents of a system with a cyclic topology and also obtain the exact rate function for the ST currents of a general system. The relationship between the rate functions for LE and ST currents is clarified and the analytical results are applied to examine the fluctuations in the product rate of a three-step reversible enzyme reaction. Furthermore, we examine various types of fluctuation theorems satisfied by LE and ST currents and clarify their ranges of applicability. We show that both the absolute and net LE currents satisfy the strong form of all types of fluctuation theorems. In contrast, the absolute ST currents do not satisfy fluctuation theorems, while the net ST currents only satisfy the weak form of fluctuation theorems under the periodic boundary condition.