Gallavotti-Cohen-Type symmetry related to cycle decompositions for Markov chains and biochemical applications

TL;DR

This paper extends the Gallavotti-Cohen fluctuation theorem to continuous-time Markov chains with multiple edges, providing new insights into cycle decompositions and applications to biochemical systems and molecular motors.

Contribution

It introduces an extended fluctuation theorem for Markov chains with multiple edges, connecting cycle decompositions to fluctuation relations in biochemical and mechanical systems.

Findings

Derived a fluctuation theorem involving cycle affinities.

Applied the theorem to molecular motors and biochemical currents.

Provided a new method for analyzing large deviations in Markov processes.

Abstract

We slightly extend the fluctuation theorem obtained in \cite{LS} for sums of generators, considering continuous-time Markov chains on a finite state space whose underlying graph has multiple edges and no loop. This extended frame is suited when analyzing chemical systems. As simple corollary we derive in a different method the fluctuation theorem of D. Andrieux and P. Gaspard for the fluxes along the chords associated to a fundamental set of oriented cycles \cite{AG2}. We associate to each random trajectory an oriented cycle on the graph and we decompose it in terms of a basis of oriented cycles. We prove a fluctuation theorem for the coefficients in this decomposition. The resulting fluctuation theorem involves the cycle affinities, which in many real systems correspond to the macroscopic forces. In addition, the above decomposition is useful when analyzing the large deviations of additive functionals of the Markov chain. As example of application, in a very general context we derive a fluctuation relation for the mechanical and chemical currents of a molecular motor moving along a periodic filament.