The Measure-theoretic Identity Underlying Transient Fluctuation Theorems

Abstract

We prove a measure-theoretic identity that underlies all transient fluctuation theorems (TFTs) for entropy production and dissipated work in inhomogeneous deterministic and stochastic processes, including those of Evans and Searles, Crooks, and Seifert. The identity is used to deduce a tautological physical interpretation of TFTs in terms of the arrow of time, and its generality reveals that the self-inverse nature of the various trajectory and process transformations historically relied upon to prove TFTs, while necessary for these theorems from a physical standpoint, is not necessary from a mathematical one. The moment generating functions of thermodynamic variables appearing in the identity are shown to converge in general only in a vertical strip in the complex plane, with the consequence that a TFT that holds over arbitrary timescales may fail to give rise to an asymptotic fluctuation theorem for any possible speed of the corresponding large deviation principle. The case of strongly biased birth-death chains is presented to illustrate this phenomenon. We also discuss insights obtained from our measure-theoretic formalism into the results of Saha et. al. on the breakdown of TFTs for driven Brownian particles.