Fluctuation theorem as a special case of Girsanov theorem

Abstract

Stochastic thermodynamics is an important development in the direction of finding general thermodynamic principles for non-equilibrium systems. We believe stochastic thermodynamics has the potential to benefit from the measure-theoretic framework of stochastic differential equations (SDEs). Toward this, in this work, we show that fluctuation theorem (FT) is a special case of the Girsanov theorem, which is an important result in the theory of SDEs. We report that by employing Girsanov transformation of measures between the forward and the reversed dynamics of a general class of Langevin dynamic systems, we arrive at the integral fluctuation relation. Following the same approach, we derive the FT also for the overdamped case. Our derivation is applicable to both transient and steady state conditions and can also incorporate diffusion coefficients varying as a function of state and time, e.g. in the context of multiplicative noise. We expect that the proposed method will be an easy route towards deriving the FT irrespective of the complexity and non-linearity of the system.