Consistent expansion of the Langevin propagator with application to entropy production

Stochastic thermodynamics is a developing theory for systems out of thermal equilibrium. It allows us to formulate a wealth of nontrivial connections between thermodynamic quantities (such as heat dissipation, excess work, and entropy production) and the statistics of trajectories in generic nonequilibrium stochastic processes. A key quantity for the derivation of these relations is the propagator — the probability to observe a transition from one point in phase space to another after a given time. Here, applying stochastic Taylor expansions, we devise a formal short-time expansion procedure for the propagator of overdamped Langevin dynamics. The three leading orders are obtained explicitly. This technique resolves the shortcomings of the common mathematical machinery used for proving stochastic-thermodynamic relations. In particular, we identify that functionals of the propagator such as the entropy production, which we refer to as ‘first derivatives of the trajectory’, require a previously-unrecognized high-order expansion of the propagator. The method presented here can be extended to arbitrarily higher orders needed to accurately compute any other functional of the propagator. We discuss applications to higher-order simulations of overdamped Langevin dynamics.