Integral fluctuation relations for entropy production at stopping times

A stopping time T is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral fluctuation relation < e(-Stot(T))> = 1 for the stochastic entropy production S-tot in a stationary physical system at stochastic stopping times T. This fluctuation relation implies the law < S-tot(T)> >= 0, which states that it is not possible to reduce entropy on average, even by stopping a stochastic process at a stopping time, and which we call the second law of thermodynamics at stopping times. This law bounds the average amount of heat and work a system can extract from its environment when stopped at a random time. Furthermore, the integral fluctuation relation implies that certain fluctuations of entropy production are universal or are bounded by universal functions. These universal properties descend from the integral fluctuation relation by selecting appropriate stopping times: for example, when T is a first-passage time for entropy production, then we obtain a bound on the statistics of negative records of entropy production. We illustrate these results on simple models of nonequilibrium systems described by Langevin equations and reveal two interesting phenomena. First, we demonstrate that isothermal mesoscopic systems can extract on average heat from their environment when stopped at a cleverly chosen moment and that the second law at stopping times bounds the average extracted heat. Second, we demonstrate that the efficiency at stopping times of an autonomous stochastic heat engine, such as Feymann's ratchet, can be larger than the Carnot efficiency and that the second law of thermodynamics at stopping times bounds its efficiency at stopping times.