Infimum Law and First-Passage-Time Fluctuation Theorem for Entropy Production

We derive an infimum law and a first-passage-time fluctuation theorem for entropy production of stochastic processes at steady state. We show that the ratio between the probability densities of the first-passage time to produce Stot of entropy and of the first-passage time to produce −Stot of entropy equals etot, with k Boltzmann’s constant. This first-passage-time fluctuation relation is valid for processes with one or higher order passages. In addition, we derive universal bounds for the infimum statistics of entropy production using the fact that at steady state e−Stot/k is a martingale process. We show that the mean value of the entropy-production infimum obeys 〈inf Stot〉 ≥ −k and derive a bound on the cumulative distribution of entropy-production infima. Our results are derived using a measure-theoretic formalism of stochastic thermodynamics. We illustrate our results with a drift-diffusion process and with numerical simulations of a Smoluchowski–Feynman ratchet.