Duration of local violations of the second law of thermodynamics along single trajectories in phase space

Abstract

We define the violation fraction ν as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends on both the number of degrees of freedom N and the duration of the time interval τ. In the large-τ and large-N limit we show that, for ergodic and microreversible systems, the mean value of ν scales as ⟨ν(N,τ)⟩∼(τN11+α)−1. The exponent α is positive and generally depends on the protocol for the external driving forces, being α=1 for a constant drive. As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. In this case we show that the scaling of ⟨ν⟩ with N and τ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.