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

TL;DR

This paper investigates the duration and frequency of violations of the second law of thermodynamics along single trajectories in phase space, revealing a universal scaling law dependent on system size and observation time.

Contribution

It introduces a scaling law for the violation fraction in ergodic systems and verifies it with a non-Gaussian entropy production model, also discussing potential breakdown near phase transitions.

Findings

Mean violation fraction scales as (τ N^{1/(1+α)})^{-1} in large systems and long times.

Scaling law holds for a non-Gaussian entropy production model with anharmonic trapping.

Potential breakdown of the scaling law near continuous phase transitions.

Abstract

We define the {\it violation fraction} $\nu$ as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends both on the number of degrees of freedom $N$ and the duration of the time interval $\tau$. In the large-$\tau$ and large-$N$ limit we show that, for ergodic and microreversible systems, the mean value of $\nu$ scales as $\langle\nu(N,\tau)\rangle\sim\big(\tau N^{\frac{1}{1+\alpha}}\big)^{-1}$. The exponent $\alpha$ is positive and generally depends on the protocol for the external driving forces, being $\alpha=1$ 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 $\langle \nu \rangle$ with $N$ and $\tau$ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.