Infinite family of second-law-like inequalities

TL;DR

This paper introduces a family of inequalities that generalize the second law for out-of-equilibrium systems using trial distributions, providing a new way to assess and optimize approximations without needing the true stationary distribution.

Contribution

It develops a set of inequalities based on trial distributions that extend the second law and offers a criterion for their accuracy and an optimization method, applicable even without knowing the true distribution.

Findings

Inequalities constrain out-of-equilibrium systems based on trial distributions.

Better approximations lead to more restrictive inequalities.

The approach extends the Hatano-Sasa fluctuation relation without requiring the true stationary distribution.

Abstract

The probability distribution function for an out of equilibrium system may sometimes be approximated by a physically motivated "trial" distribution. A particularly interesting case is when a driven system (e.g., active matter) is approximated by a thermodynamic one. We show here that every set of trial distributions yields an inequality playing the role of a generalization of the second law. The better the approximation is, the more constraining the inequality becomes: this suggests a criterion for its accuracy, as well as an optimization procedure that may be implemented numerically and even experimentally. The fluctuation relation behind this inequality, -a natural and practical extension of the Hatano-Sasa theorem-, does not rely on the a priori knowledge of the stationary probability distribution.