Detailed fluctuation theorem bounds apparent violations of the second law

TL;DR

This paper derives tight bounds on the frequency of apparent violations of the second law in small systems using detailed fluctuation theorem, with applications to heat exchange, nanoparticles, and classical particles.

Contribution

It introduces a new bound on the probability of negative entropy production based on the strong detailed fluctuation theorem, extending understanding of second law violations.

Findings

Bound verified for heat exchange between reservoirs

Bound applied to levitated nanoparticle entropy production

Bound confirmed for classical particle in a box

Abstract

The second law of thermodynamics is a statement about the statistics of the entropy production, $\langle \Sigma \rangle \geq 0$. For small systems, it is known that the entropy production is a random variable and negative values ($\Sigma < 0$) might be observed in some experiments. This situation is sometimes called apparent violation of the second law. In this sense, how often is the second law violated? For a given average $\langle \Sigma \rangle $, we show that the strong detailed fluctuation theorem implies a lower tight bound for the apparent violations of the second law. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a bosonic mode in the weak coupling approximation, a levitated nanoparticle and a classical particle in a box.