A Note on Non-equilibrium Work Fluctuations and Equilibrium Free Energies

TL;DR

This paper explores the relationship between non-equilibrium work fluctuations and equilibrium free energies, analytically and via simulations, revealing how dissipation decreases in quasistatic processes while the probability of second law violation increases.

Contribution

It provides an analytical framework linking work fluctuations, dissipation, and second law violations, supported by Monte Carlo simulations in liquid crystal systems.

Findings

Dissipation decreases as processes become more quasistatic.

Probability of second law violation approaches 0.5 in the quasistatic limit.

Work distribution becomes Gaussian in nearly quasistatic processes.

Abstract

We consider in this paper, a few important issues in non-equilibrium work fluctuations and their relations to equilibrium free energies. First we show that Jarzynski identity can be viewed as a cumulant expansion of work. For a switching process which is nearly quasistatic the work distribution is sharply peaked and Gaussian. We show analytically that dissipation given by average work minus reversible work $W_R$, decreases when the process becomes more and more quasistatic. Eventually, in the quasistatic reversible limit, the dissipation vanishes. However estimate of $p$ - the probability of violation of the second law given by the integral of the tail of the work distribution from $-\infty$ to $W_R$, increases and takes a value of $0.5$ in the quasistatic limit. We show this analytically employing Gaussian integrals given by error functions and Callen-Welton theorem that relates fluctuations to dissipation in process that is nearly quasistatic. Then we carry out Monte Carlo simulation of non-equilibrium processes in a liquid crystal system in the presence of an electric field and present results on reversible work, dissipation, probability of violation of the second law and distribution of work