Second moments of work and heat for a single particle stochastic heat engine in a breathing harmonic potential

TL;DR

This paper analyzes the second moments of work and heat in a single-particle stochastic heat engine with a breathing harmonic potential, providing integral expressions and quasi-static limit results.

Contribution

It derives explicit integral formulas for the second moments of heat and work in a stochastic heat engine model with a breathing harmonic potential, including quasi-static limit simplifications.

Findings

Expressions for variances and covariances of heat and work are obtained.

In the quasi-static limit, results simplify to functions of bath temperatures.

The coefficient of variation of work estimates the probability of work exceeding a threshold.

Abstract

We consider a simple model of a stochastic heat engine, which consists of a single Brownian particle moving in a one-dimensional periodically breathing harmonic potential. Overdamped limit is assumed. Expressions of second moments (variances and covariances ) of heat and work are obtained in the form of integrals, whose integrands contain functions satisfying certain differential equations. The results in the quasi-static limit are simple functions of temperatures of hot and cold thermal baths. The coefficient of variation of the work is suggested to give an approximate probability for the work to exceeds a certain threshold. During derivation, we get the expression of the cumulant-generating function.