Relation between fluctuations and efficiency at maximum power for small heat engines

TL;DR

This paper investigates the fluctuations in work and heat for small heat engines at maximum power, finding that the ratio of variances is well approximated by the square of the Curzon-Ahlborn efficiency, indicating a trade-off between efficiency and stability.

Contribution

It extends the understanding of fluctuation bounds to finite-time regimes and establishes an approximate relation between variance ratios and the Curzon-Ahlborn efficiency for small heat engines.

Findings

The ratio of variances $ ext{η}^{(2)}$ is approximately $ ext{η}_{ ext{CA}}^2$ at maximum power.

Numerical simulations confirm the relation $ ext{η}_{ ext{MP}}^{(2)} oughly ext{η}_{ ext{CA}}^2$.

The relation implies a trade-off between efficiency and stability in finite-time heat engines.

Abstract

We study the ratio between the variances of work output and heat input, $\eta^{(2)}$, for a class of four-stroke heat engines which covers various typical cycles. Recent studies on the upper and lower bounds of $\eta^{(2)}$ are based on the quasistatic limit and the linear response regime, respectively. We extend these relations to the finite-time regime within the endoreversible approximation. We consider the ratio $\eta_{\text{MP}}^{(2)}$ at maximum power and find that the square of the Curzon-Ahlborn efficiency, $\eta_{\text{CA}}^2$, gives a good estimate of $\eta_{\text{MP}}^{(2)}$ for the class of heat engines considered, i.e., $\eta_{\text{MP}}^{(2)} \simeq \eta_{\text{CA}}^2$. This resembles the situation where the Curzon-Ahlborn efficiency gives a good estimate of the efficiency at maximum power for various kinds of finite-time heat engines. Taking an overdamped Brownian particle in a harmonic potential as an example, we can realize such endoreversible small heat engines and give an expression of the cumulants of work output and heat input. The approximate relation $\eta_{\text{MP}}^{(2)} \simeq \eta_{\text{CA}}^2$ is verified by numerical simulations. This relation also suggests a trade-off between the efficiency and the stability of finite-time heat engines at maximum power.