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

Abstract

We study the ratio between the variances of work output and heat input, η^{(2)}, for a class of four-stroke heat engines, which covers various typical cycles. Recent studies on the upper and lower bounds of η^{(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 η_{MP}^{(2)} at maximum power and find that the square of the Curzon-Ahlborn efficiency, η_{CA}^{2}, gives a good estimate of η_{MP}^{(2)} for the class of heat engines considered, i.e., η_{MP}^{(2)}≃η_{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 η_{MP}^{(2)}≃η_{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.