On the efficiency of heat engines at the micro-scale and below

TL;DR

This paper derives a universal efficiency law for sub-micro-scale heat engines operating under stochastic thermodynamics, linking maximum power efficiency to Carnot efficiency and providing explicit optimal protocols.

Contribution

It establishes a universal efficiency law at maximum power for micro-scale heat engines and connects it to optimal mass transport solutions.

Findings

Efficiency at maximum power follows a universal law $rac{2\, ext{η}_C}{4- ext{η}_C}$.

Optimal protocols are explicitly determined by Monge--Ampère--Kantorovich algorithms.

Results extend to anisotropic and temperature-dependent mobility cases.

Abstract

We investigate the thermodynamic efficiency of sub-micro-scale heat engines operating under the conditions described by over-damped stochastic thermodynamics. We prove that at maximum power the efficiency obeys for constant isotropic mobility the universal law $\eta=2\,\eta_{C}/(4-\eta_{C})$ where $\eta_{C}$ is the efficiency of an ideal Carnot cycle. The corresponding power optimizing protocol is specified by the solution of an optimal mass transport problem. Such solution can be determined explicitly using well known Monge--Amp\`ere--Kantorovich reconstruction algorithms. Furthermore, we show that the same law describes the efficiency of heat engines operating at maximum work over short time periods. Finally, we illustrate the straightforward extension of these results to cases when the mobility is anisotropic and temperature dependent.