Entropic anomaly and maximal efficiency of microscopic heat engines

TL;DR

This paper reveals that microscopic heat engines in temperature gradients experience a significant efficiency loss due to the entropic anomaly, with maximum efficiency occurring at a finite cycle period inversely related to the gradient.

Contribution

It demonstrates that the entropic anomaly causes quasi-static engines to have vanishing efficiency in temperature gradients and provides an explicit example with a Brownian particle in a linear temperature profile.

Findings

Efficiency decreases as inverse of cycle period in temperature gradients.

Maximum efficiency occurs at a finite cycle period inversely proportional to the square root of the gradient.

Relative efficiency loss scales with the square root of the temperature gradient.

Abstract

The efficiency of microscopic heat engines in a thermally heterogenous environment is considered. We show that, as a consequence of the recently discovered entropic anomaly, quasi-static engines, whose efficiency is maximal in a fluid at uniform temperature, have in fact vanishing efficiency in presence of temperature gradients. For slow cycles the efficiency falls off as the inverse of the period. The maximum efficiency is reached at a finite value of the cycle period that is inversely proportional to the square root of the gradient intensity. The relative loss in maximal efficiency with respect to the thermally homogeneous case grows as the square root of the gradient. As an illustration of these general results, we construct an explicit, analytically solvable example of a Carnot stochastic engine. In this thought experiment, a Brownian particle is confined by a harmonic trap and immersed in a fluid with a linear temperature profile. This example may serve as a template for the design of real experiments in which the effect of the entropic anomaly can be measured.