The unlikely Carnot efficiency

Abstract

The efficiency of an heat engine is traditionally defined as the ratio of its average output work over its average input heat. Its highest possible value was discovered by Carnot in 1824 and is a cornerstone concept in thermodynamics. It led to the discovery of the second law and to the definition of the Kelvin temperature scale. Small-scale engines operate in the presence of highly fluctuating input and output energy fluxes. They are therefore much better characterized by fluctuating efficiencies. In this study, using the fluctuation theorem, we identify universal features of efficiency fluctuations. While the standard thermodynamic efficiency is, as expected, the most likely value, we find that the Carnot efficiency is, surprisingly, the least likely in the long time limit. Furthermore, the probability distribution for the efficiency assumes a universal scaling form when operating close-to-equilibrium. We illustrate our results analytically and numerically on two model systems.

Stochastic efficiency of an isothermal work-to-work converter engine

“…They have computed the large deviation function for stochastic efficiency which supported the prediction given in Ref. [45]. Gingrich et al [47] computed the finite time probability density function for stochastic efficiency of a twolevel heat engine using time-asymmetric driving in a cyclic process.…”

“…It captures the large time statistics of the efficiency of the stochastic engine. In a recent study, Verley et al [45] computed the large deviation function J(η) using FT for microscopic heat engine using two set of examples: work to work converter engine and a photoelectric device. They have shown that the Carnot efficiency is least likely in the long time limit, which is a remarkable result.…”

Efficiency at and near maximum power of low-dissipation heat engines

“…The larger these two variables are, the better engine we have. Traditionally [15,39,40] they are defined as η = W out /Q h and P = W out /t p , where Q h = ∆E I + W I is the amount of heat accepted by the engine from the hot reservoir.…”

“…The ratio of these variables yields the efficiency which is thus itself random variable. Large deviation form of the corresponding probability distribution exhibits the following universal features [40][41][42][43][44][45][46][47][48]: (i) the most probable value of efficiency corresponds to its traditional definition using averages of work and heat, (ii) in case of time-reversal symmetric driving the least probable value of efficiency equals to η C .…”

Anomalous Brownian refrigerator

“…The non self-averaging nature of fluctuations in stochastic efficiency and other quantities requires detailed understanding of full probability distributions as opposed to the average behavior [41,42]. Large deviation properties of such distributions are recently under theoretical investigations [45,46,47,48,49,50,51]. Research on fluctuation relations for heat engines [52,53,54] are being pursued.…”