General lower and upper bounds of efficiency of heat engines.

Abstract

According to thermodynamics, the efficiency of a heat engine is upper bounded by Carnot efficiency. For macroscopic systems, the Carnot efficiency is, however, achieved only for reversible processes. And, considerable attention has been paid to provide general evaluation of the efficiency at a finite speed. Recently, several upper bounds of the efficiency have been derived in the context of the trade-off among the efficiency, power, and other quantities such as the fluctuation of power. Here, we show general lower and upper bounds of the efficiency from the thermodynamic uncertainty relations for the entropy production and for the heat transfers. The lower bound is characterized by the ratio between the fluctuation of the irreversible entropy production and mean output work. The upper bound of the efficiency is described by a generalized precision of the heat transfers among the working substance, and hot and cold reservoirs. Fix the quantities except for the variance, the lower bound increases with a decrease in fluctuation, while the upper bound rises with an increase in fluctuation. We explicitly derive necessary and sufficient conditions of both the lower and upper bounds in a unified manner. Hence, our result provides an operating principle of the heat engine.