Maximum Efficiency of Heat Engines Based on a Small System: Carnot Cycle at the Nanoscale

We study the maximum efficiency of a Carnot cycle heat engine based on a small system. It is revealed that due to the finiteness of the system, irreversibility may arise when the working substance contacts with a heat bath. As a result, there is a working-substance-dependent correction to the usual Carnot efficiency, which is valid only when the working substance is in the thermodynamic limit. We derives a general and simple expression for the maximum efficiency of a Carnot cycle heat engine in terms of the relative entropy. This maximum efficiency approaches the usual Carnot efficiency asymptotically when the size of the working substance increases to the thermodynamic limit. Our study extends the Carnot's result to cases with arbitrary size working substance and demonstrates the subtlety of thermodynamics in small systems.