Universality of maximum-work efficiency of a cyclic heat engine based on a finite system of ultracold atoms

Abstract

We study the performance of a cyclic heat engine which uses a small system with a finite number of ultracold atoms as its working substance and works between two heat reservoirs at constant temperatures Th and Tc(<Th). Starting from the expression of heat capacity which includes finite-size effects, the work output is optimized with respect to the temperature of the working substance at a special instant along the cycle. The maximum-work efficiency ηmw at small relative temperature difference can be expanded in terms of the Carnot value {{boldsymbol{eta }}}_{{boldsymbol{C}}}={bf{1}}-{{boldsymbol{T}}}_{{boldsymbol{c}}}/{{boldsymbol{T}}}_{{boldsymbol{h}}}, {{boldsymbol{eta }}}^{{boldsymbol{m}}{bf{w}}}={{boldsymbol{eta }}}_{{boldsymbol{C}}}/{bf{2}}+{{boldsymbol{eta }}}_{{boldsymbol{C}}}^{{bf{2}}}({bf{1}}/{bf{8}}+{{boldsymbol{a}}}_{{bf{0}}})+{boldsymbol{ldots }}, where a0 is a function depending on the particle number N and becomes vanishing in the symmetric case. Moreover, we prove using the relationship between the temperatures of the working substance and heat reservoirs that the maximum-work efficiency, when accurate to the first order of ηC, reads {{boldsymbol{eta }}}^{{boldsymbol{m}}{bf{w}}}={{boldsymbol{eta }}}_{{boldsymbol{CA}}}+{bf{O}}(ΔT2). Within the framework of linear irreversible thermodynamics, the maximum-power efficiency is obtained as {{boldsymbol{eta }}}^{{boldsymbol{mp}}}={{boldsymbol{eta }}}_{{boldsymbol{CA}}}+{bf{O}}(ΔT2) through appropriate identification of thermodynamic fluxes and forces, thereby showing that this kind of cyclic heat engines satisfy the tight-coupling condition.