The finite quantum grand canonical ensemble and temperature from single-electron statistics for a mesoscopic device

I present a theoretical model of a quantum statistical ensemble for which, unlike in conventional physics, the total number of particles is extremely small. The thermodynamical quantities are calculated by taking a small N by virtue of the orthodicity of the canonical ensemble. The finite quantum grand partition function of a Fermi-Dirac system is calculated. The model is applied to a quantum dot coupled with a small two-dimensional electron system. Such a system consists of an alternatively singly and doubly occupied electron system confined in a quantum dot, which exchanges one electron with a small N two-dimensional electron reservoir. The analytic determination of the temperature of a (1 <-> 2) electron system and the role of ergodicity are discussed. The generalized temperature expression in the small N regime recovers the usual temperature expression form on taking the limit of N <-> infinity for the electron bath.