Finite Quantum Grand Canonical Ensemble and Temperature from Single Electron Statistics in a Mesoscopic Device

TL;DR

This paper develops a theoretical model for a quantum statistical ensemble with a very small number of particles, deriving a finite quantum grand canonical ensemble and applying it to a quantum dot system to determine temperature from single-electron statistics.

Contribution

It introduces a finite quantum grand canonical ensemble model for small N systems and applies it to quantum dots to extract temperature from single-electron data.

Findings

Finite quantum grand partition function derived for small N Fermi-Dirac systems.

Analytic expression for temperature in a (1↔2) electron quantum dot system.

Generalized temperature formula reduces to classical form as N approaches infinity.

Abstract

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 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 system consists of an alternatively single and double occupied electron system confined in a quantum dot, which exhanges one electron with a small $N$ two dimensional electron reservoir. The analytic determination of the temperature of a $(1\leftrightarrow 2)$ electron system and the role of ergodicity are discussed. The generalized temperature expression in the small $N$ regime recovers the usual temperature expression by taking the limit of $N\to\infty$ of the electron bath.