Thermodynamic limit of the free electron gas on a circle

TL;DR

This paper derives the ground state partition function of a 1D free electron gas on a circle using eigenfunction expansion, highlighting the differences between zero and finite temperature limits.

Contribution

It provides an analytic derivation of the ground state partition function and clarifies the limitations of the eigenfunction expansion approach at finite temperatures.

Findings

Analytic expression for the ground state partition function derived

Eigenfunction expansion approach is valid at zero temperature

Approximation fails at finite temperature due to non-coincident limits

Abstract

We show that for the ground state of a one dimensional free electron gas on a circle the analytic expression for the canonical ensemble partition function can be easily derived from the density matrix by assuming that the thermodynamic limit coincides with the limit of the eigenfunction expansion of the kinetic energy. This approximation fails to give the finite temperature partition function because those two limits cannot be chosen as coincident.