Thermodynamic instability and first-order phase transition in an ideal Bose gas

TL;DR

This paper rigorously demonstrates that an ideal Bose gas confined in a finite cubic box exhibits a first-order phase transition with a genuine singularity at a critical particle number, challenging the assumption that such transitions require the thermodynamic limit.

Contribution

It provides the first rigorous evidence that finite systems can undergo discontinuous phase transitions with true singularities, without relying on the thermodynamic limit or continuous approximations.

Findings

Thermodynamic instability appears at particle number ≥ 7616.

Finite ideal Bose gas can exhibit a genuine first-order phase transition.

The critical particle number is a characteristic of the cubic geometry.

Abstract

We conduct a rigorous investigation into the thermodynamic instability of ideal Bose gas confined in a cubic box, without assuming thermodynamic limit nor continuous approximation. Based on the exact expression of canonical partition function, we perform numerical computations up to the number of particles one million. We report that if the number of particles is equal to or greater than a certain critical value, which turns out to be 7616, the ideal Bose gas subject to Dirichlet boundary condition reveals a thermodynamic instability. Accordingly we demonstrate - for the first time - that, a system consisting of finite number of particles can exhibit a discontinuous phase transition featuring a genuine mathematical singularity, provided we keep not volume but pressure constant. The specific number, 7616 can be regarded as a characteristic number of 'cube' that is the geometric shape of the box.