Finite Volume Thermodynamics of an Ideal Gas in a Periodic Box

TL;DR

This paper investigates how finite volume effects influence thermodynamic properties of an ideal gas in a periodic box, revealing boundary condition impacts on energy and fluctuations but not on the equation of state.

Contribution

It provides an exact analysis of finite-size corrections in a periodic box using twisted boundary conditions, highlighting boundary effects on thermodynamic observables.

Findings

Finite volume corrections affect energy and fluctuations.

Boundary conditions influence thermodynamic observables.

Equation of state remains unchanged by boundary effects.

Abstract

Approach to the thermodynamic limit of a non-relativistic ideal gas in a periodic box is investigated. The single particle wave function obeys twisted boundary condition, $\psi(L)=e^{i\theta}\psi(0)$ for which the free particle spectrum is constructed in terms of the twist angle, $\theta$. The exact density of states is utilized to construct finite-size corrections of thermodynamic observables. Leading finite volume corrections in the free energy do not arise due to the boundary -- its implication for mixing entropy is examined. Finite volume corrections to the average energy, its fluctuations and the pressure are also examined with corrections arising exclusively through the boundary condition. However, the equation of state, the ratio of pressure to energy density, remains unmodified by the boundary.