Finite Volume Thermodynamics of an Ideal Gas in a Periodic Box

TL;DR

This work analyzes finite-volume corrections for a non-relativistic ideal gas in a periodic box with twisted boundary condition $ψ(L)=e^{iθ}ψ(0)$. By deriving the exact single-particle spectrum and reformulating the partition function via the Dirac comb density of states and Fourier methods, it obtains explicit finite-size corrections within the canonical ensemble. The leading corrections to the free energy and mixing entropy are exponentially suppressed in $L_T$, while the average energy and its fluctuations acquire boundary-dependent terms; notably, the equation of state satisfies $\frac{P_d(θ)}{(\overline{E}(θ)/L^d)}=\frac{2}{d}$, independent of the twist angle. The approach yields an undergraduate-accessible, rigorous framework for the thermodynamic-limit behavior and clarifies how boundary conditions influence finite systems without altering the fundamental equation of state.

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, $ψ(L)=e^{iθ}ψ(0)$ for which the free particle spectrum is constructed in terms of the twist angle, $θ$. 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.

Figures (4)

• Figure 1: Plot of the ratio of the approximate single particle partition function, $L/{\lambda}_{T}$, to the exact single-particle partition function, $Z_{1}(0)$, in one spatial dimension, see Eq. (\ref{['eq:single-particle-partition-function']}), as a function of box size (normalized by the de Broglie wavelength), $L/{\lambda}_{T}$.
• Figure 2: Plot of the spectrum, ${\varepsilon}_{n}(\theta)$, as a function of the twist angle, $\theta$ for $n=0,\pm1,\pm2, 3$.
• Figure 3: Contour plot of finite size correction to the average energy (normalized by its thermodynamic limit counterpart), as function of box size, $L/{\lambda}_{T}$ and the twist angle, $\theta$. (see text for discussion).
• Figure 4: Plot of the standard deviation normalized by the thermodynamic limit result as function of box size, $L/{\lambda}_{T}$ and the twist angle, $\theta$. (see text for discussion).