Approaching the Thermodynamic Limit of an Ideal Gas

TL;DR

The paper addresses how the thermodynamic limit for an ideal gas is approached when wall-induced particle correlations are included within the canonical ensemble.It compares classical and quantum wall-correlation mechanisms using a finite-range wall model and Dirichlet confinement, deriving leading finite-size corrections that scale as $N^{-1/3}$ through an effective excluded length $\ell(\beta)$ and quantifying corrections to $\overline{E}$ and energy fluctuations.The work provides explicit expansions for the partition function and energy statistics, highlighting the distinct temperature dependencies and the role of the thermal de Broglie wavelength in the quantum case.These results clarify how non-extensive wall effects vanish with increasing $N$, offer a teaching framework for undergraduate/graduate statistical mechanics, and have relevance for small-system physics and numerical simulations.

Abstract

For a gas confined in a container, particle-wall interactions produce modifications to the partition function involving the average surface density of gas particles. While such correlations have a vanishing effect in the thermodynamic limit, examining them is beneficial for a sharper understanding of how the limit is attained. We contrast a classical and a quantum model of particle-wall correlations within the canonical ensemble.

Figures (2)

• Figure 1: Cutaway of a cube of volume $L^3$ with a length $\ell$ depicted as excluded from each direction. The three interior faces shown have volume $L^2 \times \ell$ excluded, except at the three edges shown where there is an overlapping volume of $L \times \ell^2$, aside from the far corner where there is an overlapping volume of $\ell^3$.
• Figure 2: Relative width of the energy distribution compared to a Gaussian $\Gamma_E \equiv \frac{{\sigma}_E}{{\overline{E}}} / \left( \frac{{\sigma}_E}{{\overline{E}}} \right)_G$ as a function of temperature. Using a classical model of the confining potential Eq. (\ref{['eq:V0']}), the relative width Eq. (\ref{['eq:Gamma']}) is shown for three different values of the range parameter $a$, spanning $1$--$10\%$ of the linear size of the container. As the temperature is raised past $k_B T = \frac{1}{2} V_0$, the distribution changes from broader-than-Gaussian to narrower-than-Gaussian.