Velocity and energy distributions in microcanonical ensembles of hard spheres

TL;DR

This paper derives analytical velocity and energy distributions for hard sphere ensembles in microcanonical settings, showing convergence to classical distributions in the thermodynamic limit and validating results with simulations.

Contribution

It provides explicit analytical forms of velocity and energy distributions in microcanonical ensembles of hard spheres, including their convergence to Maxwell-Boltzmann distributions in the thermodynamic limit.

Findings

Distributions are beta and Dirichlet types depending on conditions.

Distributions converge to gamma distributions in the thermodynamic limit.

Simulations confirm analytical predictions with minor deviations for small N.

Abstract

In a microcanonical ensemble (constant $NVE$, hard reflecting walls) and in a molecular dynamics ensemble (constant $NVE\mathbf{PG}$, periodic boundary conditions) with a number $N$ of smooth elastic hard spheres in a $d$-dimensional volume $V$ having a total energy $E$, a total momentum $\mathbf{P}$, and an overall center of mass position $\mathbf{G}$, the individual velocity components, velocity moduli, and energies have transformed beta distributions with different arguments and shape parameters depending on $d$, $N$, $E$, the boundary conditions, and possible symmetries in the initial conditions. This can be shown marginalizing the joint distribution of individual energies, which is a symmetric Dirichlet distribution. In the thermodynamic limit the beta distributions converge to gamma distributions with different arguments and shape or scale parameters, corresponding respectively to the Gaussian, i.e., Maxwell-Boltzmann, Maxwell, and Boltzmann or Boltzmann-Gibbs distribution. These analytical results agree with molecular dynamics and Monte Carlo simulations with different numbers of hard disks or spheres and hard reflecting walls or periodic boundary conditions. The agreement is perfect with our Monte Carlo algorithm, which acts only on velocities independently of positions with the collision versor sampled uniformly on a unit half sphere in $d$ dimensions, while slight deviations appear with our molecular dynamics simulations for the smallest values of $N$.