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In a microcanonical ensemble (constant $NVE$, hard reflecting walls) and in amolecular dynamics ensemble (constant $NVE\mathbf{PG}$, periodic boundaryconditions) 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 $\mathbf{G}$, theindividual velocity components, velocity moduli, and energies have transformedbeta distributions with different arguments and shape parameters depending on$d$, $N$, $E$, the boundary conditions, and possible symmetries in the initialconditions. This can be shown marginalizing the joint distribution ofindividual energies, which is a symmetric Dirichlet distribution. In thethermodynamic limit the beta distributions converge to gamma distributions withdifferent arguments and shape or scale parameters, corresponding respectivelyto the Gaussian, i.e., Maxwell-Boltzmann, Maxwell, and Boltzmann orBoltzmann-Gibbs distribution. These analytical results agree with moleculardynamics and Monte Carlo simulations with different numbers of hard disks orspheres and hard reflecting walls or periodic boundary conditions. Theagreement is perfect with our Monte Carlo algorithm, which acts only onvelocities independently of positions with the collision versor sampleduniformly on a unit half sphere in $d$ dimensions, while slight deviationsappear with our molecular dynamics simulations for the smallest values of $N$.

Velocity and energy distributions in microcanonical ensembles of hard spheres