Distribution of energy in the ideal gas that lacks equipartition

Abstract

The energy and velocity distributions of ideal gas particles were first obtained by Boltzmann and Maxwell in the second half of the nineteenth century. In the case of a finite number of particles, the particle energy distribution was obtained by Boltzmann in 1868. However, it appears that this distribution is not valid for all vessels. A round vessel is a special case due to the additional integral of motion, the conservation of the gas angular momentum. This paper is intended to fill this gap, it provides the exact distribution of particle energy for a classical non-rotating ideal gas of a finite number of colliding particles in a round vessel. This previously unknown distribution was obtained analytically from the first principles, it includes the dependence on all the particle masses. The exact mean energies of gas particles are also found to depend on the system parameters, i.e., the distribution of energy over the degrees of freedom is not uniform. Therefore, the usual ideal gas model allows for the uneven energy partitioning, which we study here both theoretically and in simple numerical experiments.