Statistics of energy partitions for many-particle systems in arbitrary dimension

TL;DR

This paper derives formulas for the average energy components in many-particle systems of equal mass in arbitrary dimensions, supported by numerical validation for planar systems and considering random masses.

Contribution

It provides new formulas for mean energy partitions in d-dimensional systems with equal masses, extending previous work to arbitrary dimensions.

Findings

Formulas for mean energy components are derived for any dimension and particle number.

Numerical validation confirms the formulas for planar systems with 3 to 100 particles.

The study also explores systems with randomly chosen particle masses.

Abstract

In some previous articles, we defined several partitions of the total kinetic energy T of a system of N classical particles in the d-dimensional Euclidean space into components corresponding to various modes of motion. In the present paper, we propose formulas for the mean values of these components in the normalization T=1 (for any d and N) under the assumption that the masses of all the particles are equal. These formulas are proven at the "physical level" of rigor and numerically confirmed for planar systems (d=2) at N from 3 through 100. The case where the masses of the particles are chosen at random is also considered. The paper complements our article of 2008 [Russian J Phys Chem B, 2(6), 947-963] where similar numerical experiments were carried out for spatial systems (d=3) at N from 3 through 100.