Statistics of Energy Partitions for Many-Particle Systems in Arbitrary Dimension

In some previous articles, we defined several partitions of the total kinetic energy $T$ of a system of $N$ classical particles in $\mathbb{R}^d$ 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 $3\leqslant N \leqslant 100$. 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 $3\leqslant N \leqslant 100$.