Maxwell's hypothesis reconsidered

TL;DR

This paper reexamines Maxwell's assumption of independence among molecular velocity components, showing that isotropy in phase space nullifies the need for independence and that Gaussian distribution results hold under weaker assumptions.

Contribution

It demonstrates that Maxwell's independence assumption can be relaxed due to phase space isotropy, maintaining the Gaussian velocity distribution under weaker conditions.

Findings

Independence assumption is unnecessary due to phase space isotropy.

Gaussian velocity distribution remains valid under weaker assumptions.

Reevaluation of Maxwell's original derivation with relaxed assumptions.

Abstract

Maxwell's derivaion of the distributions of the velocities of molecules is based on the assumption that the velocity components in the three mutualy orthogonal directions are independent. Here we note that his assumption, the phase space is isotropic, in fact nullifies the effect of a variety of dependencies among the velocity componenets. Thus we can do away with the independence assumption. Further, we observe that his conclusion regarding distribution of the velocity components (Gaussian) remains true under a set of weaker assumptions.