The Maxwell-Boltzmann Distribution is not the Equilibrium on a Hyperboloid

TL;DR

This paper explores the geometric formulation of stochastic dynamics on Lie algebras, revealing that the classical Maxwell-Boltzmann distribution is not an equilibrium on hyperboloids, and presents an alternative integrable equilibrium solution.

Contribution

It introduces a geometric approach to Fokker-Planck equations on Lie algebras, showing the non-normalizability of Boltzmann distribution on non-unimodular algebras and providing explicit solutions on hyperboloids.

Findings

Boltzmann distribution is not normalizable on non-unimodular algebras.

An alternative equilibrium solution exists on hyperboloids, breaking rotational symmetry.

The most probable velocity on the hyperboloid is non-zero.

Abstract

We give a geometric formulation of the Fokker-Planck-Kramer equations for a particle moving on a Lie algebra under the influence of a dissipative and a random force. Special cases of interest are fluid mechanics, the Stochastic Loewner Equation and the rigid body. We find that the Boltzmann distribution, although a static solution, is not normalizable when the algebra is not unimodular. This is because the invariant measure of integration in momentum space is not the standard one. We solve the special case of the upper half-plane (hyperboloid) explicitly: there is another equilibrium solution to the Fokker-Planck equation, which is integrable. It breaks rotation invariance; moreover, the most likely value for velocity is not zero.