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

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

2009-07-14

arxiv.org/abs/0907.2401v1

Physics

Mathematical Physics

Scientific paper

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.

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