The Langevin equation on Lie algebras: Maxwell-Boltzmann is not always the equilibrium

Journal Article · Tue Dec 15 00:00:00 EST 2009 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2009.08.005· OSTI ID:21336077

Rajeev, S G ^[1]

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1. Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 (United States)

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.

• OSTI ID: 21336077
• Journal Information: Annals of Physics (New York), Vol. 324, Issue 12; Other Information: DOI: 10.1016/j.aop.2009.08.005; PII: S0003-4916(09)00149-3; Copyright (c) 2009 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
EQUILIBRIUM
FLUID MECHANICS
FOKKER-PLANCK EQUATION
GEOMETRY
INTEGRAL CALCULUS
LANGEVIN EQUATION
LIE GROUPS
MATHEMATICAL SOLUTIONS
RANDOMNESS
ROTATION
STOCHASTIC PROCESSES