Brownian motion on a manifold

Journal Article · Tue Jul 01 04:00:00 EDT 1986 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01010902· OSTI ID:6843064

Van Kampen, N G

The question of the existence and correct form of equations describing Brownian motion on a manifold cannot be answered by mathematic alone, but requires a study of the underlying physics. As in classical mechanics, manifolds enter through the transformation of variables needed to account for the presence of constraints. The constraints are either due to a physical agency that forces the motion to remain on a manifold, or they represent conserved quantities of the equation of motion themselves. Also the Brownian motion is described either by a Smoluchowski diffusion equation or by a Kramers equation. The four cases lead to the following conclusions. (i) Smoluchowski diffusion with a conserved quantity reduces to a diffusion equation on the manifold; (ii) The same is true for diffusion with a physical constraint in three dimensions, but in more dimensions it may happen that no autonomous equation on the manifold results; (iii) A Kramers equation with a conserved quantity reduces to an equation on the manifold, but in general not of the form of a Kramers equation; (iv) The Kramers equation with a physical constraint reduces to an autonomous Kramers equation on the manifold only for a special shape of that constraint. Throughout, only a certain type of physical constraints has been envisaged, and global questions are ignored. Finally, the customary heuristic construction of a Fokker-Planck equation for a mechanical system on a manifold is demonstrated for the case of Brownian rotation of rigid body, and its shortcomings are emphasized.

• Research Organization: Univ. at Utrecht, Utrecht
• OSTI ID: 6843064
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 44:1/2; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BROWNIAN MOVEMENT
DIFFERENTIAL EQUATIONS
DIFFUSION
ELEMENTARY PARTICLES
EQUATIONS
FOKKER-PLANCK EQUATION
KRAMERS THEOREM
LIOUVILLE THEOREM
MASSLESS PARTICLES
MATHEMATICAL MANIFOLDS
MATHEMATICAL MODELS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE MODELS
PHOTONS
QUANTUM MECHANICS
REFLECTION
TRANSFORMATIONS