On the master equation approach: linear and nonlinear Fokker--Planck equations
Year: 2004
Type: preprint
Abstract: We discuss the relationship between kinetic equations of the Fokker-Planck type (two linear and one non-linear) and the Kolmogorov (a.k.a. master) equations of certain N-body diffusion processes, in the context of Kac’s propagation-of-chaos limit. The linear Fokker-Planck equations are well-known, but here they are derived as a limit N → ∞ of a simple linear diffusion equation on 3N − C-dimensional N-velocity spheres of radius ∝ √ N (where C = 1 or 4 depending on whether the system conserves energy only or energy and momentum). In this case, a spectral gap separating the zero eigenvalue from the positive spectrum of the Laplacian remains as N → ∞, so that the exponential approach to equilibrium of the master evolution is passed on to the limiting Fokker-Planck evolution in R 3. The non-linear Fokker-Planck equation is known as Landau’s equation in the plasma physics literature. Its N-particle master equation, originally introduced (in the 1950s) by Balescu and Prigogine (BP), is studied here on the 3N − 4-dimensional N-velocity sphere. It is shown that the BP master equation represents a superposition of diffusion processes on certain two-dimensional sub-manifolds of R 3N determined by the conservation laws for two-particle collisions. The initial value problem for the BP master equation is proved to be well-posed and its solutions are shown to decay exponentially fast to equilibrium. However, the first non-zero eigenvalue of the BP operator is shown to vanish in the limit N → ∞. This indicates that the exponentially fast approach to equilibrium may not be passed from the finite-N master equation on to Landau’s nonlinear kinetic equation. c○2004 The authors. This paper may be reproduced for noncommercial purposes. 1 1
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2283052742