On the master equation approach to kinetic theory: linear and nonlinear Fokker--Planck equations

TL;DR

This paper explores the connection between kinetic Fokker-Planck equations and master equations for N-body diffusion processes, analyzing their spectral properties and convergence to equilibrium in the large N limit.

Contribution

It derives the Fokker-Planck equations as limits of diffusion processes on high-dimensional spheres and studies the spectral gap behavior of the associated master equations.

Findings

Linear Fokker-Planck equations are limits of diffusion on N-velocity spheres.

Spectral gap persists for linear equations, ensuring exponential convergence.

For the nonlinear Landau equation, the spectral gap vanishes as N approaches infinity.

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->infty of a simple linear diffusion equation on (3N-C)-dimensional N-velocity spheres of radius sqrt(N) (with 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->infty,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->infty. 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.