On the Master‐Equation Approach to Kinetic Theory: Linear and Nonlinear Fokker‐Planck Equations

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We discuss the relationship between kinetic equations of the Fokker‐Planck type (two linear and one nonlinear) 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 ℝ3. The nonlinear Fokker‐Plank 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 submanifolds of ℝ3N 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 nonzero 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.