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arXiv math-ph/0401027math-phmath.MP
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动力学理论的主方程方法：线性和非线性Fokker-Planck方程

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

Michael K. -H. Kiessling, Carlo Lancellotti

备注：
20 pages; based on talk at the 18th ICTT Conference. Some typos and a few minor technical fixes. Modified title slightly
发表信息：
Transp. Theor. Stat. Phys. vol.33, p.379-401 (2004)
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摘要：

我们讨论了福克-普朗克型动力学方程（两个线性方程和一个非线性方程）与柯尔莫戈罗夫（又名主）方程的某些N体扩散过程中，在卡茨的“传播混沌”的限制。线性Fokker-Planck方程是众所周知的，但在这里它们是作为半径为sqrt（N）的（3 N-C）维N-速度球上的简单线性扩散方程的极限N->infty导出的（其中C=1或4取决于系统是仅保存能量还是保存能量和动量）。在这种情况下，将拉普拉斯算子的零本征值与正谱分开的谱隙保持为N->infty，因此主演化的指数平衡态被传递到R^3中的极限福克-普朗克演化。非线性福克-普朗克方程在等离子体物理文献中被称为朗道方程。在（3 N-4）维N速度球上研究了它的N粒子主方程，该方程最初是由Balescu和Prigogine（BP）在20世纪50年代提出的。结果表明，BP主方程代表了由两粒子碰撞守恒律确定的R^{3 N}的某些二维子流形上的扩散过程的叠加。证明了BP主方程的初值问题是适定的，其解是指数衰减的。然而，BP算子的第一个非零特征值在极限N->infty中消失。这表明，指数快速接近平衡可能不会通过从有限N主方程的朗道的非线性动力学方程。

英文摘要：

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.

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