arXiv 1003.47422010-03-24cond-mat.stat-mechphysics.flu-dyn

速度空间中的反常输运：从Fokker-Planck到一般方程

Anomalous Transport in Velocity Space, from Fokker-Planck to General Equation

S. A. Trigger

备注：: 23 pages, 0 figures
DOI：: 10.1088/1751-8113/43/28/285005

摘要：

基于主方程和适当的概率转移函数（PTF），研究了动量（速度）空间中的反常扩散问题。作者最近开发的坐标空间的方法，适用于速度空间进行必要的修改。导出了动量空间中动量分布函数随时间演化的新的一般方程。这使得解决各种问题的异常运输时的概率转移函数（PTF）有一个长尾的动量空间。对于相反的情况下的PTF迅速减少作为转移动量的函数（当大转移动量被强烈抑制），所开发的方法使我们能够考虑强烈的非平衡情况下的系统演化。研究了稳态解和非稳态解。作为一个例子，玻尔兹曼型PT函数的特定情况下，重粒子和轻粒子的碰撞与确定（规定）的分布函数，这可能是强烈的非平衡，被认为是在建议的一般方法。找到了合适的扩散系数和摩擦系数，证明了在这些情况下，摩擦系数和扩散系数之间的爱因斯坦关系是不成立的。

英文摘要：

The problem of anomalous diffusion in momentum (velocity) space is considered based on the master equation and the appropriate probability transition function (PTF). The approach recently developed by the author for coordinate space, is applied with necessary modifications to velocity space. A new general equation for the time evolution of the momentum distribution function in momentum space is derived. This allows the solution of various problems of anomalous transport when the probability transition function (PTF) has a long tail in momentum space. For the opposite cases of the PTF rapidly decreasing as a function of transfer momenta (when large transfer momenta are strongly suppressed), the developed approach allows us to consider strongly non-equilibrium cases of the system evolution. The stationary and non-stationary solutions are studied. As an example, the particular case of the Boltzmann-type PT-function for collisions of heavy and light particles with the determined (prescribed) distribution function, which can be strongly non-equilibrium, is considered within the proposed general approach. The appropriate diffusion and friction coefficients are found. The Einstein relation between the friction and diffusion coefficients is shown to be violated in these cases.

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