摘要翻译:
一个质量$m$的粒子在一个质量$m\llm$的分子浴中的布朗运动的微观理论被认为超过了质量比$m/m$的最低级。相应的Langevin方程包含对耗散力的非线性修正,广义Fokker-Planck方程包含二阶以上的导数。这些方程由第一性原理导出,其系数用粒子所受微观力的相关函数表示。在有限时间的广义Rayleigh模型中,我们显式地计算了分子-粒子碰撞的系数。在低密度熔池的极限下,我们恢复了先前对二元碰撞模型的结果。在一般情况下,方程包含附加修正,浴密度为二次修正,源于有限的碰撞时间。这些修正可以维持在$(m/m)^2$量级,并使平稳分布非麦克斯韦分布。给出了相关的数值模拟结果。
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英文标题:
《Generalized Fokker-Planck equation, Brownian motion, and ergodicity》
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作者:
A.V. Plyukhin
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最新提交年份:
2008
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分类信息:
一级分类:Physics 物理学
二级分类:Statistical Mechanics 统计力学
分类描述:Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变,热力学,场论,非平衡现象,重整化群和标度,可积模型,湍流
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一级分类:Mathematics 数学
二级分类:Dynamical Systems 动力系统
分类描述:Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
微分方程和流动的动力学,力学,经典的少体问题,迭代,复杂动力学,延迟微分方程
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英文摘要:
Microscopic theory of Brownian motion of a particle of mass $M$ in a bath of molecules of mass $m\ll M$ is considered beyond lowest order in the mass ratio $m/M$. The corresponding Langevin equation contains nonlinear corrections to the dissipative force, and the generalized Fokker-Planck equation involves derivatives of order higher than two. These equations are derived from first principles with coefficients expressed in terms of correlation functions of microscopic force on the particle. The coefficients are evaluated explicitly for a generalized Rayleigh model with a finite time of molecule-particle collisions. In the limit of a low-density bath, we recover the results obtained previously for a model with instantaneous binary collisions. In general case, the equations contain additional corrections, quadratic in bath density, originating from a finite collision time. These corrections survive to order $(m/M)^2$ and are found to make the stationary distribution non-Maxwellian. Some relevant numerical simulations are also presented.
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PDF链接:
https://arxiv.org/pdf/802.0172