标题： 布朗运动的后牛顿近似下的福克-普朗克方程 显示英文标题

标题： Fokker-Planck equation for the Brownian motion in the post-Newtonian approximation

摘要： 在后牛顿玻尔兹曼方程的框架下分析了轻气体粒子和布朗重粒子的混合物，以确定布朗运动的福克-普朗克方程。 对于每种粒子种类，平衡分布函数指的是相应的后牛顿麦克斯韦-居特纳分布函数。 确定了第一和第二后牛顿近似中粘滞摩擦系数的表达式，并展示了它们与相应引力势的依赖关系。 发展了牛顿和后牛顿福克-普朗克方程中布朗运动的线性稳定性分析，其中假设扰动为小振幅的平面谐波。 从色散关系可以得出：(i) 对于小于杰恩斯波长的扰动波长，会出现两个传播模式——对应于相反方向的谐波波——以及一个不传播的模式；(ii) 对于大于杰恩斯波长的扰动波长，扰动的时间演化对应于增长或衰减，其中增长的模式对应于不稳定性。 显示英文摘要

摘要： A mixture of light-gas particles and Brownian heavy particles is analyzed within the framework of a post-Newtonian Boltzmann equation to determine the Fokker-Planck equation for the Brownian motion. For each species, the equilibrium distribution function refers to the corresponding post-Newtonian Maxwell-J\"uttner distribution function. The expressions for the friction viscous coefficient in the first and second post-Newtonian approximations are determined, and we show their dependence on the corresponding gravitational potentials. A linear stability analysis in the Newtonian and post-Newtonian Fokker-Planck equations for the Brownian motion is developed, where the perturbations are assumed to be plane harmonic waves of small amplitudes. From a dispersion relation it follows that: (i) for perturbation wavelengths smaller than the Jeans wavelength two propagating modes -- corresponding to harmonic waves in opposite directions -- and one mode that does not propagate show up; (ii) for perturbation wavelengths bigger than the Jeans wavelength the time evolution of the perturbation corresponds to a growth or a decay and the one which grows refers to the instability.