相对论扩散
Relativistic diffusion
摘要：
本文用Schay和Dudley的方法讨论了在适当时间内的相对论性扩散。本文导出了各种坐标系下的（朗之万）随机微分方程，证明了在某些坐标系下随机微分方程变为线性的。给出了动量几率分布的显式表达式，讨论了相对论性粒子在外电磁场中的扩散。我们在平行电场和磁场的情况下求解朗之万方程。我们导出了几率分布演化的动力学方程，讨论了导致平衡分布的阻力项，相对论模拟的Ornstein-Uhlenbeck过程不是唯一的。我们表明，如果阻力来自主方程的扩散近似，那么它的形式是强烈限制。导致Tsallis平衡分布的阻力满足这一限制，而Juettner分布的阻力则不满足。我们证明了相对论能量的任何函数都可以是静态电场中粒子的平衡分布。本文对含摩擦力的时间演化作了初步研究。结果表明，该问题等价于粒子在双曲面上运动的量子力学，其势由阻力决定。本文简要讨论了重离子碰撞中出现的扩散现象。
英文摘要：
We discuss a relativistic diffusion in the proper time in an approach of Schay and Dudley. We derive (Langevin) stochastic differential equations in various coordinates.We show that in some coordinates the stochastic differential equations become linear. We obtain momentum probability distribution in an explicit form.We discuss a relativistic particle diffusing in an external electromagnetic field. We solve the Langevin equations in the case of parallel electric and magnetic fields. We derive a kinetic equation for the evolution of the probability distribution.We discuss drag terms leading to an equilibrium distribution.The relativistic analog of the Ornstein-Uhlenbeck process is not unique. We show that if the drag comes from a diffusion approximation to the master equation then its form is strongly restricted. The drag leading to the Tsallis equilibrium distribution satisfies this restriction whereas the one of the Juettner distribution does not. We show that any function of the relativistic energy can be the equilibrium distribution for a particle in a static electric field. A preliminary study of the time evolution with friction is presented. It is shown that the problem is equivalent to quantum mechanics of a particle moving on a hyperboloid with a potential determined by the drag. A relation to diffusions appearing in heavy ion collisions is briefly discussed.
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