扩散的能量和动量守恒--随机力学近似--I部分
Energy and Momentum Conservation for Diffusion - A Stochastic Mechanics Approximation - Part I
摘要：
本文模拟了经典扩散的主要粒子通过热浴通过一个前极限微观表示其漂移的动量和能量转移在碰撞时间。碰撞点的线性插值路径可以近似为使用相同漂移函数的“内接”连续随机微分方程的解。采用随机力学的结果，然后表明，组合的主粒子/热浴系统不交换或辐射能量，如果主粒子的位置的概率分布来自薛定谔方程。此外，它表明，碰撞之间的主要粒子的距离和平均碰撞时间必须满足一种类型的Minkowski不变量。因此，如果主粒子通过碰撞点的碰撞前和碰撞后速度之间存在相关性，则平均行进距离可以经由洛伦兹变换与平均粒子间碰撞时间相关。最后一节表明，这种方法可以适用于所有的弹性主粒子/热浴粒子碰撞，无论是通过直接计算，涉及建模的碰撞散射或通过改变热浴的属性。
英文摘要：
This paper models the classical diffusion of a main particle through a heatbath by means of a pre-limit microscopic representation of its drifted momentum and energy transfers at collision times. The collision point linear interpolated path can be approximated by the solution to the "inscribed" continuous stochastic differential equation using the same drift function. Employing results from stochastic mechanics it is then shown that the combined main particle/heatbath system does not exchange or radiate energy if the probability distribution for the position of the main particle is derived from Schroedinger's equation. Furthermore it is shown that the main particle distance traveled between collisions and the mean inter-collision time must satisfy a type of Minkowski invariant. Hence if there is a correlation between the pre- and post-collision velocities of the main particle through a collision point then the mean distance traveled can be related to the mean inter-particle collision times via a Lorentz transformation. The last Section shows that this approach can be applied to all elastic main particle/heatbath particle collisions either via direct calculation involving modeling the collision scattering or by altering the properties of the heatbath.
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