碰撞动力学方程导出分数阶扩散极限的方法研究

Research on the Method of Deriving Fractional Diffusion Limits from Collision Dynamics Equations

摘要: 碰撞动力学方程(Boltzmann方程)是描述物质中微观粒子运动的重要方程，该方程中包含微观粒 子之间的碰撞过程，因此可以用来研究物质中的输运现象。 当物质中微观粒子(如分子或粒子)的运 动呈现出一定的长时间规律且不满足Fick扩散定律时，我们就会遇到分数阶扩散的现象。 通常情 况下，此时我们可以通过碰撞动力学方程推导出分数阶扩散的极限。 本文将探究对碰撞动力学方 程的缩放方程的估计进而导出分数阶扩散极限来实现导出分数阶扩散极限。

Abstract: The Boltzmann equation of collision dynamics is an important equation for describing the motion of microscopic particles, which includes microscopic particles. The collision process between particles can be used to study the Transport phenomenon. When the transport of microscopic particles (such as molecules or particles) exhibit a certain long-term pattern and does not satisfy Fick’s diffusion law, we will encounter the phenomenon of fractional diffusion. Normally, at this point, we can derive the limit of fractional diffusion through the collision dynamics equation. This article will explore the scaling equation for collision dynamics equations. The estimation is then used to achieve the derivation of fractional diffusion limits.