随机游走概率密度函数的一种推导方法

A derivation of random walk probability density function

随机游走模型是一类经典的物理模型,能应用在诸多物理场景之中,甚至在很多其他领域也能找到它的身影.但一般教材中对这个问题的计算较为复杂,或使用到过于高深的数学工具,或使用了没有必要引入的近似.本文在多步数的近似之下,应用中心极限定理,推导出随机游走问题在某一方向分量的概率密度函数.接着利用随机游走问题的球对称性和上一步中得到的结果,推导出径向的概率密度分布函数.

Random walks are one of the classical models studied in physics,which can be applied to multiple physical problems,even in other fields.However,most textbooks concerning this topic may use complicated math methods or introduce unnecessary approximations when calculating the result.In this paper,the random walk probability density function of one particular direction is derived using the central limit theorem under many steps approximation.The radial probability density function can be derived using the result above and introducing the isotropic postulation.