论文标题
三维随机飞行的傅立叶系列
Fourier series for the three-dimensional random flight
论文作者
论文摘要
各向同性初始条件的随机飞行的概率密度函数是通过碰撞数量和溶液的空间谐波的扩展（如傅立叶级数）获得的。该方法适用于任何维度，并针对三维情况进行详细研究。在这种情况下，还使用不同的方法发现了概率密度的函数为1和2碰撞，从而使它们从基本函数和polygarithm函数li $ _2 $方面产生。后一种方法是确切的，因为它不必像第一个方法那样截断一个系列。这提供了决定截断该系列的位置的参考。向网页提供了一个链接，读者可以在其中下载132碰撞的截断系列；在大于平均碰撞时间的100倍以上，使用高斯近似值。初始条件是沿固定方向运动的粒子的情况。
The probability density function of the random flight with isotropic initial conditions is obtained by an expansion in the number of collisions and the in the spatial harmonics of the solution, as in a Fourier series. The method holds for any dimension and is worked out in detail for the three dimensional case. In this case the probability density functions conditional to 1 and 2 collisions are also found using a different method, which yields them in terms of elementary functions and the polylogarithm function Li$_2$. The latter method is exact in the sense that one does not have to truncate a series, as in the first method. This provides a reference to decide where to truncate the series. A link is provided to a web page where the reader may download the series truncated at 132 collisions; for times larger than 100 times the average inter-collision time, the Gaussian approximations is used. The case in which the initial condition is a particle moving along a fixed direction is briefly considered.
