Fourier series for the three-dimensional random flight

TL;DR

This paper develops a Fourier series expansion method to derive the probability density function of a three-dimensional random flight with isotropic initial conditions, providing exact solutions for low collision cases and a reference for series truncation.

Contribution

It introduces a Fourier series approach for random flights in any dimension and derives exact solutions for the 3D case, including explicit formulas for low collision scenarios.

Findings

Probability density functions for 1 and 2 collisions are obtained in elementary functions.

A series truncated at 132 collisions is available online for practical use.

Gaussian approximation is valid for large times beyond 100 times the average inter-collision time.

Abstract

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.