New techniques to solve the 1-dimensional random flight

TL;DR

This paper introduces two innovative methods for solving the one-dimensional random flight problem, including an expansion in collision number and a Fourier series approach, with the latter applicable to multiple dimensions.

Contribution

The paper presents novel techniques for solving the 1D random flight, notably extending Fourier series methods to higher dimensions and deriving new mathematical identities.

Findings

Expansion in the number of collisions as a solution method

Fourier series approach applicable to multiple dimensions

New mathematical identities derived

Abstract

We present here two new techniques to solve the one-dimensional random flight. The first one is an expansion in the number of collisions. The second one is the obtention of a Fourier series. This second technique can be applied to an arbitrary number of dimensions. Some mathematical identities are obtained.