# On the free path length distribution for linear motion in an n-dimensional box

## TL;DR

This paper analyzes the distribution of free path lengths for particles in an n-dimensional box, providing explicit formulas and methods to recover box dimensions from the distribution's features.

## Contribution

It derives explicit, piecewise real analytic formulas for free path length distributions in 2D and 3D boxes, linking distribution features to box dimensions.

## Key findings

- Explicit formulas for 2D and 3D free path length distributions.
- Distribution features reveal the box's side lengths.
- As R and N tend to infinity, distributions converge to geometric intersection models.

## Abstract

We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box. If each particle travels a distance R, then, as R tends to infinity the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we give an explicit formula (piecewise real analytic) for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N tends to infinity, and give an explicit (again piecewise real analytic) formula for its probability density function. Further, in both models we can recover the side lengths of the box from the location of the discontinuities of the probability density functions.