Surprising variants of Cauchy's formula for mean chord length
Abstract
We examine isotropic and anisotropic random walks which begin on the surface of linear (N ), square (N ×N ), or cubic (N ×N ×N ) lattices and end upon encountering the surface again. The mean length of walks is equal to N and the distribution of lengths n generally scales as n^{1.5} for large n . Our results are interesting in the context of an old formula due to Cauchy that the mean length of a chord through a convex body of volume V and surface S is proportional to V /S . It has been realized in recent years that Cauchy's formula holds surprisingly even if chords are replaced by irregular insect paths or trajectories of colliding gas molecules. The random walk on a lattice offers a simple and transparent understanding of this result in comparison to other formulations based on Boltzmann's transport equation in continuum.
 Publication:

Physical Review E
 Pub Date:
 November 2019
 DOI:
 10.1103/PhysRevE.100.050103
 arXiv:
 arXiv:1908.06608
 Bibcode:
 2019PhRvE.100e0103S
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 Replaced by the published version