Diffusive properties of persistent walks on cubic lattices with application to periodic Lorentz gases

TL;DR

This paper calculates diffusion coefficients for persistent random walks on cubic lattices with memory effects and applies these results to analyze normal diffusion regimes in a 3D periodic Lorentz gas model.

Contribution

It introduces a method to compute diffusion coefficients for persistent walks with memory and applies it to a complex billiard system, bridging lattice walk theory and physical gas models.

Findings

Diffusion coefficients depend on the memory length of the walk.

Normal diffusion regimes are identified in the Lorentz gas model.

Persistent walk approximation aligns with simulation results in these regimes.

Abstract

We calculate the diffusion coefficients of persistent random walks on cubic and hypercubic lattices, where the direction of a walker at a given step depends on the memory of one or two previous steps. These results are then applied to study a billiard model, namely a three-dimensional periodic Lorentz gas. The geometry of the model is studied in order to find the regimes in which it exhibits normal diffusion. In this regime, we calculate numerically the transition probabilities between cells to compare the persistent random-walk approximation with simulation results for the diffusion coefficient.