Diffusion in infinite and semi-infinite lattices with long-range coupling

TL;DR

This paper proves that in one-dimensional lattices with long-range coupling, the diffusion of an initial pulse is ballistic over time, providing explicit formulas and numerical validation for different coupling types.

Contribution

It offers a rigorous proof of ballistic diffusion in infinite and semi-infinite lattices with various long-range couplings, including explicit MSD expressions.

Findings

Ballistic diffusion observed for infinite lattices with long-range coupling.

Explicit mean square displacement formulas derived for different coupling types.

Numerical results confirm ballistic behavior at long times for edge excitations.

Abstract

We prove that for a one-dimensional infinite lattice, with long-range coupling among sites, the diffusion of an initial delta-like pulse in the bulk, is ballistic at all times. We obtain a closed-form expression for the mean square displacement (MSD) as a function of time, and show some cases including finite range coupling, exponentially decreasing coupling and power-law decreasing coupling. For the case of an initial excitation at the edge of the lattice, we find an approximate expression for the MSD that predicts ballistic behavior at long times, in agreement with numerical results.