Diffusion of two repulsive particles in a one-dimensional lattice

TL;DR

This paper analytically solves the diffusion problem of two repulsive particles on a one-dimensional lattice, revealing anomalous drift, normal diffusion behavior, and persistent lattice effects at large times.

Contribution

It provides explicit analytical expressions for the two-particle probability function and connects lattice diffusion with continuous models, highlighting unique transient and long-term effects.

Findings

Interaction causes anomalous drift with zero velocity.

Mean square displacement grows as t^{1/2} indicating normal diffusion.

Lattice effects persist even at large times, affecting transient regimes.

Abstract

The problem of the lattice diffusion of two particles coupled by a contact repulsive interaction is solved by finding analytical expressions of the two-body probability characteristic function. The interaction induces anomalous drift with a vanishing velocity, the average coordinate of each particle growing at large times as $t^{1/2}$. The leading term of the mean square dispersions displays normal diffusion, with a diffusion constant made smaller by the interaction by the non-trivial factor $1-1/\pi$. Space continuous limit taken from the lattice calculations allows to establish connection with the standard problem of diffusion of a single fictitious particle constrained by a totally reflecting wall. Comparison between lattice and continuous results display marked differences for transient regimes, relevant with regards to high time resolution experiments, and in addition show that, due to slowly decreasing subdominant terms, lattice effects persist even at very large times.