Self-Diffusion in Simple Models: Systems with Long-Range Jumps

TL;DR

This paper reviews exact results and investigates the density dependence of self-diffusion in lattice models with long-range symmetric jumps, providing bounds and simulation insights into the behavior of the diffusion coefficient.

Contribution

It introduces bounds on the diffusion coefficient in models with long-range jumps and shows that these bounds become N-independent for large N through simulations.

Findings

Bounds on the diffusion coefficient are established for various densities.

Simulations suggest the bounds are N-independent for N ≥ 20.

The study enhances understanding of diffusion in systems with long-range jumps.

Abstract

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self diffusion coefficient, $D_N(\rho)$, in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set of $N$ neighboring sites. We obtain positive upper and lower bounds on $F_N(\rho)=N((1-\r)-[D_N(\rho)/D_N(0)])/(\rho(1-\rho))$ for $\rho\in [0,1]$. Computer simulations for the square, triangular and one dimensional lattice suggest that $F_N$ becomes effectively independent of $N$ for $N\ge 20$.