Positional Order and Diffusion Processes in Particle Systems

TL;DR

This paper explores how diffusion processes affect positional order in particle systems, deriving a relation between order and mean square displacement, and highlighting the role of swapping diffusion in preserving order.

Contribution

It introduces a cumulant expansion approach to relate positional order to diffusion, including the impact of swapping diffusion on order preservation.

Findings

Positional order parameter relates to mean square displacement via an exponential decay.

Normal diffusion destroys positional order over time.

Swapping diffusion can preserve positional order despite particle movement.

Abstract

Nonequilibrium behaviors of positional order are discussed based on diffusion processes in particle systems. With the cumulant expansion method up to the second order, we obtain a relation between the positional order parameter $\Psi$ and the mean square displacement $M$ to be $\Psi \sim \exp(- {\bf K}^2 M /2d)$ with a reciprocal vector ${\bf K}$ and the dimension of the system $d$. On the basis of the relation, the behavior of positional order is predicted to be $\Psi \sim \exp(-{\bf K}^2Dt)$ when the system involves normal diffusion with a diffusion constant $D$. We also find that a diffusion process with swapping positions of particles contributes to higher orders of the cumulants. The swapping diffusion allows particle to diffuse without destroying the positional order while the normal diffusion destroys it.