Positional Order and Diffusion Processes in Particle Systems

In particle systems, a relation between the positional order parameter $\Psi$ and the mean square displacement $M$ is derived to be $\Psi \sim \exp(-v{K}^2 M/2d)$ with a reciprocal vector $v{K}$ and the dimension of the system $d$. On the basis of the equiation, the behavior of $\Psi$ is found to be $\Psi \sim \exp(-v{K}^2 D t)$ when the system involves normal diffusion with a diffusion constant $D$. While the behavior in two-dimensional solid is predicted to be $M \sim \ln t$, numerical simulations shows a linear diffusion $M \sim t$. This can be explained by a swapping diffusion process which allows particles to diffuse without destroying the positional order.