Positional order and diffusion processes in particle systems

The relaxation of a nonequilibrium solid to a fluid is determined by observing the positional order parameter in Monte Carlo simulations, and discussed based on diffusion processes in the hard-particle systems. From the cumulant expansion up to the second order, the relation between the positional order parameter $\Psi$ and the mean square displacement $⟨u_i^2⟩$ is obtained to be $\Psi \sim \exp (-{\mathbf{K}}^2⟨u_i^2⟩∕2d)$ with a reciprocal vector $\mathbf{K}$ and the dimension of the system $d$. On the basis of this relation, the positional order should decay exponentially as $\Psi \sim \exp (-{\mathbf{K}}^{2}Dt)$ when the system involves normal diffusion with a diffusion constant $D$. A diffusion process with swapping positions of particles is also discussed. The swapping of particles contributes to the higher orders of the cumulants, and swapping positions allows particles to diffuse without destroying the positional order while the normal diffusion destroys it.

Figure 1

Time evolution of the mean square displacement $⟨u_i^2⟩$ in (a) two- and (b) three-dimensional systems. The values are normalized by the radius $\sigma$ and therefore the unit is dimensionless. The decimal logarithm are taken for both axes. The dashed lines denote the diffusion in the low density limit [16]. Number of particles $N=23288$ for two- and $N=36000$ for three-dimensional system with the periodic boundary condition.

Figure 2

Time evolution of the positional order parameter and calculated values from the diffusion for (a) two- and (b) three-dimensional systems. The solid lines are the positional order parameters and the symbols are the calculated values using Eq. (5). The densities $\rho =0.7$, 0.8, 0.9, and 1.0 are studied, and the numbers on the graphs denote each density. It shows good agreement in the region where $\Psi$ is not so small.

Figure 3

(a) Mean square displacement of the two-dimensional solid. The number of particles $N=23288$ and the density $\rho =0.92$. The solid line is $C_1\ln t$ with $C_1=1.6\times {10}^{-2}$. It varies from the logarithmic behavior at around $t={10}^4$. (b) The time evolution of the value $\ln (\Psi ∕{\Psi }^{\prime })$ in Eq. (13). The decimal logarithms are taken for both axes. The solid line $C_2{t}^{1.75}$ is drawn as a guide to the eyes with $C_2=2.2\times {10}^{-6}$. It shows that the exchanging rate increases as $E_r\sim t^{0.75}$.

Figure 4

Distribution of displacements ${\mathbf{u}}_i$ at $t=5\times {10}^5$ of the two-dimensional system with $N=2900$ and $\rho =0.92$. The lattice constant is $a=2$ and the radius of particles is $\sigma =0.89$ in this scale. The points around at the center correspond to the normal diffusion, and the six small groups around the center group correspond to the swapping diffusion.