Combining off-lattice Monte Carlo and cellular automata for the simulation of hard-sphere systems

In the present work we show how the update rule of a diffusive cellular automaton with mutual exclusion can be exploited in off-lattice Monte Carlo simulations of hard spheres to obtain a synchronous Monte Carlo sampling that satisfies the detailed balance principle.

Figure 1

(a) A two-dimensional sketch of a small lattice of cells, numbered as ${\mathbf{c}}_1,{\mathbf{c}}_2,\cdots$. Cell ${\mathbf{c}}_{13}$ is enhanced in yellow, and its Von Neumann neighboring cells are in green. In (b) we sketch the same as in (a), but with the Moore neighbors of cell ${\mathbf{c}}_{13}$ enhanced in green. In (c) the cell-to-cell boundaries are sketched as a two-dimensional grid (filled circles). We enhanced the connections (Von Neumann neighborhood) involving the boundary (in yellow) placed in between of cells ${\ell }_8$ and ${\ell }_{13}$. Of such connections, the 6 pointing to the boundaries (red circles) having access to either cell ${\ell }_8$ or cell ${\ell }_{13}$ constitute the first connection shell and are in red, whereas the 16 pointing to to the boundaries (cyan circles) having access to either the neighbors of cell ${\ell }_8$ or the neighbors of cell ${\ell }_{13}$ constitute the second connection shell and are in cyan. In (d) the enhanced connections refer to the case of the Moore neighborhood.

Figure 2

(a) The jump acceptance and (b) the average number of jump attempts when the Moore neighborhood is used, for $\gamma =0.9$ and for two different values of the number of cells per side; i.e., $z=12$ and 10, corresponding, respectively, to $\ell \approx 4.17$ and $\ell =5.0$.

Figure 3

(a) The jump acceptance and (b) the average number of jump attempts when the Von Neumann neighborhood is used, for $\gamma =0.9$ and for two different values of the number of cells per side; i.e., $z=14$ (a) and 12 (b), corresponding, respectively, to $\ell \approx 3.57$ and $\ell \approx 4.17$.

Figure 4

(a) and (b) are radial distribution functions at various values of the packing fraction, $\eta$, for a system simulated through the synchronous MC sampling with parameters $\gamma =0.9$ and $z=12$ (symbols), compared with the curves obtained through standard MC runs (lines). In (a) the solid and the dotted line refer to $\eta =0.070$ and $\eta =0.211$, respectively. In (b) the solid and the dotted line refer to $\eta =0.351$ and $\eta =0.421$, respectively. In (c) we show compressibility factors (dots) compared with the curve obtained through the Carnahan-Starling equation for the same systems (solid line).

Figure 5

Stationary distribution for the source-sink problem applied to the case of hard disks. The solid and the dotted lines refer to the case of synchronous MC and standard MC, respectively.