Fine-Lattice Discretization for Fluid Simulations: Convergence of Critical Parameters.

In simulating continuum fluids with long-range interactions, such as plasmas and electrolytes, that undergo phase separation and criticality, it is computationally advantageous to confine the particles to the sites of a lattice of fine spacing, $a_{0} $, relative to their size, $a$.$^{1,2}$ But, how does the discretization parameter, $\zeta\equiv a/a_{0}$ (typically,$^ {1} \geq 5$) affect the values of the critical temperature and density, etc.? A heuristic argument,$^{2}$ essentially exact in $d=1$ and $2$ dimensions, shows that for models with hard-core potentials, both $T_{c}(\zeta)$ and $\rho_{c}(\zeta)$ converge to their continuum limits as $1/\zeta^{(d+1)/2}$ for $d\leq 3$ when $\zeta\rightarrow\infty$. However, the behavior of the error for $d\geq 2$ (related to a classical problem in number theory) is highly erratic. Exact results for $d=1$ illuminate the issues and reveal that optimal choices for $\zeta$ can improve the rate of convergence by factors of $1/\zeta$.$^{2}$ For $d\geq 2$, the convergence of the {\em second virial coefficients} to their continuum values exhibit similar erratic behavior which transfers to $T_{c}$ and $\rho_{c}$. This can be used in to enhance extrapolation to $\zeta\rightarrow\infty$. Data for the hard-core or {\em restricted primitive model} electrolyte have thereby been used to establish that (contrary to recent suggestions) the criticality is of Ising-type --- as against classical, XY, etc.\\ 1. Y.\ C.\ Kim and M.\ E.\ Fisher, Phys.\ Rev.\ Lett.\ {\bf 92}, 185703 (2004).\\ 2. S.\ Moghaddam, Y.\ C.\ Kim and M.\ E.\ Fisher, J.\ Phys.\ Chem.\ B (2005) $~~~~$[in press].