Convergence of Fine-lattice Discretization for Near-critical Fluids

In simulating continuum model fluids that undergo phase separation and criticality, significant gains in computational efficiency may be had by confining the particles to the sites of a lattice of sufficiently fine spacing, $a_{0}$ (relative to the particle size, say $a$). But a cardinal question, investigated here, then arises, namely: How does the choice of the lattice discretization parameter, $\zeta\equiv a/a_{0}$, affect the values of interesting parameters, specifically, critical temperature and density, $T_{\scriptsize c}$ and $\rho_{\scriptsize c}$? Indeed, for small $\zeta (\lesssim 4$-$8)$ the underlying lattice can strongly influence the thermodynamic properties. A heuristic argument, essentially exact in $d=1$ and $d=2$ dimensions, indicates that for models with hard-core potentials, both $T_{\scriptsize c}(\zeta)$ and $\rho_{\scriptsize c}(\zeta)$ should converge to their continuum limits as $1/\zeta^{(d+1)/2}$ for $d\leq 3$ when $\zeta\to\infty$; but the behavior of the error is highly erratic for $d\geq 2$. For smoother interaction potentials, the convergence is faster. Exact results for $d=1$ models of van der Waals character confirm this; however, an optimal choice of $\zeta$ can improve the rate of convergence by a factor $1/\zeta$. For $d\geq 2$ models, the convergence of the {\em second virial coefficients} to their continuum limits likewise exhibit erratic behavior which is seen to transfer similarly to $T_{\scriptsize c}$ and $\rho_{\scriptsize c}$; but this can be used in various ways to enhance convergence and improve extrapolation to $\zeta = \infty$ as is illustrated using data for the restricted primitive model electrolyte.