Convergence of Fine-lattice Discretization for Near-critical Fluids

Details Convergence of Fine-lattice Discretization for Near-critical Fluids Convergence of Fine-lattice Discretization for Near-critical Fluids

2005-02-07

arxiv.org/abs/cond-mat/0502169v1

Physics

Condensed Matter

Statistical Mechanics

To appear in J. Phys. Chem. in honor of David Chandler

Scientific paper

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.

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