Fluid Coexistence close to Criticality: Scaling Algorithms for Precise Simulation

Abstract

A novel algorithm is presented that yields precise estimates of coexisting liquid and gas densities, $ρ^{\pm}(T)$, from grand canonical Monte Carlo simulations of model fluids near criticality. The algorithm utilizes data for the isothermal minima of the moment ratio $Q_{L}(T;<ρ>_{L})$ $\equiv< m^{2}>_{L}^{2}/< m^{4}>_{L}$ in $L$$ \times$$ ...$$ \times$$ L$ boxes, where $m=ρ-<ρ>_{L}$. When $L$$ \to$$ \infty$ the minima, $Q_{\scriptsize m}^{\pm}(T;L)$, tend to zero while their locations, $ρ_{\scriptsize m}^{\pm}(T;L)$, approach $ρ^{+}(T)$ and $ρ^{-}(T)$. Finite-size scaling relates the ratio {\boldmath $\mathcal Y$}$ = $$(ρ_{\scriptsize m}^{+}-ρ_{\scriptsize m}^{-})/Δρ_{\infty}(T)$ {\em universally} to ${1/2}(Q_{\scriptsize m}^{+}+Q_{\scriptsize m}^{-})$, where $Δρ_{\infty}$$ = $$ρ^{+}(T)-ρ^{-}(T)$ is the desired width of the coexistence curve. Utilizing the exact limiting $(L$$ \to $$\infty)$ form, the corresponding scaling function can be generated in recursive steps by fitting overlapping data for three or more box sizes, $L_{1}$, $L_{2}$, $...$, $L_{n}$. Starting at a $T_{0}$ sufficiently far below $T_{\scriptsize c}$ and suitably choosing intervals $ΔT_{j}$$ = $$T_{j+1}-T_{j}$$ > $0 yields $Δρ_{\infty}(T_{j})$ and precisely locates $T_{\scriptsize c}$.