Fluid Coexistence close to Criticality: Scaling Algorithms for Precise Simulation

TL;DR

This paper introduces a new finite-size scaling algorithm that accurately estimates coexisting densities near criticality in fluids using grand canonical Monte Carlo simulations, improving precision in locating critical points.

Contribution

The novel algorithm leverages minima of the moment ratio in finite systems to precisely determine coexistence densities and critical temperature, enhancing simulation accuracy near criticality.

Findings

Accurate estimates of coexisting densities near criticality.

Precise location of critical temperature.

Universal scaling relations for finite-size data.

Abstract

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