Asymmetric Fluid Criticality II: Finite-Size Scaling for Simulations

The vapor-liquid critical behavior of intrinsically asymmetric fluids is studied in finite systems of linear dimensions, $L$, focusing on periodic boundary conditions, as appropriate for simulations. The recently propounded ``complete’’ thermodynamic $(L\to \mathrm{\infty })$ scaling theory incorporating pressure mixing in the scaling fields as well as corrections to scaling $[arXiv:cond-mat\mathrm{/}0212145]$, is extended to finite $L$, initially in a grand canonical representation. The theory allows for a Yang-Yang anomaly in which, when $L\to \mathrm{\infty }$, the second temperature derivative, $(d^2{\mu }_{\sigma }\mathrm{/}d{T}^2)$, of the chemical potential along the phase boundary, ${\mu }_{\sigma }(T)$, diverges when $T\to \text{Tc}-$. The finite-size behavior of various special {\em critical loci} in the temperature-density or $(T,\rho )$ plane, in particular, the $k$-inflection susceptibility loci and the $Q$-maximal loci – derived from $Q_{L}(T,<\rho>{L}) \equiv < m^{2}>^{2}{L}/< m^{4}>{L}$where$m \equiv \rho - <\rho>{L}$ – is carefully elucidated and shown to be of value in estimating $\text{Tc}$ and $\text{rhoc}$. Concrete illustrations are presented for the hard-core square-well fluid and for the restricted primitive model electrolyte including an estimate of the correlation exponent $\nu$ that confirms Ising-type character. The treatment is extended to the canonical representation where further complications appear.