Pseudospectral Methods and Critical Phenomena

Abstract: We employ pseudospectral methods to solve the homogeneous Ornstein-Zernike (OZ) equation for a model fluid in the vicinity of the critical point. Focusing on the Mean-Spherical Approximation (MSA) as a closure to the OZ equation, we obtain numerical estimates for the critical exponents $η$, $δ$ and $γ$ for a system of hard-core Yukawa particles both in two and three dimensions. The three-dimensional MSA exponents are already well-known from an analytic solution and are recovered by our numerical methods. The two-dimensional exponents are a new output of this work. The pseudospectral method allows for rapid and highly accurate solution of liquid-state integral equation theories, and enables calculations on truely infinite domains, as needed for highly correlated states. In addition, we analyse the standard Picard iteration scheme and propose a variation of it which provides increased stability and speed of convergence.