Numerical and analytical studies of the long-ranged solutions of the Yvon–Born–Green equation

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Abstract
In this paper we give a detailed description of our numerical studies of the critical behavior of the Yvon–Born–Green (YBG) equation in various dimensions and the conclusions which we believe these studies warrent. A central issue in these studies is the relationship between the convergence of the numerical methods used to solve the YBG equation, the long-ranged nature of the solutions, and possible bifurcations in the solution structure of the equation. An understanding of these relationships is essential to the construction of reliable numerical solutions in the critical region of the YBG equation and is likely to be equally important for a large class of nonlinear integral equations with short-ranged kernels. We give some analytic models and numerical techniques which are effective in exploring these questions. From these studies the following conclusions concerning the critical behavior of the YBG equation are drawn: (1) For d=3 there is a region of density and temperature in which the correlations are long ranged but this range never becomes more than 13 hard-core radii. The compressibility remains finite, though large, in this region. There is no true critical point for d=3 but rather a region of ‘‘near critical’’ behavior. (2) For d=5, 6 there is a (stability) line in the density-temperature plane and, as that line is approached, the range of numerical solutions and the compressibility appear to increase without bound. Inside this stability line solutions cannot be found by the usual numerical techniques. All of the data near the stability line can be fit, quite accurately, with mean-field theory exponents. The thermodynamic behavior of the correlation range and the isothermal compressibility have the scaling (homogeneity) properties of a mean-field or van der Waals-type theory.