Albert Reiner:
Abstract: Combining renormalization group theoretical ideas with the integral equation approach to fluid structure and thermodynamics, the Hierarchical Reference Theory (HRT) is known to be successful even in the vicinity of the critical point and for sub-critical temperatures for a wide variety of systems. Restricting ourselves to the case of simple one-component fluids, we present analytical, semi-analytical and numerical results on the usual formulation of HRT and the customary closure reminiscent of the Lowest-Order γ-Ordered Approximation and the equivalent Optimized Random-Phase Approximation, investigating the necessary approximations' significance for the numerical procedure. In particular, we clarify the mechanism leading to a suppression of van der Waals loops and furthermore show that it gives rise to the equations' stiffness for close-to-critical and sub-critical temperatures; we also discuss the so-called decoupling assumption related to the elimination of terms proportional to third-order partial derivatives of a suitably modified free energy, and we prove the theory's instability for predominantly repulsive potentials. Applying HRT to both hard-core Yukawa and square-well fluids we confirm the trend of decreasing accuracy for narrower potentials, assess convergence and appropriateness of an approximate implementation of the core condition, consider the boundary conditions' relevance for the binodal's location and the numerical procedure, and we highlight the rôle of discontinuities in the potential's perturbational part in triggering unphysical shifts of the critical temperature predicted. The numerical investigations are carried out by means of our re-implementation of the theory in a fully modular software package relying heavily on the use of a meta-language and code construction techniques and going to great lengths to ensure the numerical soundness of the calculation.
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Fri Mar 29 15:15:06 MET 2002