Hierarchical reference theory of fluids and the critical point

We derive, for a classical fluid, a new and exact hierarchy of integro-differential equations for the free energy and direct correlation functions of successively higher order. This hierarchy gives the evolution of these quantities as the attractive part w of the interaction is turned on by the successive inclusion of the different Fourier components of w, starting from large momenta. The full hierarchy is also written as a unique functional differential equation for the free energy of a nonuniform fluid. This hierarchy is the basis of a unified theory of fluids. Near the critical point and at small momenta, our equations become equivalent to renormalization-group equations and we recover the usual expansion. A very simple approximation gives, in three dimensions, =21. 38. Suitable truncation of the hierarchy reproduces the correct low-density (virial coefficients) and high-density (optimized random-phase) limit.