Tarazona's weighted density approximation

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Inspired by the exact solution known for the system of 1-dimensional hard rods, Pedro Tarazona proposed a series of models in the 1980's [1] [2] [3] [4] in which the dependence of the Helmholtz energy function is weighted:

A_{\mathrm {hard~sphere}}^{\mathrm {excess}} [\rho ({\mathbf r})] = \int \rho ({\mathbf r}) a_{\mathrm {excess}} [\overline\rho ({\mathbf r})]{\mathrm d}{\mathbf r},

where \overline\rho ({\mathbf r}) is an average of the density distribution and where A is the Helmholtz energy function. The function a(x) should vanish at low values of its argument (so that the excess vanishes and one is left with the ideal case), and diverge at some saturation density corresponding to complete packing.