Tarazona's weighted density approximation

Inspired by the exact solution known for the system of 1D hard rods, Pedro Tarazona proposed a series of models in the 80s, in which the dependence of the free energy 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.

References

1. P. Tarazona "A density functional theory of melting", Molecular Physics 52 pp. 81-96 (1984)
2. P. Tarazona "Free-energy density functional for hard spheres", Physical Review A 31 pp. 2672 - 2679 (1985)
3. P. Tarazona "Erratum: Free-energy density functional for hard spheres", Physical Review A 32 p. 3148 (1985)
4. P. Tarazona, U. Marini Bettolo Marconi and R. Evans "Phase equilibria of fluid interfaces and confined fluids: Non-local versus local density functionals" Molecular Physics 60 pp. 573 - 595 (1987)