Smooth cut-off HRT equations

The Hierarchical Reference Theory (HRT) is, so far, the only practical RG scheme able to determine both the universal and non-universal critical properties of fluids. Although it is very successful, the current sharp cut-off formulation is not able to give, analytically or numerically, the expected discontinuity of the compressibility on the coexistence curve in any dimension smaller than four. This can be corrected by using an improved smooth cut-off formulation of the equations, in which the suppression of the long wave-length fluctuations is done by using a continuous function rather than defining a cut-off value. I analyzed the smooth cut-off HRT equations numerically, in the context of a scalarfield theory. The results are able to give the expected discontinuity of the inverse compressibility across the coexistence curve. This discontinuity has the nature of an essential singularity in three dimensions. Critical exponents were also determined which prove to be slightly more accurate than the ones predicted by the sharp cut-off formulation, although they depend on the cut-off procedure. This formulation can also be used in describing the critical behavior of fluids and work is being made for applying it to a simple Yukawa model fluid with a hard core. In fact, a valuable advantage of the smooth cut-off formulation over the sharp cut-off one is that, for such a fluid, the request that the radial distribution function vanishes inside the hard core (the so-called core condition) can be implemented exactly. In the regime where the range of the attractive tail is short with respect to the dimension of the particle this condition is essential for describing accurately the thermodynamics and the correlations and for providing reliable structural data needed for the determination of the dynamical arrest lines.