  A new functional for the entropy that is asymptotically correct both in the
high and low density limits is proposed. The new form is [
S=S^{(id)}+S^{(ln)}+S^{(r)}+S^{(c)} ] where the new term S^{(c)} depends on the
p-bodies density fluctuations $\alpha_p$ and has the form [ S^{(c)}= <N> {ln
2-1+\sum_{p=2}^\infty \frac{(\ln 2) ^p}{p!}\alpha_p-[ \exp
(\alpha_2-1)-\alpha_2]} +\hat S ], where $\hat S$ renormalizes the ring
approximation S^{(r)}. This result is obtained by analyzing the functional
dependence of the most general expression of the entropy: Two main results for
S^{(c)} are proven: i) In the thermodynamic limit, only the functional
dependence on the one body distribution function survives and ii) by summing to
infinite order the leading contributions in the density a new numerical
expression for the entropy is proposed with a new renormalized ring
approximation included. The relationship of these results to the incompressible
approximation to entropy is discussed.
