0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.25 0.3 0.35 0.4 F$_{im}$ (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.3 0.35 0.4 0.45 0.5 F$_{im}$ (b) FIG. 6: (a) Fraction of imaginary modes against density for fixed temperatures T ∗ =0.20,0.30,0.40,0.55 from bottom to top. (b) Fraction of imaginary modes against density for fi xed temperatures T ∗ =0.80,1.0,1.20,1.40,1.60.1.80,2.0,2.2 from bottom to top. 1 1.5 2 2.5 3 r/ -5 0 5 10 U$^{*}$(r) -dU$^{*}$/dr * d$^{2}$U$^{*}$/dr * 2 F$_{i}$ FIG. 7: Potential, force and second derivative of the potential in units of ϵ and F$_{im}$ ∗ 15 . 0 − 1 . 0 for T ∗ = 1 . 0 versus reduced distance. real branch while the S$_{im}$ contribution is given by S$_{im}$ = S − S$_{r}$ . In order to get a better understanding of the role played by each contribution to the entropy, Fig. (8) shows the be- havior with density for a fixed temperature of s ∗ $_{im}$, s ∗ $_{r}$, s ∗ id and s ∗ $_{ex}$. It is immediately evident that s ∗ r shows a very similar density dependence to s ∗ $_{id}$, even though the numerical value of s ∗ r is significantly lower. Other than a small inflection at ρ ∗ ≈ 0 . 1, the contribution of the real INM modes to the entropy carries virtually no signature of the structural, entropy or 14