0 0.1 0.2 0.3 -3 -2 -1 0 s * FIG. 2: The pair correlation entropy, s ∗ 2 (solid lines), and the excess entropy, s ∗ ex $^{30}$, (dashe d lines) against density for fixed temperatures for T ∗ = 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 from bottom to top. (c) show the scaling of the reduced diffusivity, D$_{R}$ , with the excess, s ∗ $_{ex}$, and pair, s ∗ $_{2}$, entropy. Using the Rosenfeld macroscopic reduction parameters for the length as ρ − 1 / 3 and the ther- mal velocity as ( k$_{B}$T/m ) 1 / $^{2}$, the dimensionless difusivity is defined as The scaling of the reduced diffusivity, D$_{R}$ with with s ∗ ex is excellent with D$_{R}$ = Ae Bs ∗ ex where A = 0 . 95 and B = 0 . 98. The scaling with the pair entropy, s ∗ $_{2}$, shows a weak isochore- dependence and the line of best fit is obtained with A = 0 . 68 and B = 1 . 35. The comparison between the Fig. (3)(b) and Fig. (3)(c) indicates that for our anomalous fluid, the diffusivity scaling with the pair correlation entropy is not as universal as with the excess entropy. This is likely to be a consequence of the presence of two, density-dependent length scales, in such water-like fluids when compared to simple liquids, such as the Lennard-Jones fluid. Next, we present the results from our instantaneous normal mode (INM) analysis of the simulation for the core-softened fluid. The INM spectra along the ρ ∗ = 0 . 11 isochore for various temperatures is illustrated in Fig. 4(a) while the INM spectra along the T ∗ = 0 . 20 10 B. Instantaneous Normal Mode Analysis D$_{R}$ ≡ D ρ 1 / 3 ( k$_{B}$T/m ) 1 / 2 . (17)