In the present work, we use the above equations to estimate S$_{r}$ and S$_{im}$ ; the per particle values of these quantities in units of k$_{B}$ are labeled s ∗ r and s ∗ $_{im}$. To our knowledge, this decomposition of the entropy has not been used previously. A somewhat parallel approach was used by Goddard et al. to treat the entropy of a liquid as a sum of contributions from a harmonic component and a hard-sphere fluid component. 60 The excess entropy is defined as the difference between the entropy of the real fluid and that of the ideal gas at the same temperature and density. Figure 2 illustrates the density dependence of the excess (represented in the figure by dashed lines) and pair correlation entropy (represented in the figure by solid lines) for four different isotherms. The values of the thermodynamic excess entropy, s ∗ $_{ex}$, have been taken from Mittal et al. work. 30 The s ∗ $_{ex}$( ρ ) curves at low temperatures show a pronounced excess entropy anomaly, corre- sponding to a rise in excess entropy on isothermal compression. Such an entropy anomaly is characteristic of water-like liquids 13,27–29 and contrasts with the behaviour of simple liquids where free volume arguments are sufficient to justify a monotonic decrease in entropy on isothermal compression. Figure 2 also compares s ∗ $_{ex}$( ρ ) and s ∗ $_{2}$( ρ ) curves at four tempera- tures. It is evident that s ∗ 2 essentially captures the anomalous behavior present in s ∗ $_{ex}$. The effect of the higher-order multiparticle correlations terms in s ∗ ex is to generate a downward shift in the values of the entropy and to attenuate the entropy anomaly. In the case of simple liquids, the residual multiparticle entropy (RMPE), Δ s ∗ = s ∗ ex − s ∗ $_{2}$, is typically of the order of 10-15% of s ∗ ex for a fairly wide range of densities. Clearly in the case of the core-softened modeled fluids, the residual multiparticle entropy contribution is larger in magnitude and more strongly density- dependent. The anomalous pair entropy regime at a given tempera- ture is an interval of densities ρ$_{s}$$_{2}$ max < ρ < ρ$_{s}$$_{2}$$_{min}$ within which ( ∂S$_{2}$/∂ρ )$_{T}$ > 0. This can be identified from the locus of extrema in s$_{2}$ ( ρ ) shown in Fig. 10. We now consider the scaling relationship between the diffusivity and the excess entropy. The diffusivity as a function of density for different isotherms is shown in Fig. (3)(a). Clear maxima and minima in the D ( ρ ) curves can be identified at low temperatures. Fig. (3)(b)- 9 IV. RESULTS A. Excess Entropy and Diffusivity