content_image/342900.jpg
<img src='content_image/342900.jpg'>

ξ$_{c}$ = ρ 1 / $^{3}$L/ 2 , where L = V 1 / $^{3}$. For a completely uncorrelated system (ideal gas) g = 1 and t vanishes. In a crystal, the presence of long-range translational ( g ̸ = 1) implies that t depends on choice of ξ$_{c}$ . In simple liquids, t increases with isothermal compression. In liquids with water-like anomalies at low temperatures, t shows a non-monotonic behaviour. At a given temperature T , a structurally anomalous regime can be defined between densities ρ$_{t}$$_{−}$$_{max}$ ( T ) and ρ$_{t}$$_{−}$$_{min}$ ( T ) corresponding to locations of the maxima and minima in the translational order. In this structurally anomalous regime, shown using dot-dashed lines in Fig. 1, an increase in density induces a decrease in translational order. The nested structures of the anomalous regions is evident with the structurally anomalous regime enclosing the diffusion anomalous region which in turn encloses the region of density anomaly. 

## III. INSTANTANEOUS NORMAL MODES ANALYSIS 

In this section, we define the instantaneous normal mode spectrum and explore how it can be used to connect the energy landscape of a liquid with its thermodynamic entropy. The potential energy of configuration r near r 0 can be written as a Taylor expansion of the 

