Signatures of dynamic heterogeneity from a generalized hydrodynamic model of the supercooled liquid

In this paper we study the four point correlation function $\chi_4$ of collective density fluctuations in a nonequilibrium liquid. The equilibration is controlled by a modified stretched exponential behavior ( $\exp [-{(t_\mathrm{w}/\tau)}^\beta]$ ) having the relaxation time $\tau$ dependent on the aging time $t_\mathrm{w}$. Similar aging behavior has been seen experimentally in supercooled liquids. The basic equations of fluctuating nonlinear hydrodynamics (FNH) are solved here numerically to obtain $\chi_4$ for equilibrium and non equilibrium states. We also identify a dynamic length scale $\xi$ from the equilibrated function. $\xi(T)$ grows with fall of temperature $T$. From a broader perspective, we demonstrate here that the characteristic signatures of dynamical heterogeneities in a supercooled liquid, observed previously in computer simulations of the dynamics of a small number of particles, are also present in the coarse grained equations of generalized hydrodynamics.