Time-dependent correlations in a supercooled liquid from nonlinear fluctuating hydrodynamics.

We solve numerically the equations of nonlinear fluctuating hydrodynamics (NFH). A coarse graining of the density field is applied at each time step to avoid instabilities which otherwise plague the algorithm at long times. The equilibrium correlation of the density fluctuations at different times obtained directly from the solutions of the NFH equations are shown here to be in quantitative agreement with corresponding molecular dynamics simulation data. Low-order perturbative treatment of the these NFH equations obtains the mode coupling model. The latter has been widely studied for understanding the slow dynamics characteristic of the supercooled state. A crucial aspect of this theory is a rounded version of a possible ergodic-nonergodic transition in the supercooled liquid at a temperature T(c) between melting point T(m) and the glass transition temperature T(g). In the present work we demonstrate numerically the role of strongly coupled density fluctuations in giving rise to slow dynamics and how the 1/ρ nonlinearity in the NFH equations of motion is essential in restoring the ergodic behavior in the liquid. The relaxation data indicate that at moderate supercooling near T(c), the time temperature superposition holds. The relaxation gets increasingly stretched with increased supercooling. The relaxation time τ shows an initial power-law divergence approaching a transition temperature T(c) generally identified as the mode coupling temperature. From the direct solutions we obtain a value for T(c) much lower than that typically estimated from solution of low-order integral equations of mode coupling theory. This is in agreement with the trend seen in computer simulations.