We compute analytically and numerically the four-point correlation function that characterizes nontrivial cooperative dynamics in glassy systems within several models of glasses: elastoplastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR’s), diffusing defects, and kinetically constrained models (KCM’s). Some features of the four-point susceptibility ${\chi }_4\left(t\right)$ are expected to be universal: at short times we expect a power-law increase in time as $t^4$ due to ballistic motion ($t^2$ if the dynamics is Brownian) followed by an elastic regime (most relevant deep in the glass phase) characterized by a $t$ or $\sqrt{t}$ growth, depending on whether phonons are propagative or diffusive. We find in both the $\beta$ and early $\alpha$ regime that ${\chi }_4\sim t^{\mu }$, where $\mu$ is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of ${\chi }_4$ is reached at a time $t=t^{*}$ of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power law ${\chi }_4\left(t^{*}\right)\sim t^{*\lambda }$. The value of the exponents $\mu$ and $\lambda$ allows one to distinguish between different mechanisms. For example, freely diffusing defects in $d=3$ lead to $\mu =2$ and $\lambda =1$, whereas the CRR scenario rather predicts either $\mu =1$ or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic behavior of ${\chi }_4\left(t^{*}\right)$. MCT leads to $\mu =b$ and $\lambda =1∕\gamma$, where $b$ and $\gamma$ are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time scales accessible to numerical simulations, we find that the exponent $\mu$ is rather small, $\mu <1$, with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple diffusive defect models, KCM’s with noncooperative defects, and CRR’s. Experimental and numerical determination of ${\chi }_4\left(t\right)$ for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.

• Received 15 December 2004

DOI:https://doi.org/10.1103/PhysRevE.71.041505