Dynamical susceptibility of glass formers: Contrasting the predictions of theoretical scenarios

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 ξ ₄(t) 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 √t growth, depending on whether phonons are propagative or diffusive. We find in both the β and early α regime that ξ ₄∼t ^μ, where μ is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of ξ 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 ξ ₄(t*) ∼t* ^λ The value of the exponents μ and λ allows one to distinguish between different mechanisms. For example, freely diffusing defects in d=3 lead to μ=2 and λ= 1, whereas the CRR scenario rather predicts either μ,=1 or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic behavior of ξ ₄(t*). MCT leads to μ=b and λ=1/γ, where b and γ 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 μ is rather small, μ<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 ξ ₄(t) for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.