Aging and linear response in the Hébraud–Lequeux model for amorphous rheology

We analyse the aging dynamics of the Hébraud–Lequeux model, a self-consistent stochastic model for the evolution of local stress in an amorphous material. We show that the model exhibits initial-condition dependent freezing: the stress diffusion constant decays with time as $D\sim 1/t^2$ during aging so that the cumulative amount of memory that can be erased, which is given by the time integral of D(t), is finite. Accordingly the shear stress relaxation function, which we determine in the long-time regime, only decays to a plateau and becomes progressively elastic as the system ages. The frequency-dependent shear modulus exhibits a corresponding overall decay of the dissipative part with system age, while the characteristic relaxation times scale linearly with age as expected. (paper)