Aging and linear response in the H\'ebraud-Lequeux model for amorphous rheology

We analyse the aging dynamics of the H\'ebraud-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.