  We present a linearized shear-transformation-zone (STZ) theory of glassy
dynamics in which the internal STZ transition rates are characterized by a
broad distribution of activation barriers. For slowly aging or fully aged
systems, the main features of the barrier-height distribution are determined by
the effective temperature and other near-equilibrium properties of the
configurational degrees of freedom. Our theory accounts for the wide range of
relaxation rates observed in both structural glasses and soft glassy materials
such as colloidal suspensions. We find that the frequency dependent loss
modulus is not just a superposition of Maxwell modes. Rather, it exhibits an
$\alpha$ peak that rises near the viscous relaxation rate and, for nearly
jammed, glassy systems, extends to much higher frequencies in accord with
experimental observations. We also use this theory to compute strain recovery
following a period of large, persistent deformation and then abrupt unloading.
We find that strain recovery is determined in part by the initial
barrier-height distribution, but that true structural aging also occurs during
this process and determines the system's response to subsequent perturbations.
In particular, we find by comparison with experimental data that the initial
deformation produces a highly disordered state with a large population of low
activation barriers, and that this state relaxes quickly toward one in which
the distribution is dominated by the high barriers predicted by the
near-equilibrium analysis. The nonequilibrium dynamics of the barrier-height
distribution is the most important of the issues raised and left unresolved in
this paper.
