  We review a new theory of viscoelasticity of a glass-forming viscous liquid
near and below the glass transition. In our model we assume that each point in
the material has a specific viscosity, which varies randomly in space according
to a fluctuating activation free energy. We include a Maxwellian elastic term
and assume that the corresponding shear modulus fluctuates as well with the
same distribution as that of the activation barriers. The model is solved in
coherent-potential approximation (CPA), for which a derivation is given. The
theory predicts an Arrhenius-type temperature dependence of the viscosity in
the vanishing-frequency limit, independent of the distribution of the
activation barriers. The theory implies that this activation energy is
generally different from that of a diffusing particle with the same
barrier-height distribution. If the distribution of activation barriers is
assumed to have Gaussian form, the finite-frequency version of the theory
describes well the typical low-temperature alpha relaxation peak of glasses.
Beta relaxation can be included by adding another Gaussian with center at much
lower energies than that responsible for the alpha relaxation. At high
frequencies our theory reduces to the description of an elastic medium with
spatially fluctuating elastic moduli (heterogeneous elasticity theory), which
explains the occurrence of the boson-peak-related vibrational anomalies of
glasses.
