  Mesoscopic models play an important role in our understanding of the
deformation and flow of amorphous materials. One such description, based on the
Shear Transformation Zone (STZ) theory, has recently been re-formulated within
a non-equilibrium thermodynamics framework, and found to be consistent with it.
We show here that a similar interpretation can be made for the Soft Glassy
Rheology (SGR) model. Conceptually this means that the "noise temperature" x,
proposed phenomenologically in the SGR model to control the dynamics of a set
of slow mesoscopic degrees of freedom, can consistently be interpreted as their
actual thermodynamic temperature. (Because such modes are slow to equilibrate,
this generally does not coincide with the temperature of the fast degrees of
freedom and/or heat bath.) If one chooses to make this interpretation, the
thermodynamic framework significantly constrains extensions of the SGR approach
to models in which x is a dynamical variable. We assess in this light some such
extensions recently proposed in the context of shear banding.
