  We show that, at least at a mean-field level, the effect of structural
disorder in sheared amorphous media is very dissimilar depending on the thermal
or athermal nature of their underlying dynamics. We first introduce a toy
model, including explicitly two types of noise (thermal versus athermal).
Within this interpretation framework, we argue that mean-field athermal
dynamics can be accounted for by the so-called H{\'e}braud-Lequeux (HL) model,
in which the mechanical noise stems explicitly from the plastic activity in the
sheared medium. Then, we show that the inclusion of structural disorder, by
means of a distribution of yield energy barriers, has no qualitative effect in
the HL model, while such a disorder is known to be one of the key ingredients
leading kinematically to a finite macroscopic yield stress in other mean-field
descriptions, such as the Soft-Glassy-Rheology model. We conclude that the
statistical mechanisms at play in the emergence of a macroscopic yield stress,
and a complex stationary dynamics at low shear rate, are different in thermal
and athermal amorphous systems.
