On the density of shear transformation zones in amorphous solids

We study the stability of amorphous solids, focusing on the distribution P(x) of the local stress increase x that would lead to an instability. We argue that this distribution is singular P(x)x^{\theta}, where the exponent {\theta} is non-zero if the elastic interaction between rearranging regions is non-monotonic, and increases with the interaction range. For a class of finite dimensional models we show that stability implies a lower bound on {\theta}, which is found to lie near saturation. For quadrupolar interactions these models yield {\theta} ~ 0.6 for d=2 and \theta ~ 0.4 in d=3 where d is the spatial dimension, accurately capturing previously unresolved observations in atomistic models, both in quasi-static flow and after a fast quench.