Avalanche-size distributions in mean-field plastic yielding models.

We discuss the size distribution N(S) of avalanches occurring at the yielding transition of mean-field (i.e., Hebraud-Lequeux) models of amorphous solids. The size distribution follows a power law dependence of the form N(S)∼S(-τ). However (contrary to what is found in its depinning counterpart), the value of τ depends on details of the dynamic protocol used. For random triggering of avalanches we recover the τ=3/2 exponent typical of mean-field models, which, in particular, is valid for the depinning case. However, for the physically relevant case of external loading through a quasistatic increase of applied strain, a smaller exponent (close to 1) is obtained. This result is rationalized by mapping the problem to an effective random walk in the presence of a moving absorbing boundary.