Universal exponents of plastic avalanches in sheared amorphous solids
In plain words
A slowly sheared glass flows in bursts, or avalanches, of all sizes. The numbers describing how often avalanches of each size occur differ from simple mean-field theory, and an exact theory in two or three dimensions is missing.
Precise statement
In steady athermal quasistatic flow, avalanche sizes S follow $P(S)\sim S^{-\tau}$ with cutoff $S_c\sim L^{d_f}$, and the density of local distances to instability $P(x)\sim x^{\theta}$ at small $x$. Determine $\tau$, $d_f$ and $\theta$ in $d=2\ \text{and}\ 3$, which numerically lie below the mean-field value $\tau=3/2$, and derive them from a controlled theory of elastoplastic dynamics with long-range Eshelby interactions, consistent with the known scaling relations among them.
What would settle it
A renormalization-group or exact calculation of $\tau$, $d_{f}$ and $\theta$ in finite $d$ that matches high-precision particle and elastoplastic simulations.
Status in the literature
Unverified note
Scaling relations linking $\tau, d_{f} \text{ and } \theta$ were established in 2014, and simulations report $\tau$ of roughly 1.2 to 1.4 in $d = 2, 3$, but a finite-d theory of the exponents is lacking (2026).