Statistics of static avalanches in a random pinning landscape

We study the minimum-energy configuration of a $d$-dimensional elastic interface in a random potential tied to a harmonic spring. As a function of the spring position, the center of mass of the interface changes in discrete jumps, also called shocks or “static avalanches.” We obtain analytically the distribution of avalanche sizes and its cumulants within an $ϵ=4-d$ expansion from a tree and one-loop resummation using functional renormalization. This is compared with exact numerical minimizations of interface energies for random-field disorder in $d=2,3$. Connections to dynamic avalanches are mentioned.

Figure 1

In mean field, diagrams generated by Eq. (8) at $\alpha =0$ (one example shown here) have a tree structure, up to simple one-loop corrections to disorder, i.e., ${\Delta }^{\prime }\left(0^{+}\right)$ (shaded in gray). As in [17], the solid lines are propagators and the dashed lines are disorder correlators. Intuitively, solid lines represent successive local events (e.g., spin flips) in the avalanche and the dashed lines represent correlations between them due to disorder.

Figure 2

(Color online) $\tilde{Z}\left(\lambda \right)$ for RF, $d=2$. The MF and one-loop analytical curves are given in the text and $\lambda$ being rescaled to reproduce the numerically measured second moment. While the MF result differs substantially from the numerical measurement, the one-loop curve, i.e., Eq. (11) with $\alpha$ given by Eq. (10) setting $ϵ=2$ and ${\zeta }_1=1/3$, coincides for all negative $\lambda$, and almost up to the singularity for $\lambda >0$. Changing $ϵ$ from 2 to 2.1 or 1.9 already results in a visible disagreement for $\lambda <0$.

Figure 3

(Color online) Numerically computed normalized avalanche distribution $P\left(S\right)$, for random-field disorder and $d=2$ $\left(\zeta =2/3\right)$, multiplied by $S^{\tau }$ with $\tau =1.25$ [from Eq. (16)] to emphasize deviations from the power law $P\left(S\right)\sim S^{-1.25}$. Error bars are $3\sigma$ errors for $P\left(S\right)$ and the size of the box for $S$. The solid curve is a one-parameter fit to Eq. (13), with ${\mathcal{S}}_m=3500$, $\tau =1.25$, $\alpha$ given by Eq. (10) with ${\zeta }_1=1/3,ϵ=2$, and the corresponding values for $B,C$ given in the text. We use the measured value for $⟨S⟩$ in Eq. (13) hence there is no additional free parameter. The dashed line is a constant (guide to the eyes). Inset: blowup of main plot. The best fit to a pure power law would give $\tau =1.23$ and to a power law times exponential $\tau =1.2$.

Figure 4

(Color online) Distribution of intervals between jumps, in units of the step-size $\delta w=0.002$. Error bars are $3\sigma$ errors.