Distribution of Maximum Velocities in Avalanches Near the Depinning Transition

Abstract

We report exact predictions for universal scaling exponents and scaling functions associated with the distribution of the maximum collective avalanche propagation velocities v(m) in the mean field theory of the interface depinning transition. We derive the extreme value distribution P(v(m)|T) for the maximum velocities in avalanches of fixed duration T and verify the results by numerical simulation near the critical point. We find that the tail of the distribution of maximum velocity for an arbitrary avalanche duration, v(m), scales as P(v(m))~v(m)(-2) for large v(m). These results account for the observed power-law distribution of the maximum amplitudes in acoustic emission experiments of crystal plasticity and are also broadly applicable to other systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.

A stochastic model for the temporal aspects of flow intermittency in micropillar compression

“…10. This behavior is consistent with recent predictions in mean field theory (LeBlanc et al, 2012). In particular, such theory also derived a value of scaling exponent $2 for those larger velocities.…”

“…7b, the calculated burst time durations show statistics consistent with the experimental data (Papanikolaou et al, 2012a). Despite the good agreement of the calculations from the present model with the predictions in mean-field theory (LeBlanc et al, 2012), it is still an open problem as to what extent it can describe the pillar size with the diameters smaller than $10 lm. For example, recent reports of compression tests of Au nano-and microcrystals have also shown that the distribution of burst velocity follows a power law behavior (Maaß et al, 2013), but the values of the scaling exponent are different from the present results.…”

Universal fluctuations and extreme statistics of avalanches near the depinning transition

“…The main results for fixed durations at the critical point have been recently reported in Ref. [29]. Here, we present the details of the calculations and extend the work to also include the effects of nonzero sweep-rate c and the maximum velocity statistics for fixed avalanche size.…”

“…We have already seen in Ref. [29] that the driving rate dependence of the power law P (v m ) ∼ v −µ m at criticality is µ = 2 −c. In Fig.…”

“…[29] we derived the value of the exponent µ from a mean field theory of the depinning transition. In this paper, we present the calculations that led to this result in more detail and extend the results to include the effect of a nonzero driving rate and cutoff scaling functions.…”