Spatial shape of avalanches in the Brownian force model

TL;DR

This paper analyzes the spatial structure of avalanches in the Brownian force model, deriving exact distributions, predicting shapes, and studying fluctuations, with validation against numerical simulations.

Contribution

It provides an exact formula for avalanche jump distributions and predicts avalanche shapes and fluctuations in the BFM, extending understanding of interface dynamics.

Findings

Exact joint probability distribution of avalanche jumps

Predicted spatial shape of large avalanches

Quantified fluctuations and asymmetry in avalanche shapes

Abstract

We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.