Distribution of joint local and total size and of extension for avalanches in the Brownian force model
Mathieu Delorme, Pierre Le Doussal, Kay Jörg Wiese
CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure, PSL Research University,
24 rue Lhomond, 75005 Paris, France.
Abstract
The Brownian force model (BFM) is a mean-field model for the local velocities during
avalanches in elastic interfaces of internal space dimension $d$, driven in a random medium.
It is exactly solvable via a non-linear differential equation.
We study avalanches following a kick, i.e. a step in the driving force. We first recall the calculation of
the distributions of the global size (total swept area) and of the local jump size
for an arbitrary kick amplitude. We extend this calculation to the joint density
of local and global sizes within a single avalanche, in the limit of an infinitesimal kick.
When the interface is driven by a single point we find new exponents $\tau_0=5/3$
and $\tau=7/4$, depending on whether the force or the displacement is imposed.
We show that the extension of a single avalanche along one internal direction (i.e. the total length
in $d=1$) is finite and we calculate its distribution, following either a
local or a global kick. In all cases it exhibits a divergence $P(\ell) \sim \ell^{-3}$ at small $\ell$.
Most of our results are tested in a numerical simulation
in dimension $d=1$.
arXiv:1601.04940 [pdf]
Phys. Rev. E 93 (2016) 052142 [pdf]
Copyright (C)
by Kay Wiese. Last edited May 31, 2016.