First-principle derivation of static avalanche-size distribution

Abstract

We study the energy minimization problem for an elastic interface in a random potential plus a quadratic well. As the position of the well is varied, the ground state undergoes jumps, called shocks or static avalanches. We introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements. The tree-level calculation, i.e. mean-field limit, is obtained by solving a saddle-point equation. Graphically, it can be interpreted as a the sum of all tree graphs. The 1-loop corrections are computed using results from the functional renormalization group. At the upper critical dimension the shock statistics is described by the Brownian Force model (BFM), the static version of the so-called ABBM model in the non-equilibrium context of depinning. This model can itself be treated exactly in any dimension and its shock statistics is that of a Levy process. Contact is made with classical results in probability theory on the Burgers equation with Brownian initial conditions. In particular we obtain a functional extension of an evolution equation introduced by Carraro and Duchon, which recursively constructs the tree diagrams in the field theory.

33 pages, 24 figures

References in corpus (10)

• Burgers Turbulence
• Crackling dynamics in material failure as the signature of a self-organized dynamic phase transition
• Width distribution of contact lines on a disordered substrate
• Creep motion of an elastic string in a random potential
• Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
• Distribution of velocities in an avalanche
• Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model
• Universal distribution of threshold forces at the depinning transition
• Interference in disordered systems: A particle in a complex random landscape
• Shock statistics in higher-dimensional Burgers turbulence

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