Avalanche shape and exponents beyond mean-field theory

Abstract

Elastic systems, such as magnetic domain walls, density waves, contact lines, and cracks, are all pinned by substrate disorder. When driven, they move via successive jumps called avalanches, with power law distributions of size, duration and velocity. Their exponents, and the shape of an avalanche, defined as its mean velocity as function of time, have recently been studied. They are known approximatively from experiments and simulations, and were predicted from mean-field models, such as the Brownian force model (BFM), where each point of the elastic interface sees a force field which itself is a random walk. As we showed in EPL 97 (2012) 46004, the BFM is the starting point for an $\epsilon =d_{\mathrm{c}}-d$ expansion around the upper critical dimension, with $d_{\mathrm{c}}=4$ for short-ranged elasticity, and $d_{\mathrm{c}}=2$ for long-ranged elasticity. Here we calculate analytically the $\mathcal{O}(\epsilon )$, i.e. 1-loop, correction to the avalanche shape at fixed duration $T$, for both types of elasticity. The exact expression is well approximated by ${⟨\dot{u}(t=xT)⟩}_T\simeq [Tx(1-x){]}^{\gamma -1}\exp \left(\mathcal{A}\left[\frac{1}{2}-x\right]\right)$, $0<x<1$. The asymmetry $\mathcal{A}\approx -0.336(1-d/d_{\mathrm{c}})$ is negative for $d$ close to $d_{\mathrm{c}}$, skewing the avalanche towards its end, as observed in numerical simulations in $d=2$ and $3$. The exponent $\gamma =(d+\zeta )/z$ is given by the two independent exponents at depinning, the roughness $\zeta$ and the dynamical exponent $z$. We propose a general procedure to predict other avalanche exponents in terms of $\zeta$ and $z$. We finally introduce and calculate the shape at fixed avalanche size, not yet measured in experiments or simulations.

6 pages, 2 figures

References in corpus (29)

• Collective Transport: From Superconductors to Earthquakes
• Dynamics of a ferromagnetic domain wall: avalanches, depinning transition and the Barkhausen effect
• The abelian sandpile and related models
• Hysteresis, Avalanches, and Disorder Induced Critical Scaling: A Renormalization Group Approach
• Beyond power laws: Universality in the average avalanche shape
• Scaling exponents for Barkhausen avalanches in polycrystals and amorphous ferromagnets
• Renormalization of pinned elastic systems: how does it work beyond one loop ?
• 2-loop Functional Renormalization Group Theory of the Depinning Transition
• Two-dimensional scaling properties of experimental fracture surfaces
• Driven Interface Depinning in a Disordered Medium
• Scaling exponents for fracture surfaces in homogenous glass and glassy ceramics
• Exactly solvable model of avalanches dynamics for Barkhausen crackling noise
• Avalanche-size distribution at the depinning transition: A numerical test of the theory
• Functional Renormalization Group and the Field Theory of Disordered Elastic Systems
• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Self-organized criticality in the hysteresis of the Sherrington - Kirkpatrick model
• Size distributions of shocks and static avalanches from the Functional Renormalization Group
• Height fluctuations of a contact line: a direct measurement of the renormalized disorder correlator
• Avalanches and the Renormalization Group for Pinned Charge-Density Waves
• Statistics of static avalanches in a random pinning landscape
• Non-steady relaxation and critical exponents at the depinning transition
• Avalanche dynamics of elastic interfaces
• Avalanches in mean-field models and the Barkhausen noise in spin-glasses
• Introduction to the Sandpile Model
• First-principle derivation of static avalanche-size distribution
• Distribution of velocities in an avalanche
• The Bak-Tang-Wiesenfeld sandpile model around the upper critical dimension
• Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model
• Statistics of Avalanches with Relaxation, and Barkhausen Noise: A Solvable Model

Cited by in corpus (25)

• Universality of Avalanche Exponents in Plastic Deformation of Disordered Solids
• Quantitative scaling of magnetic avalanches
• A simple unified view of branching process statistics: random walks in balanced logarithmic potentials
• Time-series thresholding and the definition of avalanche size
• Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles
• Spatial clustering of depinning avalanches in presence of long-range interactions
• Deformation and flow of amorphous solids: An updated review of mesoscale elastoplastic models
• Asymmetric damage avalanche shape in quasi-brittle materials and sub-avalanches (aftershocks) clusters
• Field-theoretic approach to the universality of branching processes
• Seismic-like organization of avalanches in a driven long-range elastic string as a paradigm of brittle cracks
• Distribution of joint local and total size and of extension for avalanches in the Brownian force model
• The Spatial Shape of Avalanches
• Avalanches in Tip-Driven Interfaces in Random Media
• Universal correlations between shocks in the ground state of elastic interfaces in disordered media
• Correlations between avalanches in the depinning dynamics of elastic interfaces
• Universality in the mean spatial shape of avalanches
• Magnetic domain dynamics in an insulating quantum ferromagnet
• Depinning in the quenched Kardar-Parisi-Zhang class I: Mappings, simulations and algorithm
• Distribution of velocities in an avalanche, and related quantities: Theory and numerical verification
• Clusters in an epidemic model with long-range dispersal
• Controlling crackling dynamics by triggering low intensity avalanches
• Roughness and critical force for depinning at 3-loop order
• Effect of thresholding on avalanches and their clustering for interfaces with long-range elasticity
• Equivalence of mean-field avalanches and branching diffusions: From the Brownian force model to the super-Brownian motion
• On dissipation in crackling noise systems