Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles

Abstract

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modelled as an elastic system subject to quenched disorder. Its field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group flow involves a function, the disorder correlator ∆(w), therefore termed the functional renormalization group (FRG). ∆(w) is a physical observable, the auto-correlation function of the centre of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of these mappings requires specific techniques, which we develop, including modelling of discrete stochastic systems via coarsegrained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and nonlinear surface growth with additional KPZ terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random field magnets. Emphasis is given to numerical and experimental tests of the theory.

Emergence of a random field at the yielding transition of a mean-field Elasto-Plastic model

“…Then, from Eqs. (11,12), one finds that R ∞ (x) = 0 and P ∞ (x) = 1/ < x > ∞ x g(x )dx . After some lengthy algebra, the disconnected susceptibility in the stationary state can be simply expressed as…”

“…[19], together with its solution for an exponential distribution of random jumps g(x). For a general distribution g(x) it can be recast as P y (x) = P 0 (x + y) + g(x)F 0 (y) + y 0 dy F 0 (y )R y−y (x), (11) with F 0 (x) =…”

“…where P y (x) is given by Eqs. (11,12). (The notation m anticipates on the analogy with the magnetization in the RFIM.)…”

“…The loading curve, i.e., the average stress versus applied strain σ(γ), is then obtained by combining Eqs. ( 14), ( 16), (11) and (12) 1: Average stress σ versus applied strain γ for the mean-field EPM with a distribution of random jumps chosen as an exponential and that of the initial local stress (or stability) distribution as a linear combination of two exponentials 19 . As the strength ∆0 of the disorder increases (for fixed parameters of the random jump distribution), one passes from a regime with a discontinuous stress jump to a continuous behavior with a stress overshoot to finally a monotonically increasing regime.…”

Microscopic interplay of temperature and disorder of a one-dimensional elastic interface

“…A successful theoretical tool to study interfaces is provided by the disordered elastic systems framework [18][19][20][21][22][23]: interfaces are modeled as elastic objects evolving in a quenched disordered landscape and subject to thermal noise. Remarkably, this minimal description is enough to account for many key statistical features of the geometry and dynamics of both static or driven interfaces [24].…”