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 modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center 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 the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar–Parisi–Zhang (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.

The Relation Between Dissipation and Memory in Two‐Fluid Displacements in Disordered Media

“…As a result the macroscopic variables defining the system response exhibit hysteresis upon cyclic solicitations (Bertotti & Mayergoyz, 2006). This scenario is generic to other driven disordered systems such as elastic lines, disordered magnets, and granular packings (Wiese, 2022).…”

Shannon information entropy, soliton clusters and Bose-Einstein condensation in log gravity

“…• The statistical formulation of Burgers turbulence and its relation to (elastic) manifolds pinned by quenched disorder (see for instance [66] for a recent review) have previously been studied from the perspective of directed polymers [67], disordered trees and traveling waves [68], contexts within which the quenched disorder appears to be provided by the pinning of a fraction of the particles. We infer that the original unstable aspect of the theory due to the appearance of the logarithmic mode at the critical point corresponds to the instability of the seed (n = 1) soliton as collective fields.…”

Anchored advected interfaces, Oslo model, and roughness at depinning

“…Interfaces subject to quenched disorder describe a variety of physical phenomena [1,2], such as domain walls in magnets [3][4][5], contact-line depinning [6], fracture [7,8], or earthquakes [9]. Two situations have to be distinguished: equilibrium and depinning.…”

“…Though many numerical studies exist [10][11][12], and a field theory was developed [2,13,14], only a few exact results are available. A notable exception in equilibrium is the roughness exponent ζ d=1 RB = 2/3 for a 1 + 1 dimensional directed polymer in randombond (RB) disorder, itself related to the KPZ universality class [1,15] with roughness 1/2 and dynamic exponent z = 1/ζ d=1 RB = 3/2 [16].…”