Mean-field theories for depinning and their experimental signatures

Mean-field theory is an approximation replacing an extended system by a few variables. For depinning of elastic manifolds, these are the position u of its center of mass and the statistics of the forces F(u). There are two proposals how to model the latter: as a random walk (ABBM model), or as uncorrelated forces at integer u (discretized particle model, DPM). While for many experiments the ABBM model (in the literature misleadingly equated with mean-field theory) makes quantitatively correct predictions for the distributions of velocities, or avalanche size and duration, the microscopic disorder force-force correlations cannot grow linearly, and thus unboundedly as a random walk, with distance. Even the effective (renormalized) disorder forces which do so at small distances are bounded at large distances. To describe both regimes, we model forces as an Ornstein-Uhlenbeck process. The latter has the statistics of a random walk at small scales, and is uncorrelated at large scales. By connecting to results in both limits, we solve the model largely analytically, allowing us to describe in all regimes the distributions of velocity, avalanche size, and duration. To establish experimental signatures of this transition, we study the response function, and the correlation function of position u, velocity <(u)over dot>, and forces F under slow driving with velocity v > 0. While at v = 0 force or position correlations have a cusp at the origin and then decay at least exponentially fast to zero, this cusp is rounded at a finite driving velocity. We give a detailed analytic analysis for this rounding by velocity, which allows us, given experimental data, to extract the timescale of the response function, and to reconstruct the force-force correlator at v = 0. The latter is the central object of the field theory, and as such contains detailed information about the universality class in question. We test our predictions by careful numerical simulations extending over up to ten orders in magnitude.

Automated summary

Mean-field theory approximates extended systems using a few variables, such as the position and statistics of forces for elastic manifolds. Two models, ABBM and DPM, can be used to simulate force distributions. However, microscopic disorder force-force correlations cannot grow linearly and must be bounded, leading to the need for an Ornstein-Uhlenbeck process to account for both small and large scales. Experimental signatures of this transition can be studied through response and correlation functions.