Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model

TL;DR

This paper provides an exact solution for the ABBM model describing particle motion in a Brownian landscape, predicting non-stationary dynamics and avalanche statistics, with applications to experimental systems like magnets and earthquakes.

Contribution

It derives explicit formulas for non-stationary driving in the ABBM model, extending understanding beyond quasi-static regimes and connecting to field theory and renormalization-group methods.

Findings

Explicit formula for generating functional of velocities and positions

Derived velocity distribution after a driving velocity quench

Obtained joint avalanche size and duration distributions

Abstract

We obtain an exact solution for the motion of a particle driven by a spring in a Brownian random-force landscape, the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model. Many experiments on quasi-static driving of elastic interfaces (Barkhausen noise in magnets, earthquake statistics, shear dynamics of granular matter) exhibit the same universal behavior as this model. It also appears as a limit in the field theory of elastic manifolds. Here we discuss predictions of the ABBM model for monotonous, but otherwise arbitrary, time-dependent driving. Our main result is an explicit formula for the generating functional of particle velocities and positions. We apply this to derive the particle-velocity distribution following a quench in the driving velocity. We also obtain the joint avalanche size and duration distribution and the mean avalanche shape following a jump in the position of the confining spring. Such non-stationary driving is easy to realize in experiments, and provides a way to test the ABBM model beyond the stationary, quasi-static regime. We study extensions to two elastically coupled layers, and to an elastic interface of internal dimension d, in the Brownian force landscape. The effective action of the field theory is equal to the action, up to 1-loop corrections obtained exactly from a functional determinant. This provides a connection to renormalization-group methods.