Prediction of depinning transitions in interface models using Gini and Kolkata indices

Abstract

The dynamics of driven interfaces through disordered media is a common framework for a myriad of diverse systems starting from mode-I fracture, vortex lines in superconductors, magnetic domain walls to invading fluid in a porous medium to name a few. The dynamics, for a slow enough drive, progress through punctuated equilibria, showing an intermittent, scale free response statistics that are characterized by only a few parameters. For a strong enough driving, on the other hand, the interface is depinnind i.e., can move continuously through the embedded medium. This depinning transition, often characterized as a critical phenomenon, is a crucial point for the system in question, since it often represent a drastic change in the system (catastrophic failure in the case of fracture, magnetisation switching in the case of domain walls, breakthrough point in the case of fluid invasion etc.). In this work, we outline a framework that can give a precursory signal of the imminent depinning transition by monitoring the variations in sizes or the inequality of the intermittent responses of a system seen prior to the depinning point. In particular, we use measures traditionally used to quantify economic inequality, the Gini index and the Kolkata index, for the case of the unequal responses of pre-critical systems. The crossing point of these two indices serves as a precursor to imminent depinning. Given a scale free size distributions of the responses, we calculate the expressions for these indices, evaluate their crossing points and give a recipe for forecasting depinning transitions. We compare these with simulation results from the Edwards-Wilkindon, KPZ and fiber bundle model interface with variable interaction strength. The results are applicable for any interface dynamics undergoing depinning transition.