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

The intermittent dynamics of driven interfaces through disordered media and its subsequent depinning for large enough driving force is a common feature for a myriad of diverse systems, starting from mode-I fracture, vortex lines in superconductors, and magnetic domain walls to invading fluid in a porous medium, to name a few. 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 that are seen prior to the depinning point. In particular, we use measures traditionally used to quantify economic inequality, i.e., the Gini index and the Kolkata index, for the case of the unequal responses of precritical systems. The crossing point of these two indices serves as a precursor to imminent depinning. Given a scale-free size distribution of the responses, we calculate the expressions for these indices, evaluate their crossing points, and give a recipe for forecasting depinning transitions. We apply this method to the Edwards-Wilkinson, Kardar-Parisi-Zhang, and fiber bundle model interface with variable interaction strengths and quenched disorder. The results are applicable for any interface dynamics undergoing a depinning transition. The results also explain previously observed near-universal values of Gini and Kolkata indices in self-organized critical systems.

Figure 1

(a) The time series of avalanche sizes (in logarithmic scale) is shown along with the time variations of $g$ and $k$ for the quenched Edwards-Wilkinson model as a prototype. Here the crossing of $g$ and $k$ occurs prior to the depinning transition point (the last avalanche in the series). Therefore, the crossing of $g$ and $k$ can serve as a good indicator of an imminent depinning transition. (b) $g_{min}$ and $k_{min}$ are the values where $g$ and $k$ come closest to each other at the earliest time. As can be seen, for $\delta >2$, they do not cross. For $\delta <2$, except at $\delta =1.5$, $g_{min}=k_{min}<1$, which means $g$ and $k$ are expected to cross each other prior to depinning. The points indicate the values at which different models are expected to exhibit this precursory warning. It is seen that different models (EW, KPZ, FBM, and the square of avalanches in FBM) show $g_{min}$ and $k_{min}$ values that are nearly universal at about 0.87, which are numerically verified later. The reason for considering the square of avalanches for the FBM interface is discussed later.

Figure 2

(a) The avalanche size (prior to depinning) distribution $P\left(S\right)$ for the fiber bundle interface model is plotted for the different values of $\gamma =0$, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5 for the system size $L=10000$. (b) The variation of the size distribution exponent $\delta$ with $\gamma$. It can be seen that the avalanche size distribution exponent becomes steeper than 2 for $\gamma <2$.

Figure 3

The time variation of $g$ and $k$ is shown for the different values of $\gamma =0$, 1, 2, 3, 4, and 5 for the system size $L=10000$ in the FBM interface. Here low $\gamma$ corresponds to the mean-field limit and high $\gamma$ corresponds to the localized load-sharing regime. (a) The crossing of $g$ and $k$ is seen when the $\gamma$ value is high since the size distribution exponent value is lower there (see Fig. 2). (b) The crossing of $g$ and $k$ is always seen for all values of $\gamma$ when we calculate $g$ and $k$ by taking the square of the avalanche sizes (i.e., $S^2$) because the size distribution of $S^2$ will have a smaller exponent value than that of $S$. However, for the $\gamma$ values for which crossing was already happening for $S$, with $S^2$ the crossing happens slightly earlier. Note that for visual clarity, all curves, except $\gamma =0$, are shifted in time.

Figure 4

The variation of $g$ and $k$ plotted with time for the two models, Edward-Wilkinson and Kardar-Parisi-Zhang, for the system size $L=1000$. In the case of these two models, the crossing of $g$ and $k$ for different distributions of pinning forces (see text) is shown. Insets: The size distributions of the avalanches prior to depinning. The exponent values are always less than 2.

Figure 5

The probability distributions of the $\frac{({\sigma }_c-{\sigma }^{*})}{{\sigma }_c}$ are shown for the three different models (quenched Edwards-Wilkinson model, quenched Kardar-Parisi-Zhang model, and fiber bundle interface model) for different threshold distributions. Here, ${\sigma }_c$ is the critical load, i.e., the depinning point, and ${\sigma }^{*}$ is the point where $g$ and $k$ cross. In the EW model, we consider two sets: in set 1, we take the threshold distribution to be uniform in the range ($-2,2$), and in set 2, we choose the threshold distribution to be uniform in the range ($-2.5,2.5$). In the KPZ model, in set 1, we take the threshold distribution to be uniform in the range ($-5,5$), and in set 2, we take the threshold distribution to be uniform in the range ($-6,6$). In the FBM, we take the two values of $\gamma =4,5$ and the threshold distribution to be uniform in (0,1). As can be seen, the distribution shows the peak in between 0.1 and 0.3, which means $g$ and $k$ cross near $10\%$ to $30\%$ away from the depinning transition point and therefore give a reliable indication of the imminent depinning transition. Here the averages are done over 10 000 ensembles.

Figure 6

A schematic diagram of the Lorenz function $\mathcal{L}\left(p\right)$, where $\mathcal{L}\left(p\right)$ denotes the cumulative fraction of the avalanche mass contained in the smallest $p$ fraction of avalanches. If all avalanches were equal, this would be a diagonal straight line, called the equality line (indicated in the figure). The area between the equality line and the Lorenz curve (shaded area), therefore, is a measure of the inequality in the avalanche sizes. Two quantitative measures of such inequality are extracted from here: the ratio of the shaded area and that under the equality line (called the Gini index, $g$) and the crossing point of the opposite diagonal—from (0,1) to (1,0), shown in the dashed line, and the Lorenz curve, giving the Kolkata index $k$. The $1-k$ fraction of the largest avalanches contains a $k$ fraction of the cumulative avalanche mass.

Figure 7

(a) The time variation of the avalanche series (in logarithmic scale) along with $g$ and $k$ are plotted for the Edward-Wilkinson model (as a prototype). (b) The same avalanches along with $g$ and $k$ are shown when the avalanches are arranged in the ascending order of their sizes. Note that in the second case, the time is not the actual time order of the occurrences of the avalanches. The two figures show that as far as the crossing point values of $g$ and $k$ are concerned, it is a fair approximation to considered the size-ordered series for analytical estimates.