Height distribution of elastic interfaces in quenched random media

TL;DR

The paper addresses depinning of driven elastic interfaces in quenched random media and reveals that the local height distribution $P(h_i)$ exhibits negative skewness at depinning, a feature linked to universal aspects of the critical state. By analyzing local, long-range, and mean-field elastic interactions, the authors show that negative skewness at depinning is robust in the thermodynamic limit and insensitive to the system size $L$ and spring stiffness $K$, with $K$ setting the avalanche cutoff $s^*$. They further study how $P(h_i)$ evolves as the external force $F_{ ext{ext}}$ is ramped from zero under force-controlled driving, finding an initial symmetric distribution that becomes positively skewed before a rapid transition to negative skewness near $F_c$, signaling the onset of depinning. The results connect skewness properties to avalanche statistics and provide a morphological signature of the depinning transition that could extend to other avalanche phenomena in driven disordered systems.

Abstract

Elastic interfaces in quenched random media driven by external forces exhibit a continuous depinning phase transition between pinned and moving phases at a critical external force. Recent work [Phys. Rev. Lett. 129, 175701 (2022)] has shown that the distribution of local interface heights at depinning displays negative skewness. Here, by considering local, long-range and fully-coupled (mean-field) elasticity, we expand on this result by demonstrating the robustness of the negative skewness at depinning when approaching the thermodynamic limit and considering different values of the spring stiffness controlling the avalanche cutoff. Additionally, we investigate the evolution of the height distribution as the external force is ramped up from zero, approaching the critical force from below. Starting from a symmetric height distribution at zero force, the distribution initially develops positive skewness increasing with the external force, followed by a steep drop to the negative value characteristic of the critical point as the depinning transition is reached.

Figures (2)

• Figure 1: Examples of rough interface configurations at the depinning threshold obtained from the discrete-time models with quasistatic constant velocity driving, with (a) local, (b) long-range and (c) mean-field interactions. The system size is $L=512$ for all three cases. The black horizontal lines indicate the mean interface heights.
• Figure 2: (a) Distributions $P(\tilde{h})$ of the scaled interface heights for the discrete-time models with quasistatic constant velocity driving (i.e., at depinning) compared to the standard normal distribution $N(0,1)$ (top) and difference $N(0,1)-P(\tilde{h})$ (bottom) for the local, long-range and mean-field models in (a), (b) and (c), respectively, indicating negative skewness of $P(\tilde{h})$. (d) The roughness exponent $\zeta$ estimated from the scaling of $W^{2}(L)$ for the different models.