Height distribution of elastic interfaces in quenched random media

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.

💡 Research Summary

The paper investigates how the distribution of local heights of a driven elastic interface in a quenched random medium evolves as the system approaches the depinning transition, focusing on the skewness of this distribution. Three types of elastic interactions are considered: local (short‑range), long‑range (LR), and fully‑coupled or mean‑field (MF). The authors employ two complementary numerical protocols. In the first, a quasistatic constant‑velocity driving scheme is used: the interface is kept as close as possible to the depinning threshold by adjusting an external force F_ext with a “driving spring” of stiffness K; when the interface stops moving, F_ext is increased just enough to trigger a single site update, and then the force is reduced according to (\dot F_{\rm ext}=-K L\sum_i v_i). This protocol allows precise sampling of the critical point while controlling the avalanche‑size cutoff through K. System sizes range from (L=64) to (L=2^{15}=32768), and K is varied over more than two orders of magnitude (8×10⁻⁵ – 1.024×10⁻²).

In the second protocol, a continuous‑time dynamics with a linear mobility law is used to study the dependence on the external force itself. Here the force is ramped linearly from zero to the critical value (F_c) at a very slow rate ((\dot F_{\rm ext}=10^{-5})). The interface heights are real‑valued and updated with an Euler integrator, allowing both forward and backward motion. This setup makes it possible to monitor the full evolution of the height distribution as the driving force increases.

For each model and each protocol the authors compute the scaled height variable (\tilde h=(h_i-\langle h\rangle)/\sigma_h) and its probability density (P(\tilde h)). At the depinning threshold, all three interaction types display a pronounced negative tail compared with a standard normal distribution, indicating negative skewness. Quantitatively, the skewness values are (\gamma\approx-0.08) (local), (\gamma\approx-0.13) (LR), and (\gamma\approx-0.32) (MF). Importantly, these values are essentially independent of system size L, confirming that the negative skewness survives in the thermodynamic limit. Varying the spring stiffness K changes the magnitude only slightly; the sign and overall size of the skewness remain robust, showing that the avalanche‑cutoff scale does not affect the asymmetry of the height distribution.

When the force is ramped from zero, the situation is markedly different. At (F_{\rm ext}=0) the height distribution is symmetric (γ≈0). As the force grows, only a fraction of the interface advances, producing a positive skewness that increases with (F_{\rm ext}). However, as the force approaches the critical value from below, the skewness undergoes a rapid drop to the characteristic negative value observed at the critical point. This sharp crossover, occurring in a narrow window just before depinning, is identified as a new morphological signature of the onset of the depinning transition.

The authors also connect the observed negative skewness to previously reported features of avalanche statistics. In particular, the negative tail of (P(\tilde h)) is linked to the “bump” that appears in the cutoff scaling function (f(x)) of the avalanche‑size distribution (P(s)=s^{-\tau}f(s/s^*)). This bump has been predicted by functional renormalization‑group calculations and is thought to arise from the same broken symmetry that generates the height‑distribution asymmetry.

Overall, the study establishes four key results: (1) negative skewness of the height distribution at depinning is universal across local, long‑range, and mean‑field elasticity; (2) the skewness does not depend on system size or on the stiffness of the driving spring, implying that it persists in the thermodynamic limit; (3) the evolution of skewness with the external force shows a characteristic positive‑to‑negative transition that marks the approach to the critical point; and (4) the asymmetry of the height distribution is intimately tied to non‑trivial features of avalanche statistics, reinforcing its role as a fundamental observable of depinning criticality.

By providing a comprehensive numerical analysis that bridges the gap between critical‑point statistics and the full force‑controlled trajectory, the paper offers a solid foundation for future experimental investigations of height‑distribution asymmetry in driven disordered systems, as well as for theoretical extensions that may incorporate more complex disorder or higher‑dimensional interfaces.