Anomalous scaling of heterogeneous elastic lines: a new picture from sample to sample fluctuations

TL;DR

This work analyzes a one-dimensional elastic line with quenched internal disorder, modeled by random springs drawn from p(k) = μ k^{μ-1}. By deriving exact equilibrium distributions and the spectral density of the coupling matrix, the authors map out how μ controls the transition from Edwards-Wilkinson (μ>1) to anomalous (μ<1) scaling, with a jump-dominated interpretation of the latter. They show that disorder-averaged observables depend sensitively on boundary conditions in the anomalous regime and provide precise finite-time scaling forms tied to the low-λ behavior of the spectrum, including a Lévy-type statistics for sums of Pareto variables. Their results, supported by numerics, challenge previous local-roughness interpretations and offer a probabilistic framework for understanding intermittent large jumps in heterogeneous elastic manifolds, with potential relevance to experiments in imbibition and related systems.

Abstract

We study a discrete model of an heterogeneous elastic line with internal disorder, submitted to thermal fluctuations. The monomers are connected through random springs with independent and identically distributed elastic constants drawn from $p(k)\sim k^{μ-1}$ for $k\to0$. When $μ>1$, the scaling of the standard Edwards-Wilkinson model is recovered. When $μ<1$, the elastic line exhibits an anomalous scaling of the type observed in many growth models and experiments. Here we derive and use the exact expression for the exact probability distribution of the line shape at equilibrium, as well as the spectral properties of the matrix containing the random couplings, to fully characterize the sample to sample fluctuations. Our results lead to novel scaling predictions that partially disagree with previous works, but which are corroborated by numerical simulations. We also provide a novel interpretation of the anomalous scaling in terms of the abrupt jumps in the line's shape that dominate the average value of the observable.

Figures (8)

• Figure 1: Illustration of a random spring chain for $L$ monomers fixed at the origin and free on the other hand.
• Figure 2: Example of three interfaces for $\mu=1.5, 0.75, 0.4$, Dirichlet boundary conditions and zero mean. Decreasing $\mu$, the interfaces display larger and larger jumps which are at the origin of the anomalous behaviour.
• Figure 3: Interfaces ($\mu=0.75$) with a given realization of spring are evolved for a long time $t \sim L^z$ using the same noise. The disorder is typical but the smallest spring value is divided by a factor $10$. Top: fixed case (two Dirichlet boundary conditions) (blue) versus free case (orange). A single anomalously weak spring is enough to rip the interface with free end, but not with two fixed ends. Bottom: fixed case with a single weak spring (blue) versus with two very weak springs (orange, in this case the two smallest springs are divided by a factor $10$). Here we see that only the orange interface is ripped.
• Figure 4: Histogram of $G(x)$ for $x\ll L$ in the equilibrated regime for $\mu=0.4$ (Top) and $\mu=0.75$ (Bottom). $N_s=10^7$ values of $G$ are obtained by direct evaluation of Eq. \ref{['Gavg']} and Eq. \ref{['eq:Gfixed']}, with $L=10^4$ and $x=100$, in the free case (light green, light blue) and the fixed case (dark green, dark blue). For each distribution, the typical behaviour is located around $G \sim x^{1/\mu}$ ($2 \zeta=1/\mu$). The slow power law decay $\sim x/ G ^{1+\mu}$ as well as the fast power law decay, for the fixed case, $\sim xL/ G ^{1+ 2\mu}$ are well visible. In this latter case, the crossover occurs for $G \sim L^{1/\mu}$.
• Figure 5: Cumulative PDF of the eigenvalues $\mathcal{N}_L (\lambda)$ for $\mu=0.4$ and $N_s=10^8$ realisations (top) and for $\mu = 0.75$ (bottom) for $L=500$ for free case (blue) and free case (green). The black dashed line corresponds to the analytical predictions: the predicted asymptotics and $\mathcal{N}_\infty(\lambda) =\int_0^\lambda \rho(s) \mathrm{d} s$.