RESUMO
We study the roughening of d-dimensional directed elastic interfaces subject to quenched random forces. As in the Larkin model, random forces are considered constant in the displacement direction and uncorrelated in the perpendicular direction. The elastic energy density contains an harmonic part, proportional to (∂_{x}u)^{2}, and an anharmonic part, proportional to (∂_{x}u)^{2n}, where u is the displacement field and n>1 an integer. By heuristic scaling arguments, we obtain the global roughness exponent ζ, the dynamic exponent z, and the harmonic to anharmonic crossover length scale, for arbitrary d and n, yielding an upper critical dimension d_{c}(n)=4n. We find a precise agreement with numerical calculations in d=1. For the d=1 case we observe, however, an anomalous "faceted" scaling, with the spectral roughness exponent ζ_{s} satisfying ζ_{s}>ζ>1 for any finite n>1, hence invalidating the usual single-exponent scaling for two-point correlation functions, and the small gradient approximation of the elastic energy density in the thermodynamic limit. We show that such d=1 case is directly related to a family of Brownian functionals parameterized by n, ranging from the random-acceleration model for n=1 to the Lévy arcsine-law problem for n=∞. Our results may be experimentally relevant for describing the roughening of nonlinear elastic interfaces in a Matheron-de Marsilly type of random flow.
RESUMO
We study the slow stochastic dynamics near the depinning threshold of an elastic interface in a random medium by solving a particularly suited model of hopping interacting particles that belongs to the quenched-Edwards-Wilkinson depinning universality class. The model allows us to compare the cases of uniformly activated and Arrhenius activated hops. In the former case, the velocity accurately follows a standard scaling law of the force and noise intensity with the analog of the thermal rounding exponent satisfying a modified "hyperscaling" relation. For the Arrhenius activation, we find, both numerically and analytically, that the standard scaling form fails for any value of the thermal rounding exponent. We propose an alternative scaling incorporating logarithmic corrections that appropriately fits the numerical results. We argue that this anomalous scaling is related to the strong correlation between activated hops that, alternated with deterministic depinning-like avalanches, occur below the depinning threshold. We rationalize the spatiotemporal patterns by making an analogy of the present model in the near-threshold creep regime with some well-known models with extremal dynamics, particularly the Bak-Sneppen model.