Depinning exponents of the driven long-range elastic string

We perform a high-precision calculation of the critical exponents for the long-range elastic string driven through quenched disorder at the depinning transition, at zero temperature. Large-scale simulations are used to avoid finite-size effects and to enable high precision. The roughness, growth, and velocity exponents are calculated independently, and the dynamic and correlation length exponents are derived. The critical exponents satisfy known scaling relations and agree well with analytical predictions.Comment: 6 pages, 5 figure

Depinning of elastic manifolds

We compute roughness exponents of elastic d-dimensional manifolds in (d+1)-dimensional embedding spaces at the depinning transition for d=1,...,4. Our numerical method is rigorously based on a Hamiltonian formulation; it allows to determine the critical manifold in finite samples for an arbitrary convex elastic energy. For a harmonic elastic energy, we find values of the roughness exponent between the one-loop and the two-loop functional renormalization group result, in good agreement with earlier cellular automata simulations. We find that the harmonic model is unstable with respect both to slight stiffening and to weakening of the elastic potential. Anharmonic corrections to the elastic energy allow us to obtain the critical exponents of the quenched KPZ class.Comment: 4 pages, 4 figure

Seismic cycles, size of the largest events, and the avalanche size distribution in a model of seismicity

We address several questions on the behavior of a numerical model recently introduced to study seismic phenomena, that includes relaxation in the plates as a key ingredient. We make an analysis of the scaling of the largest events with system size, and show that when parameters are appropriately interpreted, the typical size of the largest events scale as the system size, without the necessity to tune any parameter. Secondly, we show that the temporal activity in the model is inherently non-stationary, and obtain from here justification and support for the concept of a "seismic cycle" in the temporal evolution of seismic activity. Finally, we ask for the reasons that make the model display a realistic value of the decaying exponent $b$ in the Gutenberg-Richter law for the avalanche size distribution. We explain why relaxation induces a systematic increase of $b$ from its value $b\simeq 0.4$ observed in the absence of relaxation. However, we have not been able to justify the actual robustness of the model in displaying a consistent $b$ value around the experimentally observed value $b\simeq 1$.Comment: 11 pages, 10 figure

Universal interface width distributions at the depinning threshold

We compute the probability distribution of the interface width at the depinning threshold, using recent powerful algorithms. It confirms the universality classes found previously. In all cases, the distribution is surprisingly well approximated by a generalized Gaussian theory of independant modes which decay with a characteristic propagator G(q)=1/q^(d+2 zeta); zeta, the roughness exponent, is computed independently. A functional renormalization analysis explains this result and allows to compute the small deviations, i.e. a universal kurtosis ratio, in agreement with numerics. We stress the importance of the Gaussian theory to interpret numerical data and experiments.Comment: 4 pages revtex4. See also the following article cond-mat/030146

Monte Carlo Dynamics of driven Flux Lines in Disordered Media

We show that the common local Monte Carlo rules used to simulate the motion of driven flux lines in disordered media cannot capture the interplay between elasticity and disorder which lies at the heart of these systems. We therefore discuss a class of generalized Monte Carlo algorithms where an arbitrary number of line elements may move at the same time. We prove that all these dynamical rules have the same value of the critical force and possess phase spaces made up of a single ergodic component. A variant Monte Carlo algorithm allows to compute the critical force of a sample in a single pass through the system. We establish dynamical scaling properties and obtain precise values for the critical force, which is finite even for an unbounded distribution of the disorder. Extensions to higher dimensions are outlined.Comment: 4 pages, 3 figure

Roughness at the depinning threshold for a long-range elastic string

In this paper, we compute the roughness exponent zeta of a long-range elastic string, at the depinning threshold, in a random medium with high precision, using a numerical method which exploits the analytic structure of the problem (`no-passing' theorem), but avoids direct simulation of the evolution equations. This roughness exponent has recently been studied by simulations, functional renormalization group calculations, and by experiments (fracture of solids, liquid meniscus in 4He). Our result zeta = 0.390 +/- 0.002 is significantly larger than what was stated in previous simulations, which were consistent with a one-loop renormalization group calculation. The data are furthermore incompatible with the experimental results for crack propagation in solids and for a 4He contact line on a rough substrate. This implies that the experiments cannot be described by pure harmonic long-range elasticity in the quasi-static limit.Comment: 4 pages, 3 figure

Creep dynamics of elastic manifolds via exact transition pathways

We study the steady state of driven elastic strings in disordered media below the depinning threshold. In the low-temperature limit, for a fixed sample, the steady state is dominated by a single configuration, which we determine exactly from the transition pathways between metastable states. We obtain the dynamical phase diagram in this limit. At variance with a thermodynamic phase transition, the depinning transition is not associated with a divergent length scale of the steady state below threshold, but only of the transient dynamics. We discuss the distribution of barrier heights, and check the validity of the dynamic phase diagram at small but finite temperatures using Langevin simulations. The phase diagram continues to hold for broken statistical tilt symmetry. We point out the relevance of our results for experiments of creep motion in elastic interfaces.Comment: 14 pages, 18 figure

Two-parameter quantum general linear supergroups

The universal R-matrix of two-parameter quantum general linear supergroups is computed explicitly based on the RTT realization of Faddeev--Reshetikhin--Takhtajan.Comment: v1: 14 pages. v2: published version, 9 pages, title changed and the section on central extension remove

Three-frequency resonances in dynamical systems

We investigate numerically and experimentally dynamical systems having three interacting frequencies: a discrete mapping (a circle map), an exactly solvable model (a system of coupled ordinary differential equations), and an experimental device (an electronic oscillator). We compare the hierarchies of three-frequency resonances we find in each of these systems. All three show similar qualitative behaviour, suggesting the existence of generic features in the parameter-space organization of three-frequency resonances.Comment: See home page http://lec.ugr.es/~julya