Quenched Edwards-Wilkinson Equation Research Articles

Depinning in the quenched Kardar-Parisi-Zhang class. I. Mappings, simulations, and algorithm.

Depinning of elastic systems advancing on disordered media can usually be described by the quenched Edwards-Wilkinson equation(qEW). However, additional ingredients such as anharmonicity and forces that cannot be derived from a potential energy may generate a different scaling behavior at depinning. The most experimentally relevant is the Kardar-Parisi-Zhang (KPZ) term, proportional to the square of the slope at each site, which drives the critical behavior into the so-called quenched KPZ (qKPZ) universality class. We study this universality class both numerically and analytically: by using exact mappings we show that at least for d=1,2 this class encompasses not only the qKPZ equationitself, but also anharmonic depinning and a well-known class of cellular automata introduced by Tang and Leschhorn. We develop scaling arguments for all critical exponents, including size and duration of avalanches. The scale is set by the confining potential strength m^{2}. This allows us to estimate numerically these exponents as well as the m-dependent effective force correlator Δ(w), and its correlation length ρ:=Δ(0)/|Δ^{'}(0)|. Finally, we present an algorithm to numerically estimate the effective (m-dependent) elasticity c, and the effective KPZ nonlinearity λ. This allows us to define a dimensionless universal KPZ amplitude A:=ρλ/c, which takes the value A=1.10(2) in all systems considered in d=1. This proves that qKPZ is the effective field theory for all these models. Our work paves the way for a deeper understanding of depinning in the qKPZ class, and in particular, for the construction of a field theory that we describe in a companion paper.

Depinning Exponents of Thin Film Domain Walls Depend on Disorder Strength.

Domain wall dynamics in ferromagnets is complicated by internal degrees of freedom of the domain walls. We develop a model of domain walls in disordered thin films with perpendicular magnetic anisotropy capturing such features, and use it to study the depinning transition. For weak disorder, excitations of the internal magnetization are rare, and the depinning transition takes on exponent values of the quenched Edwards-Wilkinson equation. Stronger disorder results in disorder-dependent exponents concurrently with nucleation of an increasing density of Bloch lines within the domain wall.

Nonsteady dynamic properties of a domain wall for the creep state under an alternating driving field.

With Monte Carlo simulations, the nonsteady dynamic properties of a domain wall have been systematically investigated for the thermally activated creep state under an alternating driving field. Taking the driven random-field Ising model in two dimensions as an example, two distinct growth stages of the domain interface are identified with both the correlation length and roughness function. One stage belongs to the universality class of the random depositions, and the other to that of the quenched Edwards-Wilkinson equation. In the latter case, due to the dynamic effect of overhangs, the domain interface may exhibit an intrinsic anomalous scaling behavior, different from that of the quenched Edwards-Wilkinson equation.

Thermal rounding of the depinning transition in ultrathin Pt/Co/Pt films

We perform a scaling analysis of the mean velocity of extended magnetic domain walls driven in ultrathin Pt/Co/Pt ferromagnetic films with perpendicular anisotropy, as a function of the applied external field for different film-thicknesses. We find that the scaling of the experimental data around the thermally rounded depinning transition is consistent with the universal depinning exponents theoretically expected for elastic interfaces described by the one-dimensional quenched Edwards-Wilkinson equation. In particular, values for the depinning exponent $\beta$ and thermal rounding exponent $\psi$ are tested and the present analysis of the experimental data is compatible with $\beta=0.33$ and $\psi=0.2$, in agreement with numerical simulations.

Pinning of interfaces in random media

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards–Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).

Spatial fluctuations in transient creep deformation

We study the spatial fluctuations of transient creep deformation of materials as a functionof time, both by digital image correlation (DIC) measurements of paper samples and bynumerical simulations of a crystal plasticity or discrete dislocation dynamics model. Thismodel has a jamming or yielding phase transition, around which power law or Andradecreep is found. During primary creep, the relative strength of the strain rate fluctuationsincreases with time in both cases—the spatially averaged creep rate obeys the Andrade lawϵt ∼ t − 0.7, while the time dependence of the spatial fluctuations of the local creep rates is given byΔϵt ∼ t − 0.5. A similar scaling for the fluctuations is found in the logarithmic creep regime that istypically observed for lower applied stresses. We review briefly some classical theories ofAndrade creep from the point of view of such spatial fluctuations. We consider thesephenomenological, time-dependent creep laws in terms of a description based on anon-equilibrium phase transition separating evolving and frozen states of the systemwhen the externally applied load is varied. Such an interpretation is discussedfurther by the data collapse of the local deformations in the spirit of absorbingstate/depinning phase transitions, as well as deformation–deformation correlations andthe width of the cumulative strain distributions. The results are also comparedwith the order parameter fluctuations observed close to the depinning transitionof the 2d linear interface model or the quenched Edwards–Wilkinson equation.

A Discrete Growth Model of the Quenched Edwards-Wilkinson Equation in d = 2 + 1

A Discrete Growth Model of the Quenched Edwards-Wilkinson Equation in d = 2 + 1

Anisotropy and universality: the Oslo model, the rice pile experiment, and the quenched Edwards-Wilkinson equation.

We show that any amount of anisotropy moves the Oslo model to another known universality class, the exponents of which can be derived exactly. This amounts to an exact solution of the quenched Edwards-Wilkinson equation with a drift term. We argue that anisotropy is likely to be experimentally relevant and may explain why consistent exponents have not been extracted in the rice pile experiments.

Oslo rice pile model is a quenched Edwards-Wilkinson equation.

The Oslo rice pile model is a sandpile-like paradigmatic model of self-organized criticality (SOC). In this paper it is shown that the Oslo model is in fact exactly a discrete realization of the much studied quenched Edwards-Wilkinson equation (qEW) [Nattermann et al., J. Phys. II France 2, 1483 (1992)]. This is possible by choosing the correct dynamical variable and identifying its equation of motion. It establishes for the first time an exact link between SOC models and the field of interface growth with quenched disorder. This connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of SOC.

Growth of a driven interface in isotropic and anisotropic random media

We introduce a simple stochastic model for a driven interface in a random medium, in which we can control the degree of the anisotropy of a random medium. When there is no anisotropy of a random medium, the motion of a growing interface in our model can be well described by the quenched Edwards-Wilkinson equation. When there is anisotropy of a random medium, however, the motion of a growing interface can be described by the quenched Kardar-Parisi-Zhang (KPZ) equation. In the two interfaces, apart from one growing in an isotropic medium and the other growing in an anisotropic medium, the growth rule of our model is the same. Our results support the fact that the anisotropy of a random medium is a source of the KPZ nonlinearity.

Self-organized growth model for a driven interface in random media.

We introduce a simple self-organized growth model in the quenched Edwards-Wilkinson universality class. The roughness and growth exponents are obtained for the model as alpha approximately 1.15 and beta approximately 0.89, respectively. These values of exponents are in good agreement with previous results obtained from numerical integrations or discretized model simulations of the quenched Edwards-Wilkinson equation. The velocity of a driven interface is found to be independent of the slope of the tilted substrate.