Localization in disordered media, anomalous roughening, and coarsening dynamics of faceted surfaces

Kardar-Parisi-Zhang universality class in the synchronization of oscillator lattices with time-dependent noise

Systems of oscillators subject to time-dependent noise typically achieve synchronization for long times when their mutual coupling is sufficiently strong. The dynamical process whereby synchronization is reached can be thought of as a growth process in which an interface formed by the local phase field gradually roughens and eventually saturates. Such a process is here shown to display the generic scale invariance of the one-dimensional Kardar-Parisi-Zhang universality class, including a Tracy-Widom probability distribution for phase fluctuations around their mean. This is revealed by numerical explorations of a variety of oscillator systems: rings of generic phase oscillators and rings of paradigmatic limit-cycle oscillators, like Stuart-Landau and van der Pol. It also agrees with analytical expectations derived under conditions of strong mutual coupling. The nonequilibrium critical behavior that we find is robust and transcends the details of the oscillators considered. Hence, it may well be accessible to experimental ensembles of oscillators in the presence of, e.g., thermal noise.

Fragility of Kardar-Parisi-Zhang universality class in the presence of temporally correlated noise

We study numerically a family of surface growth models that are known to be in the universality class of the Kardar-Parisi-Zhang equation when driven by uncorrelated noise. We find that, in the presence of noise with power-law temporal correlations with exponent θ, these models exhibit critical exponents that differ both quantitatively and qualitatively from model to model. The existence of a threshold value for θ below which the uncorrelated fixed point is dominant occurs for some models but not for others. In some models the dynamic exponent z(θ) is a smooth decreasing function, while it has a maximum in other cases. Despite all models sharing the same symmetries, critical exponents turn out to be strongly model dependent. Our results clearly show the fragility of the universality class concept in the presence of long-range temporally correlated noise.

Finite-size and finite-time scaling for kinetic rough interfaces

We consider discrete models of kinetic rough interfaces that exhibit space-time scale invariance in height-height correlation. We use the generic scaling theory of Ramasco et al. [Phys. Rev. Lett. 84, 2199 (2000)0031-900710.1103/PhysRevLett.84.2199] to confirm that the dynamical structure factor of the height profile can uniquely characterize the underlying dynamics. We apply both finite-size and finite-time scaling methods that systematically allow an estimation of the critical exponents and the scaling functions, eventually establishing the universality class accurately. The finite-size scaling analysis offers an alternative way to characterize the anomalous rough interfaces. As an illustration, we investigate a class of self-organized interface models in random media with extremal dynamics. The isotropic version shows a faceted pattern and belongs to the same universality class (as shown numerically) as the Sneppen model (version A). We also examine an anisotropic version of the Sneppen model and suggest that the model belongs to the universality class of the tensionless Kardar-Parisi-Zhang (tKPZ) equation in one dimension.

Anomalous phase diagram of the elastic interface with nonlocal hydrodynamic interactions in the presence of quenched disorder

We investigate the influence of quenched disorder on the steady states of driven systems of the elastic interface with nonlocal hydrodynamic interactions. The generalized elastic model (GEM), which has been used to characterize numerous physical systems such as polymers, membranes, single-file systems, rough interfaces, and fluctuating surfaces, is a standard approach to studying the dynamics of elastic interfaces with nonlocal hydrodynamic interactions. The criticality and phase transition of the quenched generalized elastic model are investigated numerically and the results are presented in a phase diagram spanned by two tuning parameters. We demonstrate that in the one-dimensional disordered driven GEM, three qualitatively different behavior regimes are possible with a proper specification of the order parameter (mean velocity) for this system. In the vanishing order parameter regime, the steady-state order parameter approaches zero in the thermodynamic limit. A system with a nonzero mean velocity can be in either the continuous regime, which is characterized by a second-order phase transition, or the discontinuous regime, which is characterized by a first-order phase transition. The focus of this research is to investigate the critical scaling features near the pinning-depinning threshold. The behavior of the quenched generalized elastic model at the critical depinning force is explored. Near the depinning threshold, the critical exponent is obtained numerically.

Universal interface fluctuations in the contact process

We study the interface representation of the contact process at its directed-percolation critical point, where the scaling properties of the interface can be related to those of the original particle model. Interestingly, such a behavior happens to be intrinsically anomalous and more complex than that described by the standard Family-Vicsek dynamic scaling Ansatz of surface kinetic roughening. We expand on a previous numerical study by Dickman and Muñoz [Phys. Rev. E 62, 7632 (2000)10.1103/PhysRevE.62.7632] to fully characterize the kinetic roughening universality class for interface dimensions d=1,2, and 3. Beyond obtaining scaling exponent values, we characterize the interface fluctuations via their probability density function (PDF) and covariance, seen to display universal properties which are qualitatively similar to those recently assessed for the Kardar-Parisi-Zhang (KPZ) and other important universality classes of kinetic roughening. Quantitatively, while for d=1 the interface covariance seems to be well described by the KPZ, Airy_{1} covariance, no such agreement occurs in terms of the fluctuation PDF or the scaling exponents.

Roughening of the anharmonic elastic interface in correlated random media

We study the roughening properties of the anharmonic elastic interface in the presence of temporally correlated noise. The model can be seen as a generalization of the anharmonic Larkin model, recently introduced by Purrello, Iguain, and Kolton [Phys. Rev. E 99, 032105 (2019)2470-004510.1103/PhysRevE.99.032105], to investigate the effect of higher-order corrections to linear elasticity in the fate of interfaces. We find analytical expressions for the critical exponents as a function of the anharmonicity index n, the noise correlator range θ∈[0,1/2], and dimension d. In d=1 we find that the interface becomes faceted and exhibits anomalous scaling for θ>1/4 for any degree of anharmonicity n>1. Analytical expressions for the anomalous exponents α_{loc} and κ are obtained and compared with a numerical integration of the model. Our theoretical results show that anomalous roughening cannot exist for this model in dimensions d>1.

Faceted patterns and anomalous surface roughening driven by long-range temporally correlated noise

We investigate Kardar-Parisi-Zhang (KPZ) surface growth in the presence of power-law temporally correlated noise. By means of extensive numerical simulations of models in the KPZ universality class we find that, as the noise correlator index increases above some threshold value, the surface exhibits anomalous kinetic roughening of the type described by the generic scaling theory of Ramasco et al. [Phys. Rev. Lett. 84, 2199 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.2199]. Remarkably, as the driving noise temporal correlations increase, the surface develops a characteristic pattern of macroscopic facets that completely dominates the dynamics in the long time limit. We argue that standard scaling fails to capture the behavior of KPZ subject to long-range temporally correlated noise. These phenomena are not not described by the existing theoretical approaches, including renormalization group and self-consistent approaches.

Roughening of the anharmonic Larkin model

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.

Exponential localization of singular vectors in spatiotemporal chaos

In a dynamical system the singular vector (SV) indicates which perturbation will exhibit maximal growth after a time interval tau . We show that in systems with spatiotemporal chaos the SV exponentially localizes in space. Under a suitable transformation, the SV can be described in terms of the Kardar-Parisi-Zhang equation with periodic noise. A scaling argument allows us to deduce a universal power law tau(-gamma) for the localization of the SV. Moreover the same exponent gamma characterizes the finite- tau deviation of the Lyapunov exponent in excellent agreement with simulations. Our results may help improve existing forecasting techniques.

Complex dynamics during metal dissolution: from intrinsic to faceted anomalous scaling

The kinetic roughening of dissolving polycrystalline pure iron has been studied. A depth analysis of surface images has shown two consecutive growth regimes characterized by different scaling anomalous properties: an initial intrinsic anomalous scaling evolving in the thick film limit towards the theoretically conjectured faceted anomalous scaling. This represents the first experimental evidence of such scaling as well as of such transition. The dynamics presented here may account for the striped surface pattern observed during the evolution of metals or alloys in a large number of processes.

Dynamics of perturbations in disordered chaotic systems

We study the time evolution of perturbations in spatially extended chaotic systems in the presence of quenched disorder. We find that initially random perturbations tend to exponentially localize in space around static pinning centers that are selected by the particular configuration of disorder. The spatiotemporal behavior of typical perturbations deltau(x,t) is analyzed in terms of the Hopf-Cole transform h(x,t) identical withlnmid R:deltau(x,t)mid R: . Our analysis shows that the associated surface h(x,t) self-organizes into a faceted structure with scale-invariant correlations. Scaling analysis of critical roughening exponents reveals that there are three different universality classes for error propagation in disordered chaotic systems that correspond to different symmetries of the underlying disorder. Our conclusions are based on numerical simulations of disordered lattices of coupled chaotic elements and equations for diffusion in random potentials. We propose a phenomenological stochastic field theory that gives some insights on the path for a generalization of these results for a broad class of disordered extended systems exhibiting space-time chaos.

Anomalous kinetic roughening during anodic dissolution of polycrystalline Fe

Dynamics of surface roughness during polycrystalline pure iron electrodissolution is investigated at constant current density by means of ex situ atomic force microscopy. The scaling of the local surface width reveals that surface kinetic roughening is anomalous with both the exponents of local roughness, alpha(loc) , and growth, beta , close to 1 pointing out that interface evolution is unstable. We show that this anomalous unstable behavior results from the development of a faceted surface structure exposing different crystal orientations. The presence of smooth faceted walls is consistent with the value alpha(local) approximately 1 , whereas the difference in the dissolution rates on the different crystallographic planes account for the nonlocal effects causing the unstable growth. Results are discussed in the context of a recently reported anomalous scaling which accounts for dynamics of self-organized depinning models displaying faceted interfaces. The influence of the electrode potential on the dissolution rates of the different crystallographic planes, along with its effect on the mobility of metal adatoms, are discussed to be behind the complex behavior of local roughness when the current density is varied.