Edwards-Wilkinson Universality Class Research Articles

Bifractality in the one-dimensional Wolf-Villain model.

We introduce a multifractal optimal detrended fluctuation analysis to study the scaling properties of the one-dimensional Wolf-Villain (WV) model for surface growth. This model produces coarsened surface morphologies for long timescales (up to 10^{9} monolayers) and its universality class remains an open problem. Our results for the multifractal exponent τ(q) reveal an effective local roughness exponent consistent with a transient given by the molecular beam epitaxy (MBE) growth regime and Edwards-Wilkinson (EW) universality class for negative and positive q values, respectively. Therefore, although the results corroborate that long-wavelength fluctuations belong to the EW class in the hydrodynamic limit, as conjectured in the recent literature, a bifractal signature of the WV model with an MBE regime at short wavelengths was observed.

Q-TASEP with position-dependent slowing

We introduce a new interacting particle system on Z, slowed t-TASEP. It may be viewed as a q-TASEP with additional position-dependent slowing of jump rates, depending on a parameter t, which leads to discrete asymptotic fluctuations at large time. If on the other hand t→1 as time→∞, we prove A law of large numbers for particle positions, A central limit theorem, with convergence to the fixed-time Gaussian marginal of a stationary solution to SDEs derived from the particle jump rates, and A bulk limit to a certain explicit stationary Gaussian process on R, with scaling exponents characteristic of the Edwards-Wilkinson universality class in (1+1) dimensions. The proofs relate slowed t-TASEP to a certain Hall-Littlewood process, and use contour integral formulas for observables of this process. Unlike most previously studied Macdonald processes, this one involves only local interactions, resulting in asymptotics characteristic of (1+1)-dimensional rather than (2+1)-dimensional systems.

Edwards-Wilkinson depinning transition in random Coulomb potential background.

The quenched Edwards-Wilkinson growth of the 1+1 interface is considered in the background of the correlated random noise. We use random Coulomb potential as the background long-range correlated noise. A depinning transition is observed in a critical driving force F[over ̃]_{c}≈0.037 (in terms of disorder strength unit) in the vicinity of which the final velocity of the interface varies linearly with time. Our data collapse analysis for the velocity shows a crossover time t^{*} at which the velocity is size independent. Based on a two-variable scaling analysis, we extract the exponents, which are different from all universality classes we are aware of. Especially noting that the dynamic and roughness exponents are z_{w}=1.55±0.05, and α_{w}=1.05±0.05 at the criticality, we conclude that the system is different from both Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) universality classes. Our analysis shows therefore that making the noise long-range correlated, drives the system out of the EW universality class. The simulations on the tilted lattice show that the nonlinearity term (λ term in the KPZ equations) goes to zero in the thermodynamic limit.

Universal critical exponents of the magnetic domain wall depinning transition

Magnetic-field-driven domain wall dynamics in a ferrimagnetic GdFeCo thin film with perpendicular magnetic anisotropy is studied using low-temperature magneto-optical Kerr microscopy. Measurements performed in a practically athermal condition allow for the direct experimental determination of the velocity ($\ensuremath{\beta}=0.30\ifmmode\pm\else\textpm\fi{}0.03$) and correlation length ($\ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3$) exponents of the depinning transition. The whole family of exponents characterizing the transition is deduced, providing evidence that the depinning of magnetic domain walls is better described by the quenched Edwards-Wilkinson universality class.

Current statistics in the q-boson zero range process

We obtain exact formulas of the first two cumulants of particle current in the q-boson zero range process on a ring via exact perturbative solution of the TQ-equation. The result is represented as an infinite sum of double contour integrals. We perform the asymptotic analysis of the large system size limit N → ∞ of the expressions obtained. For |q| ≠ 1 the leading terms of the second cumulant reproduce the N 3/2 scaling expected for models in the Kardar–Parisi–Zhang universality class. The scaling corresponds to the crossover between the Kardar–Parisi–Zhang and Edwards–Wilkinson universality classes. Under this scaling the sum converges to an integral, resulting in the crossover scaling function derived previously for the asymmetric simple exclusion process and conjectured to be universal.

Common universal behavior of magnetic domain walls driven by spin-polarized electrical current and magnetic field

We explore the universal behaviors of a magnetic domain wall driven by the spin-transfer torque of an electrical current, in a ferromagnetic (Ga,Mn)(As,P) thin film with perpendicular magnetic anisotropy. For a current transverse to the domain wall, the dynamics of the thermally activated creep regime and the depinning transition are found to be compatible with a self-consistent universal description of magnetic-field-induced domain-wall dynamics. This common universal behavior, characteristic of the so-called quenched Edwards-Wilkinson universality class, is confirmed by an independent analysis of domain-wall roughness. Complementary investigations reveal the directional properties of interaction between current and domain walls which result in the instability of their transverse orientation.

Random deposition model with surface relaxation in higher dimensions

A random deposition model with surface relaxation, so-called the Family model is studied in higher dimensions. In three dimensions, the surface width W(t) characterizing the roughness of a surface grows as 2blogt at the beginning and becomes saturated at 2alogL for t≫Lz, where L is the system size. The dynamic exponent z=1.99(2) is estimated from the relation z=a∕b and a nice data collapse of the scaling plot W2(L,t)∼logL2agt∕Lz is given with z=2. In four dimensions, the surface width approaches an intrinsic width Wint with a small correction term W2(L,t)=Wint2−L2αft∕Lz, where z≈1.97 and negative exponent α≈−0.52 are obtained. Our results support that the Family model belongs to the Edwards–Wilkinson universality class even in higher dimensions.

Learning universality and scaling from simple deposition models

We use deposition models of kinetic roughening of a growing surface to introduce the concepts of universality and scaling and to analyze the qualitative and quantitative role of different parameters. In particular, we focus on two classes of models where the deposition is accompanied by a local relaxation process within a distance δ. The models are in the Edwards-Wilkinson universality class, but the role of δ is nontrivial.

Finite-size effects in the short-time height distribution of the Kardar–Parisi–Zhang equation

We use the optimal fluctuation method to evaluate the short-time probability distribution of height at a single point, , of the evolving Kardar–Parisi–Zhang (KPZ) interface on a ring of length 2L. The process starts from a flat interface. At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards–Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of . At large the faster-decaying tail has a double structure: it is L-independent, , at intermediately large , and L-dependent, , at very large . The transition between these two regimes is sharp and, in the large limit, behaves as a fractional-order phase transition. The transition point depends on . At small , the double structure of the faster tail disappears, and only the very large-H tail, , is observed. The slower-decaying tail does not show any L-dependence at large , where it coincides with the slower tail of the GOE Tracy–Widom distribution. At small this tail also has a double structure. The transition between the two regimes occurs at a value of height which depends on . At the transition behaves as a mean-field-like second-order phase transition. At the slower tail behaves as , whereas at it coincides with the slower tail of the GOE Tracy–Widom distribution.

From dynamical scaling to local scale-invariance: a tutorial

Dynamical scaling arises naturally in various many-body systems far from equilibrium. After a short historical overview, the elements of possible extensions of dynamical scaling to a local scale-invariance will be introduced. Schr\"odinger-invariance, the most simple example of local scale-invariance, will be introduced as a dynamical symmetry in the Edwards-Wilkinson universality class of interface growth. The Lie algebra construction, its representations and the Bargman superselection rules will be combined with non-equilibrium Janssen-de Dominicis field-theory to produce explicit predictions for responses and correlators, which can be compared to the results of explicit model studies. At the next level, the study of non-stationary states requires to go over, from Schr\"odinger-invariance, to ageing-invariance. The ageing algebra admits new representations, which acts as dynamical symmetries on more general equations, and imply that each non-equilibrium scaling operator is characterised by two distinct, independent scaling dimensions. Tests of ageing-invariance are described, in the Glauber-Ising and spherical models of a phase-ordering ferromagnet and the Arcetri model of interface growth.

Pinning-depinning transition in a stochastic growth model for the evolution of cell colony fronts in a disordered medium.

We study a stochastic lattice model for cell colony growth, which takes into account proliferation, diffusion, and rotation of cells, in a culture medium with quenched disorder. The medium is composed of sites that inhibit any possible change in the internal state of the cells, representing the disorder, as well as by active medium sites that do not interfere with the cell dynamics. By means of Monte Carlo simulations we find that the velocity of the growing interface, which is taken as the order parameter of the model, strongly depends on the density of active medium sites (ρ_{A}). In fact, the model presents a (continuous) second-order pinning-depinning transition at a certain critical value of ρ_{A}^{crit}, such as, for ρ_{A}>ρ_{A}^{crit}, the interface moves freely across the disordered medium, but for ρ_{A}<ρ_{A}^{crit} the interface becomes irreversible pinned by the disorder. By determining the relevant critical exponents, our study reveals that within the depinned phase the interface can be rationalized in terms of the Kardar-Parisi-Zhang universality class, but when approaching the critical threshold, the nonlinear term of the Kardar-Parisi-Zhang equation tends to vanish and then the pinned interface belongs to the quenched Edwards-Wilkinson universality class.

Schramm–Loewner evolution theory of the asymptotic behaviors of (2+1)-dimensional Wolf–Villain model

Abstract Although extensive analytical and numerical work has focus on investigating the (2+1)-dimensional Wolf–Villain (WV) model, some problems concerning its asymptotical behaviors, such as the universality class to which it belongs remain controversial. The Schramm–Loewner evolution (SLE) theory is an attractive approach to describe the fluctuation phenomena and random processes, and has also been applied to the analysis of the stochastic growth of surfaces. In this work, we applied SLE theory to the analysis of the saturated surface to conduct in-depth research, with a new perspective, on the asymptotical behaviors of the (2+1)-dimensional WV model. On the basis of the analysis of the saturated surface contour lines, we determine that the diffusion coefficient calculated for (2+1)-dimensional WV model is κ = 2.91 ± 0.01 , and that for Family model is κ = 2.88 ± 0.01 . Accordingly we can conclude from the view of SLE theory, that the (2+1)-dimensional WV model, similar to the Family model, also belongs to the Edwards–Wilkinson universality class.

Edwards–Wilkinson fluctuations in the Howitt–Warren flows

We study current fluctuations in a one-dimensional interacting particle system known as the dual smoothing process that is dual to random motions in a Howitt–Warren flow. The Howitt–Warren flow can be regarded as the transition kernels of a random motion in a continuous space–time random environment. It turns out that the current fluctuations of the dual smoothing process fall in the Edwards–Wilkinson universality class, where the fluctuations occur on the scale t1/4 and the limit is a universal Gaussian process. Along the way, we prove a quenched invariance principle for a random motion in the Howitt–Warren flow. Meanwhile, the centered quenched mean process of the random motion also converges on the scale t1/4, where the limit is another universal Gaussian process.

A bifractal nature of reticular patterns induced by oxygen plasma on polymer films.

Plasma etching was demonstrated to be a promising tool for generating self-organized nano-patterns on various commercial films. Unfortunately, dynamic scaling approach toward fundamental understanding of the formation and growth of the plasma-induced nano-structure has not always been straightforward. The temporal evolution of self-aligned nano-patterns may often evolve with an additional scale-invariance, which leads to breakdown of the well-established dynamic scaling law. The concept of a bifractal interface is successfully applied to reticular patterns induced by oxygen plasma on the surface of polymer films. The reticular pattern, composed of nano-size self-aligned protuberances and underlying structure, develops two types of anomalous dynamic scaling characterized by super-roughening and intrinsic anomalous scaling, respectively. The diffusion and aggregation of short-cleaved chains under the plasma environment are responsible for the regular distribution of the nano-size protuberances. Remarkably, it is uncovered that the dynamic roughening of the underlying structure is governed by a relaxation mechanism described by the Edwards-Wilkinson universality class with a conservative noise. The evidence for the basic phase, characterized by the negative roughness and growth exponents, has been elusive since its first theoretical consideration more than two decades ago.

Nonuniversal effects in mixing correlated-growth processes with randomness: interplay between bulk morphology and surface roughening.

To construct continuum stochastic growth equations for competitive nonequilibrium surface-growth processes of the type RD+X that mixes random deposition (RD) with a correlated-growth process X, we use a simplex decomposition of the height field. A distinction between growth processes X that do and do not create voids in the bulk leads to the definition of the effective probability p(eff) of the process X that is a measurable property of the bulk morphology and depends on the activation probability p of X in the competitive process RD+X. The bulk morphology is reflected in the surface roughening via nonuniversal prefactors in the universal scaling of the surface width that scales in p(eff). The equation and the resulting scaling are derived for X in either a Kardar-Parisi-Zhang or Edwards-Wilkinson universality class in (1+1) dimensions and are illustrated by an example of X being a ballistic deposition. We obtain full data collapse on its corresponding universal scaling function for all p∈(0;1]. We outline the generalizations to (1+n) dimensions and to many-component competitive growth processes.

Fluctuating Interfaces Subject to Stochastic Resetting

We study one-dimensional fluctuating interfaces of length L, where the interface stochastically resets to a fixed initial profile at a constant rate r. For finite r in the limit L→∞, the system settles into a nonequilibrium stationary state with non-Gaussian interface fluctuations, which we characterize analytically for the Kardar-Parisi-Zhang and Edwards-Wilkinson universality class. Our results are corroborated by numerical simulations. We also discuss the generality of our results for a fluctuating interface in a generic universality class.

Coarse-grained approach for universality classification of discrete models

Discrete and continuous models belonging to a universality class share the same linearities and (or) nonlinearities. In this work, we propose a new approach to calculate coarse-grained coefficients of the continuous differential equation from discrete models. We apply small constant translations in a test space and show how to obtain these coefficients from the transformed average interface growth velocity. Using the examples of the ballistic deposition (BD) model and the restricted solid-on-solid (RSOS) model, both belonging to the Kardar–Parisi–Zhang (KPZ) universality class, we demonstrate how to apply our approach to calculate analytically the corresponding coefficients of the KPZ equation. Our analytical nonlinear coefficients are in agreement with numerical results obtained by Monte Carlo tilted simulations. In addition to the BD and the RSOS, we study a competitive RSOS model that shows crossover between the KPZ and Edwards–Wilkinson universality classes.

Nonequilibrium Cluster Diffusion During Growth and Evaporation in Two Dimensions

The diffusion of growing or evaporating two-dimensional clusters is investigated. At equilibrium, it is well known that the mean square displacement (MSD) of the cluster center of mass is linear in time. In nonequilibrium conditions, we find that the MSD exhibits a nonlinear time dependence, leading to three regimes: (i)during curvature-driven evaporation, the MSD shows a square-root singularity close to the collapse time; (ii)in slow growth or evaporation, the dynamics is in the Edwards-Wilkinson universality class, and the MSD shows a logarithmic behavior; (iii)far from equilibrium, the dynamics belongs to the Kardar-Parisi-Zhang universality class and the MSD shows a power-law behavior with a characteristic exponent 1/3. These results agree with kinetic MonteCarlo simulations, and can be generalized to other universality classes.

Current fluctuations for stochastic particle systems with drift in one spatial dimension

This review article discusses limit distributions and vari- ance bounds for particle current in several dynamical stochastic systems of particles on the one-dimensional integer lattice: independent particles, in- dependent particles in a random environment, the random average process, the asymmetric simple exclusion process, and a class of totally asymmetric zero range processes. The first three models possess linear macroscopic flux functions and lie in the Edwards-Wilkinson universality class with scaling exponent 1=4 for current fluctuations. For these we prove Gaussian limits for the current process. The latter two systems belong to the Kardar-Parisi- Zhang class. For these we prove the scaling exponent 1=3 in the form of upper and lower variance bounds.

Spectrum of a multi-species asymmetric simple exclusion process on a ring

The spectrum of the Hamiltonian (Markov matrix) of a multi-species asymmetric simple exclusion process on a ring is studied. The dynamical exponent concerning the relaxation time is found to coincide with the one-species case. It implies that the system belongs to the Kardar–Parisi–Zhang or Edwards–Wilkinson universality classes depending on whether the hopping rate is asymmetric or symmetric, respectively. Our derivation exploits a poset structure of the particle sectors, leading to a new spectral duality and inclusion relations. The Bethe ansatz integrability is also demonstrated.