Anomalous dynamic scaling of the non-local growth equations

The anomalous dynamic scaling behavior of the d+1 dimensional non-local growth equations is investigated based on the scaling approach. The growth equations studied include the non-local Kardar-Parisi-Zhang (NKPZ), non-local Sun-Guo-Grant (NSGG), and non-local Lai-Das Sarma-Villain (NLDV) equations. The anomalous scaling exponents in both the weak- and strong-coupling regions are obtained, respectively. Our results show that non-local interactions can affect anomalous scaling properties of surface fluctuations.     

Evidence for the anomalous scaling behaviour of the molecular-beam epitaxy growth equation

According to the scaling idea of local slope, we investigate numerically and analytically anomalous dynamic scaling behaviour of (1+1)-dimensional growth equation for molecular-beam epitaxy. The growth model includes the linear molecular-beam epitaxy (LMBE) and the nonlinear Lai--Das Sarma--Villain (LDV) equations. The anomalous scaling exponents in both the LMBE and the LDV equations are obtained, respectively. Numerical results are consistent with the corresponding analytical predictions.     

含时空关联噪声生长方程的标度奇异性分析

基于动力学重整化群理论研究表面界面生长动力学标度奇异性问题, 得到含时空关联噪声的表面生长方程标度奇异指数的一般结果,并将此方法应用于几种典型的局域生长方程——Kardar-Parisi-Zhang（KPZ）方程、线性生长方程、Lai-Das Sarma-Villain（LDV）方程.结果表明,在长波长极限下局域生长方程的动力学标度奇异性与最相关项、基底维数以及噪声有关,并且若出现标度奇异性,只会是超粗化（super rough）奇异标度行为,而不是内禀（intrinsically）奇异标度行为. 关键词： 标度奇异性 动力学重整化群理论 时空关联噪声     

含有广义守恒律的生长方程标度奇异性的直接标度分析

利用直接标度分析方法研究一个含有广义守恒律生长方程的标度奇异性,得到强弱耦合区域的奇异标度指数.作为其特殊情况,这个方程包含Kardar-Parisi-Zhang（KPZ）方程、 Sun-Guo-Grant（SGG）方程以及分子束外延（MBE）生长方程,并能对其进行统一的研究.研究发现, KPZ方程和SGG方程,无论在弱耦合还是在强耦合区域内都遵从自仿射Family -Vicsek正常标度规律;而MBE 方程在弱耦合区域内服从正常标度,在强耦合区域内能呈现内禀奇异标度行为.这里所得到生长方程的奇异标度性质与利用重正化群理论、数值模拟以及实验相符很好. 关键词： 标度奇异性 强耦合 弱耦合     

Generic dynamic scaling in kinetic roughening

We study the dynamic scaling hypothesis in invariant surface growth. We show that the existence of power-law scaling of the correlation functions (scale invariance) does not determine a unique dynamic scaling form of the correlation functions, which leads to the different anomalous forms of scaling recently observed in growth models. We derive all the existing forms of anomalous dynamic scaling from a new generic scaling ansatz. The different scaling forms are subclasses of this generic scaling ansatz associated with bounds on the roughness exponent values. The existence of a new class of anomalous dynamic scaling is predicted and compared with simulations.     

Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

Abstract

We analyze asymptotic scaling properties of a model class of anomalous reaction-diffusion (ARD) equations. Numerical experiments show that solutions to these have, for large t, well defined scaling properties. We suggest a general framework to analyze asymptotic symmetry properties; this provides an analytical explanation of the observed asymptotic scaling properties for the considered ARD equations.     

Anomalous scaling exponents in nonlinear models of turbulence

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.     

Interplay between carrier and impurity concentrations in annealed Ga1-xMnxAs: intrinsic anomalous hall effect

Investigating the scaling behavior of annealed Ga1-xMnxAs anomalous Hall coefficients, we note a universal crossover regime where the scaling behavior changes from quadratic to linear. Furthermore, measured anomalous Hall conductivities in the quadratic regime when properly scaled by carrier concentration remain constant, spanning nearly a decade in conductivity as well as over 100 K in T_[C] and comparing favorably to theoretically predicated values for the intrinsic origins of the anomalous Hall effect. Both qualitative and quantitative agreements strongly point to the validity of new equations of motion including the Berry phase contributions as well as the tunability of the anomalous Hall effect.     

Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multiscaling

We develop a consistent closure procedure for the calculation of the scaling exponents ζ _n of the nth-order correlation functions in fully developed hydro-dynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ _n . This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ _n . Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.     

Influence of Velocity Shell Correlations on Anomalous Scaling Exponents of Passive Scalars in Shell Models

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of passive scalars of turbulence. Different to the original problem, the distribution function of the prescribed random velocity field is multi-dimensional normal and delta-correlated in time. Here, our random velocity field is spatially correlative. For comparison, we also give the result obtained by the Gaussian random velocity field without spatial correlation. The anomalous scaling exponents H(p) of passive scalar advected by two kinds of random velocity above are determined for structure function up to p=15 by numerical simulations of the random shell model with Runge-Kutta methods to solve the stochastic differential equations. We observed that the H(p)'s obtained by the multi-dimensional normal distribution random velocity are more anomalous than those obtained by the independent Gaussian random velocity.     

Scaling Exponents for Active Scalars

We provide bounds for Dirichlet quotients and for generalized structure functions for 3D active scalars and Navier–Stokes equations. These bounds put constraints on the possible extent of anomalous scaling.     

Reduced Two-Field Equations and Anomalous Transport Scaling in Low-Temperature Plasmas

After a review of more or less systematic approaches for normal as well as anomalous particle transport phenomena across a magnetic field, simplified macroscopic models are discussed. A reduced two-field model for collision-dominated low-temperature plasmas is presented. Although it looks similar to its high-temperature analogue (known as the Hasegawa-Wakatani equations), it has significant peculiarities which are discussed in detail. Some basic results following from the reduced two-field model, e.g. anomalous transport scaling close to the onset of collisional driftwave instability, are discussed.     

Anomalous scaling in a non-Gaussian random shell model for passive scalars

In this paper, we have introduced a shell-model of Kraichnan's passive scalar problem. Different from the original problem, the prescribed random velocity field is non-Gaussian and $\delta$ correlated in time, and its introduction is inspired by She and L\'{e}v\^{e}que (Phys. Rev. Lett. {\bf 72}, 336 (1994)). For comparison, we also give the passive scalar advected by the Gaussian random velocity field. The anomalous scaling exponents $H(p)$ of passive scalar advected by these two kinds of random velocities above are determined for structure function with values of $p$ up to 15 by Monte Carlo simulations of the random shell model, with Gear methods used to solve the stochastic differential equations. We find that the $H(p)$ advected by the non-Gaussian random velocity is not more anomalous than that advected by the Gaussian random velocity. Whether the advecting velocity is non-Gaussian or Gaussian, similar scaling exponents of passive scalar are obtained with the same molecular diffusivity.     

Anomalous roughening of Hele-Shaw flows with quenched disorder

The kinetic roughening of a stable oil-air interface moving in a Hele-Shaw cell that contains a quenched columnar disorder (tracks) has been studied. A capillary effect is responsible for the dynamic evolution of the resulting rough interface, which exhibits anomalous scaling. The three independent exponents needed to characterize the anomalous scaling are determined experimentally. The anomalous scaling is explained in terms of the initial acceleration and subsequent deceleration of the interface tips in the tracks coupled by mass conservation. A phenomenological model that reproduces the measured global and local exponents is introduced.     

On the Weak Solutions to the Maxwell–Landau–Lifshitz Equations and to the Hall–Magneto–Hydrodynamic Equations

In this paper we deal with weak solutions to the Maxwell–Landau–Lifshitz equations and to the Hall–Magneto–Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall–Magneto–Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.     

Growth dynamics of ZnPc and TiOPc thin films: Effect of crystallinity on anomalous scaling behavior

The surface morphology evolution and scaling behavior of zinc phthalocyanine (ZnPc) and titanyl phthalocyanine (TiOPc) thin films have been studied using atomic force microscopy, X-ray diffraction and height difference correlation function analysis. In contrast to the large growth exponent (β) values and anomalous scaling behavior previously reported for other crystalline molecular thin films, significantly small β and anomaly values were observed for amorphous TiOPc thin films. The relatively small anomaly value of ZnPc thin films, though larger than that of TiOPc thin films, is also rationalized by the lack of crystallographic ordering at the initial stage of growth.     

Scaling behavior of the time-fractional equations for molecular-beam epitaxy growth: scaling analysis versus numerical stimulations

The scaling behavior of the time-fractional molecular-beam epitaxy (TFMBE) equations in 1+1 dimensions is investigated by numerical simulations and scaling analysis, respectively. The growth equations studied include the time-fractional linear molecular-beam epitaxy (TFLMBE) and the time-fractional Lai-Das Sarma-Villain (TFLDV). Growth exponents are obtained using the two methods. The analytical results are consistent with the corresponding numerical solutions based on Caputo-type fractional derivative.     

Anomalous dynamics in the Ising chain

We analyze a 1D Ising system with anomalous distributions of nearest neighbor interactions and show that the single-spin-flip dynamics exhibit breakdown of dynamic scaling. The results are obtained by a real-space numerical method applied to the exact equations of motion and they may be explained by domain wall motion arguments reformulated in terms of extreme value statistics.     

Anomalous roughness in dimer-type surface growth

We point out how geometric features affect the scaling properties of nonequilibrium dynamic processes, by a model for surface growth where particles can deposit and evaporate only in dimer form, but dissociate on the surface. Pinning valleys (hilltops) develop spontaneously and the surface facets for all growth (evaporation) biases. More intriguingly, the scaling properties of the rough one dimensional equilibrium surface are anomalous. Its width, W approximately Lalpha, diverges with system size L as alpha = 1 / 3 instead of the conventional universal value alpha = 1 / 2. This originates from a topological nonlocal evenness constraint on the surface configurations.     

1+1维Wolf-Villain模型奇异标度行为的数值模拟

采用Kinetic Monte Carlo(KMC)方法对描述分子束外延生长(MBE)的1+1维Wolf-Villain模型进行大尺寸和长生长时间的数值模拟研究,以消除渡越行为的影响.计算得到整体和局域标度指数.结果显示,在所模拟的空间和时间尺度范围内,1+1维Wolf-Villain模型仍呈现出固有奇异标度行为.这一结论与López等人最近的理论分析结果不一致.