Anomalous Scaling of Surface Growth Equations with Spatially and Temporally Correlated Noise

Based on the scaling idea of local slopes by López et al. [Phys. Rev. Lett. 94 (2005) 166103], we investigate anomalous dynamic scaling of (d+1)-dimensional surface growth equations with spatially and temporally correlated noise. The growth equations studied include the Kardar-Parisi-Zhang (KPZ), Sun-Guo-Grant (SGG), and Lai-Das Sarma-Villain (LDV) equations. The anomalous scaling exponents in both the weak- and strong-coupling regions are obtained, respectively.     

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     

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.     

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     

Anomalous scaling in epitaxial growth on vicinal surfaces: meandering and mounding instabilities in a linear growth equation with spatiotemporally correlated noise

We have undertaken an extensive analytical and kinetic Monte Carlo study of the (2+1) dimensional discrete growth model on a vicinal surface. A non-local, phenomenological continuum equation describing surface growth in unstable systems with anomalous scaling is presented. The roughness produced by unstable growth is first studied considering various effects in surface diffusion processes (corresponding to temperature, flux, diffusion anisotropy). We found that the thermally activated roughness is well-described by a generalized Lai–Das Sarma–Villain model with non linear growth continuum equation and uncorrelated noise. The corresponding critical exponents are computed analytically for the first time and show a continuous variation in agreement with simulation results of a solid-on-solid model. However, the roughness related to the meandering instability is found, unexpectedly, to be well described by a linear continuum equation with spatiotemporally correlated noise.     

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2.The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.     

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.     

Application of an electrostatically actuated cantilevered carbon nanotube with an attached mass as a bio-mass sensor

In the present paper, another latent capability of SWCNT as a mass sensor is investigated. The relationship between the resonant frequency, dynamic pull-in voltage at the resonance frequency shift, and the attached mass is established by using the nonlocal Euler–Bernoulli beam theory. Using this relationship, a general closed-form nonlinear sensor-equation has been derived for the detection of the mass attached to the SWCNT. The aim of this study and present model is to show the sensitivity of the Cantilevered SWCNT to the values and positions of attached mass. Moreover, the results indicate that by increasing the value of attached mass and considering a single non-local scaling parameter (e₀), the values of dynamic pull-in voltage at the resonance frequency shift are decreased. Because of the small scaling parameter (e₀), the mass sensitivity of carbon nanotube increases, when the position of the attached mass is in the tip of a Cantilevered SWCNT length. The authority and the accuracy of these formulas are examined with other pull-in sensor equations in literatures. The results demonstrate that the new sensor equation can be applied for CNT-based mass sensors with rational accuracy.     

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.     

Exclusive processes in quantum chromodynamics: Evolution equations for hadronic wavefunctions and the form factors of mesons

The predictions of quantum chromodynamics for meson form factors at large momentum transfer are given. Evolution equations are derived which determine the structure of hadronic wavefunctions at short distances from their form at large distances. The eigenvalues of the evolution equations appear as exponents in anomalous logarithm corrections to the nominal power law of form factors determined by dimensional counting. The results lead to detailed tests of the spin and scaling structure of QCD at short distances. The predictions for the charged pion, kaon and rho form factors and the γ → π⁰ transition form factor of the photon are absolutely normalized at asymptotic momentum transfer.     

Computing the non-linear anomalous diffusion equation from first principles

We investigate asymptotically the occurrence of anomalous diffusion and its associated family of statistical evolution equations. Starting from a non-Markovian process à la Langevin we show that the mean probability distribution of the displacement of a particle follows a generalized non-linear Fokker-Planck equation. Thus we show that the anomalous behavior can be linked to a fast fluctuation process with memory from a microscopic dynamics level, and slow fluctuations of the dissipative variable. The general results can be applied to a wide range of physical systems that present a departure from the Brownian regime.     

Perturbative fragmentation of a vector color particle into bound states involving a heavy antiquark

Vector-particle fragmentation into possible S-wave bound states involving a heavy antiquark is considered for high-energy processes at high transverse momenta, and the relevant fragmentation function is calculated in the leading order of perturbative QCD for various patterns of the anomalous magnetic moment. One-loop equations describing the q ² evolution of the fragmentation-function moments that is caused by hard-gluon emission from the vector particle are derived. The integrated probabilities of fragmentation are obtained. The distribution of the bound state in the transverse momentum defined with respect to the fragmentation axis is calculated in the scaling limit.     

Dynamical scaling and kinetic roughening of single valued fronts propagating in fractal media

We consider the dynamical scaling and kinetic roughening of single-valued interfaces propagating in 2D fractal media. Assuming that the nearest-neighbor height difference distribution function of the fronts obeys Lévy statistics with a well-defined algebraic decay exponent, we consider the generalized scaling forms and derive analytic expressions for the local scaling exponents. We show that the kinetic roughening of the interfaces displays intrinsic anomalous scaling and multiscaling in the relevant correlation functions. We test the predictions of the scaling theory with a variety of well-known models which produce fractal growth structures. Results are in excellent agreement with theory. For some models, we find interesting crossover behavior related to large-scale structural instabilities of the growing aggregates. Received 22 May 2002 Published online 19 November 2002     

The importance of non-local shadowing for the topography evolution of a-C:H films grown by toluene based plasma enhanced chemical vapor deposition

The topography evolution of hydrogenated diamond-like carbon coatings deposited through toluene based capacitively coupled plasma enhanced chemical vapor deposition has been studied experimentally and with continuum growth models. The experimentally observed mound formation and surprisingly large growth exponents (β≈ 0.9±0.1) cannot be reproduced by familiar local stochastic differential equations that are successfully used for other thin film deposition techniques. Here we introduce a novel numerical approach to simulate a continuum growth model that takes into account non-local shadowing effects. We show that the major characteristics of the experimentally observed topography evolution can be accurately represented by this model.     

Active and passive scalar intermittent statistics in turbulent atmospheric convection

We study the small-scale statistics of active and passive scalar fields, obtained from 3D large-eddy simulations of the atmospheric boundary layer turbulence. The velocity field is anisotropic and inhomogeneous, due to the action of both buoyancy and shear. We focus on scalar field rare fluctuations dominated by the so-called fronts. Temperature, coupled to the velocity field by the Boussinesq equations, exhibits anomalous scaling and saturation of the scaling exponents to a constant value, due to the presence of thermal fronts. Although qualitatively similar, the small-scale statistics of a passive tracer advected by the convective flow shows quantitative differences: the large fluctuations of the tracer concentration field distribute differently and appear to be less intermittent than the temperature ones. To better understand these results, the role of boundaries in this problem is discussed.     

Non-local continuity equations for self-dual SU(N) Yang-Mills fields

Using a manifestly gauge invariant formulation, an infinite set of classical non-local continuity equations is constructed for the self-dual SU(N) Yang-Mills fields.