Universality class of fluctuating pulled fronts

It has recently been proposed that fluctuating "pulled" fronts propagating into an unstable state should not be in the standard Kardar-Parisi-Zhang (KPZ) universality class for rough interface growth. We introduce an effective field equation for this class of problems, and show on the basis of it that noisy pulled fronts in d+1 bulk dimensions should be in the universality class of the ((d+1)+1)D KPZ equation rather than of the (d+1)D KPZ equation. Our scenario ties together a number of heretofore unexplained observations in the literature, and is supported by previous numerical results.     

Determination of cross over effects in lattice models from the local height difference distribution

Growth of interfaces during vapor deposition is analyzed on a discrete lattice. It leads to finding distribution of local heights, measurable for any lattice model. Invariance in the change of this distribution in time is used to determine the cross over effects in various models. The analysis is applied to the discrete linear growth equation and Kardar-Parisi-Zhang (KPZ) equation. A new model is devised that shows early convergence to the KPZ dynamics. Various known conservative and non conservative models are tested on a one dimensional substrate by comparing the growth results with the exact KPZ and linear growth equation results. The comparison helps in establishing the condition that determines the presence of cross over effect for the given model. The new model is used in (2+1) dimensions to predict close to the true value of roughness constant for KPZ equation.     

Fluctuation and relaxation properties of pulled fronts: A scenario for nonstandard kardar-parisi-zhang scaling

We argue that while fluctuating fronts propagating into an unstable state should be in the standard Kardar-Parisi-Zhang (KPZ) universality class when they are pushed, they should not when they are pulled: The 1/t velocity relaxation of deterministic pulled fronts makes it unlikely that the KPZ equation is their proper effective long-wavelength low-frequency theory. Simulations in 2D confirm the proposed scenario, and yield exponents beta approximately 0.29+/-0.01, zeta approximately 0.40+/-0.02 for fluctuating pulled fronts, instead of the (1+1)D KPZ values beta = 1/3, zeta = 1/2. Our value of beta is consistent with an earlier result of Riordan et al., and with a recent conjecture that the exponents are the (2+1)D KPZ values.     

The Most Effective Model for Describing the Universal Behavior of a Noisy Kuramoto–Sivashinsky Equation as a Paradigmatic Model

We study a noisy Kuramoto–Sivashinsky (KS) equation which describes unstable surface growth and chemical turbulence. It has been conjectured that the universal long-wavelength behavior of the equation, which is characterized by scale-dependent parameters, is described by a Kardar–Parisi–Zhang (KPZ) equation. We consider this conjecture by analyzing a renormalization-group equation for a class of generalized KPZ equations. We then uniquely determine the parameter values of the KPZ equation that most effectively describes the universal long-wavelength behavior of the noisy KS equation.     

Structural complexity of disordered surfaces: Analyzing the porous silicon SFM patterns

This paper introduces a relative structural complexity measure for the characterization of disordered surfaces. Numerical solutions of 2d+1 KPZ equation and scanning force microscopy (SFM) patterns of porous silicon samples are analyzed using this methodology. The results and phenomenological interpretation indicate that the proposed measure is efficient for quantitatively characterize the structural complexity of disordered surfaces (and interfaces) observed and/or simulated in nano, micro and ordinary scales.     

Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence

We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1 dimensions [Takeuchi and Sano in Phys. Rev. Lett. 104:230601, 2010; Takeuchi et al. in Sci. Rep. 1:34, 2011]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.     

Coupled Kardar-Parisi-Zhang Equations in One Dimension

Over the past years our understanding of the scaling properties of the solutions to the one-dimensional KPZ equation has advanced considerably, both theoretically and experimentally. In our contribution we export these insights to the case of coupled KPZ equations in one dimension. We establish equivalence with nonlinear fluctuating hydrodynamics for multi-component driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the two-component AHR model. Its steady state is computed using the matrix product ansatz. Thereby all coefficients appearing in the coupled KPZ equations are deduced from the microscopic model. Time correlations in the steady state are simulated and we confirm not only the scaling exponent, but also the scaling function and the non-universal coefficients.     

Zero Tension Kardar-Parisi-Zhang Equation in (d + 1)–Dimensions

The joint probability distribution function (PDF) of the height and its gradients is derived for a zero tension d + 1-dimensional Kardar-Parisi-Zhang (KPZ) equation. It is proved that the height's PDF of zero tension KPZ equation shows lack of positivity after a finite time t _c . The properties of zero tension KPZ equation and its differences with the case that it possess an infinitesimal surface tension is discussed. Also potential relation between the time scale t _c and the singularity time scale t _c.v→0 of the KPZ equation with an infinitesimal surface tension is investigated.     

Matrix generalizations of some dynamic field theories

We introduce matrix generalizations of the Navier-Stokes (NS) equation for fluid flow, and the Kardar-Parisi-Zhang (KPZ) equation for interface growth. The underlying field, velocity for the NS equation, or the height in the case of KPZ, is promoted to a matrix that transforms as the adjoint representation of SU (N). Perturbative expansions simplify in the N → ∞ limit, dominated by planar graphs. We provide the results of a one-loop analysis, but have not succeeded in finding the full solution of the theory in this limit.     

Effects of memory on scaling behaviour of Kardar--Parisi--Zhang equation

In order to describe the time delay in the surface roughing process the Kardar-Parisis-Zhang (KPZ) equation with memory effects is constructed and analysed using the dynamic renormalization group and the power counting mode coupling approach by Chattopadhyay [2009 Phys. Rev. E 80 011144]. In this paper, the scaling analysis and the classical self-consistent mode-coupling approximation are utilized to investigate the dynamic scaling behaviour of the KPZ equation with memory effects. The values of the scaling exponents depending on the memory parameter are calculated for the substrate dimensions being 1 and 2, respectively. The more detailed relationship between the scaling exponent and memory parameter reveals the significant influence of memory effects on the scaling properties of the KPZ equation.     

Small-scale properties of the KPZ equation and dynamical symmetry breaking

A functional integral technique is used to study the ultraviolet or short distance properties of the Kardar–Parisi–Zhang (KPZ) equation with white Gaussian noise. We apply this technique to calculate the one-loop effective potential for the KPZ equation. The effective potential is (at least) one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, but non-renormalizable in 4 or higher space dimensions. This potential is intimately related to the probability distribution function (PDF) for the spacetime averaged field. For the restricted class of field configurations considered here, the KPZ equation exhibits dynamical symmetry breaking (DSB) via an analog of the Coleman–Weinberg mechanism in 1 and 2 space dimensions, but not in 3 space dimensions.     

Effective potential for the massless KPZ equation

In previous work we have developed a general method for casting a classical field theory subject to Gaussian noise (that is, a stochastic partial differential equation (SPDE)) into a functional integral formalism that exhibits many of the properties more commonly associated with quantum field theories (QFTs). In particular, we demonstrated how to derive the one-loop effective potential. In this paper we apply the formalism to a specific field theory of considerable interest, the massless KPZ equation (massless noisy Burgers equation), and analyze its behavior in the ultraviolet (short-distance) regime. When this field theory is subject to white noise we can calculate the one-loop effective potential and show that it is one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, and fails to be ultraviolet renormalizable in higher dimensions. We show that the one-loop effective potential for the massless KPZ equation is closely related to that for λφ⁴ QFT. In particular, we prove that the massless KPZ equation exhibits one-loop dynamical symmetry breaking (via an analog of the Coleman–Weinberg mechanism) in 1 and 2 space dimensions, and that this behavior does not persist in 3 space dimensions.     

1+1维含噪声Kuramoto-Sivashinsky方程表面宽度分布率的数值计算

通过对1+1维含噪声Kuramoto-Sivashinsky(KS)方程进行数值计算,得到其在饱和状态下的表面宽度分布率并与Kardar-Parisi-Zhang(KPZ)方程进行比较.结果表明,1+1维含噪声KS方程的表面宽度分布率标度函数受有限尺寸效应影响较小,并与KPZ方程具有相近的表面宽度分布率标度函数.     

On the renormalization of the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation of nonlinear stochastic growth in d dimensions is studied using the mapping onto a system of directed polymers in a quenched random medium. The polymer problem is renormalized exactly in a minimally subtracted perturbation expansion about d = 2. For the KPZ roughening transition in dimensions d > 2, this renormalization group yields the dynamic exponent z = 2 and the roughness exponent χ = 0, which are exact to all orders in ε ≡ (2 − d)/2. The expansion becomes singular in d = 4. If this singularity persists in the strong-coupling phase, it indicates that d = 4 is the upper critical dimension of the KPZ equation. Further implications of this perturbation theory for the strong-coupling phase are discussed. In particular, it is shown that the correlation functions and the coupling constant defined in minimal subtraction develop an essential singularity at the strong-coupling fixed point.     

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     

The role of pinning and instability in a class of non-equilibrium growth models

We study the dynamics of a growing crystalline facet where the growth mechanism is controlled by the geometry of the local curvature. A continuum model, in (2+1) dimensions, is developed in analogy with the Kardar-Parisi-Zhang (KPZ) model is considered for the purpose. Following standard coarse graining procedures, it is shown that in the large time, long distance limit, the continuum model predicts a curvature independent KPZ phase, thereby suppressing all explicit effects of curvature and local pinning in the system, in the “perturbative” limit. A direct numerical integration of this growth equation, in 1+1 dimensions, supports this observation below a critical parametric range, above which generic instabilities, in the form of isolated pillared structures lead to deviations from standard scaling behaviour. Possibilities of controlling this instability by introducing statistically “irrelevant" (in the sense of renormalisation groups) higher ordered nonlinearities have also been discussed. Received 23 April 2002 / Received in final form 24 July 2002 Published online 31 October 2002 RID="a" ID="a"e-mail: akc@mpipks-dresden.mpg.de     

Phase Separation Driven by a Fluctuating Two-Dimensional Self-Affine Potential Field

We study phase separation in a system of hard-core particles driven by a fluctuating two-dimensional self-affine potential landscape which evolves through Kardar–Parisi–Zhang (KPZ) dynamics. We find that particles tend to cluster together on a length scale which grows in time. The final phase-separated steady state is characterized by an unusual cusp singularity in the scaled correlation function and a broad distribution for the order parameter. Unlike the one-dimensional case studied earlier, the cluster-size distribution is asymmetric between particles and holes, reflecting the broken reflection symmetry of the KPZ dynamics, and has a contribution from an infinite cluster in addition to a power law part. A study of the surface in terms of coarse-grained depth variables helps understand many of these features.     

Stationary Correlations for the 1D KPZ Equation

We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the initial condition the two-sided Brownian motion (BM) with respect to the space variable. Developing techniques for dealing with this initial condition in the replica analysis, we elucidate some exact nature of the height fluctuation for the KPZ equation. In particular, we obtain an explicit representation of the probability distribution of the height in terms of the Fredholm determinants. Furthermore from this expression, we also get the exact expression of the space-time two-point correlation function.     

Dynamics of rough interfaces in chemical vapor deposition: experiments and a model for silica films

We study the surface dynamics of silica films grown by low pressure chemical vapor deposition. Atomic force microscopy measurements show that the surface reaches a scale invariant stationary state compatible with the Kardar-Parisi-Zhang (KPZ) equation in three dimensions. At intermediate times the surface undergoes an unstable transient due to shadowing effects. By varying growth conditions and using spectroscopic techniques, we determine the physical origin of KPZ scaling to be a low value of the surface sticking probability, related to the surface concentration of reactive groups. We propose a stochastic equation that describes the qualitative behavior of our experimental system.     

Singularity Time Scale of the Kardar–Parisi–Zhang Equation in 2+1 Dimensions

A master equation for the Kardar–Parisi–Zhang (KPZ) equation in 2+1 dimensions is developed. In the fully nonlinear regime we determine the finite time scale of the singularity formation in terms of the characteristics of forcing. The exact probability density function of the one point height field is obtained correspondingly.