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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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     

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.     

Superdiffusivity of the 1D Lattice Kardar-Parisi-Zhang Equation

The continuum Kardar-Parisi-Zhang equation in one dimension is lattice discretized in such a way that the drift part is divergence free. This allows to determine explicitly the stationary measures. We map the lattice KPZ equation to a bosonic field theory which has a cubic anti-hermitian nonlinearity. Thereby it is established that the stationary two-point function spreads superdiffusively.     

DYNAMIC SCALING OF GROWING SURFACES WITH GROWTH INHOMOGENEITIES OF SCREENED COULOMBIC FUNCTION

The dynamic scaling properties of growing surfaces with growth inhomogeneities are studied by applying a dynamic renormalization-group analysis to the generalized Kardar-Parisi-Zhang(hereafter abbreviated to KPZ) equation, which contains an additional term of growth inhomogeneities. In a practical crystal growth process, these growth inhomogeneities can be induced by surface impurities and defects and are modeled by a screened Coulomb function in this paper. Our results show that the existence of the growth inhomogeneities can significantly change the dynamic scaling properties of a growing surface and can lead to a rougher surface.     

Renormalization of Generalized KPZ Equation

Journal of Statistical Physics - We use Renormalization Group to prove local well posedness for a generalized KPZ equation introduced by H. Spohn in the context of stochastic hydrodynamics. The...     

Surface tension driven flow on a thin reaction front

Surface tension driven convection affects the propagation of chemical reaction fronts in liquids. The changes in surface tension across the front generate this type of convection. The resulting fluid motion increases the speed and changes the shape of fronts as observed in the iodate-arsenous acid reaction. We calculate these effects using a thin front approximation, where the reaction front is modeled by an abrupt discontinuity between reacted and unreacted substances. We analyze the propagation of reaction fronts of small curvature. In this case the front propagation equation becomes the deterministic Kardar-Parisi-Zhang (KPZ) equation with the addition of fluid flow. These results are compared to calculations based on a set of reaction-diffusion-convection equations.     

A fluctuating lattice Boltzmann scheme for the one-dimensional KPZ equation

Based on the well-known mapping between the Burgers equation with noise and the Kardar–Parisi–Zhang (KPZ) equation for fluctuating interfaces, we develop a fluctuating lattice Boltzmann (LB) scheme for growth phenomena, as described by the KPZ formalism. A very simple LB-KPZ scheme is demonstrated in 1+1 spacetime dimensions, and is shown to reproduce the scaling exponents characterizing the growth of one-dimensional fluctuating interfaces.     

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.     

Recent results on multiplicative noise

Recent developments in the analysis of Langevin equations with multiplicative noise (MN) are reported. In particular, we (i) present numerical simulations in three dimensions showing that the MN equation exhibits, like the Kardar-Parisi-Zhang (KPZ) equation, both a weak coupling fixed point and a strong coupling phase, supporting the proposed relation between MN and KPZ; (ii) present a dimensional and mean-field analysis of the MN equation to compute critical exponents; (iii) show that the phenomenon of the noise-induced ordering transition associated with the MN equation appears only in the Stratonovich representation and not in the Ito one; and (iv) report the presence of a first-order-like phase transition at zero spatial coupling, supporting the fact that this is the minimum model for noise-induced ordering transitions.