The most effective model for describing the universal behavior of unstable surface growth

TL;DR

This paper investigates whether the universal long-wavelength behavior of a noisy Kuramoto-Sivashinsky equation, modeling unstable surface growth, can be effectively described by a Kardar-Parisi-Zhang equation through renormalization-group analysis.

Contribution

It uniquely identifies the KPZ parameters that best capture the universal behavior of the noisy KS equation, advancing understanding of surface growth models.

Findings

KPZ effectively describes the universal behavior of the noisy KS equation

Renormalization-group analysis determines optimal KPZ parameters

Supports conjecture linking KS and KPZ universality classes

Abstract

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.

Equations

$${\partial }_{t}h$$

$$⟨\eta (x,t)\eta (x^{{}^{\prime }},t^{{}^{\prime }})⟩$$

$$\mathcal{X}({\Lambda }^{\prime };\mathcal{X}({\Lambda }_0))=\mathcal{X}({\Lambda }^{\prime };\mathcal{X}(\Lambda ;\mathcal{X}({\Lambda }_0))).$$

$$-\Lambda \frac{d\mathcal{X}}{d\Lambda }={\Psi }_{\mathcal{X}}(\mathcal{X}),$$

$$Z[J,\tilde{J}]$$

$$\displaystyle+\int_{-\infty}^{\infty}d\omega\int_{-\Lambda_{0}}^{\Lambda_{0}}dk\biggl{(}J(k,\omega)h(-k,-\omega)+\tilde{J}(k,\omega)i\tilde{h}(-k,-\omega)\biggr{)}\biggr{]},$$

$$S[h,i\tilde{h};{\Lambda }_0]=$$

$$+\frac{{\lambda }_0}{2}{\int }_{-\infty }^{\infty }\frac{d{\omega }_{1}d{\omega }_2}{(2\pi {)}^2}{\int }_{-{\Lambda }_0}^{{\Lambda }_0}\frac{d{k}_{1}d{k}_2}{(2\pi {)}^2}{k}_1{k}_{2}i\tilde{h}(-k_1-k_2,-{\omega }_1-{\omega }_2)h(k_1,{\omega }_1)h(k_2,{\omega }_2),$$

$$G_0^{-1}(k,\omega )=\left(\begin{array}{cc}0& i\omega +{\nu }_0{k}^2+K_0{k}^4\\ -i\omega +{\nu }_0{k}^2+K_0{k}^4& -2(D_0+D_{d0}{k}^2)\end{array}\right).$$

$$A^{<}(k,\omega )$$

$$A^{>}(k,\omega )$$

$$Z[J^{<},{\tilde{J}}^{<}]=$$

$$\displaystyle+\int_{-\infty}^{\infty}d\omega\int_{-\Lambda}^{\Lambda}dk\biggl{(}J^{<}(k,\omega)h^{<}(-k,-\omega)+\tilde{J}^{<}(k,\omega)i\tilde{h}^{<}(-k,-\omega)\biggr{)}\biggr{]}.$$

$$\displaystyle\exp\biggl{[}-S[h^{<},i\tilde{h}^{<};\Lambda]\biggr{]}=\int\mathcal{D}[h^{>},i\tilde{h}^{>}]\exp\biggl{[}-S[h^{<}+h^{>},i\tilde{h}^{<}+i\tilde{h}^{>};\Lambda_{0}]\biggr{]}.$$

$$(G^{-1}{)}_{\tilde{h}h}(k_1,{\omega }_1;\Lambda )\delta ({\omega }_1+{\omega }_2)$$

$$(G^{-1}{)}_{\tilde{h}\tilde{h}}(k_1,{\omega }_1;\Lambda )\delta ({\omega }_1+{\omega }_2)$$

$${\Gamma }_{\tilde{h}hh}(k_1,{\omega }_1;k_2,{\omega }_2;\Lambda )$$

$$\equiv {\frac{{\delta }^{3}S[h^{<},i{\tilde{h}}^{<};\Lambda ]}{\delta (i{\tilde{h}}^{<}(k_1,{\omega }_1))\delta (h^{<}(k_2,{\omega }_2))\delta (h^{<}(k_3,{\omega }_3))}|}_{h^{<},i{\tilde{h}}^{<}=0}.$$

$$\nu (\Lambda )$$

$$K(\Lambda )$$

$$-2D(\Lambda )$$

$$-2D_d(\Lambda )$$

$$\lambda (\Lambda )$$

$$\lambda (\Lambda )={\lambda }_0.$$

$$(G^{-1}{)}_{\tilde{h}h}(k,\omega ;\Lambda )=$$

$$\displaystyle+\lambda_{0}^{2}\int^{\infty}_{-\infty}\frac{d\Omega}{2\pi}\int_{\Lambda\leq|q|\leq\Lambda_{0}}\frac{dq}{2\pi}\biggl{[}kq(k-q)^{2}(G_{0})_{\tilde{h}h}(q,\Omega)C_{0}(k-q,\omega-\Omega)$$

$$\displaystyle+kq^{2}(k-q)(G_{0})_{\tilde{h}h}(k-q,\omega-\Omega)C_{0}(q,\Omega)\biggr{]},$$

$$(G^{-1}{)}_{\tilde{h}\tilde{h}}(k,\omega ;\Lambda )=$$

$$-2{\lambda }_0^2{\int }_{-\infty }^{\infty }\frac{d\Omega }{2\pi }{\int }_{\Lambda \le |q|\le {\Lambda }_0}\frac{dq}{2\pi }{q}^2(k-q{)}^2{C}_0(q,\Omega )C_0(k-q,\omega -\Omega ),$$

$$C_0(k,\omega )\equiv 2(D_0+D_{d0}{k}^2)|(G_0{)}_{\tilde{h}h}(k,\omega ){|}^2.$$

$${\int }_{\Lambda \le |q|\le {\Lambda }_0}\frac{dq}{2\pi }$$

$$\ne {\int }_{\Lambda \le |q|\le {\Lambda }_0}\frac{dq}{2\pi }{q}^2(k-q)(G_0{)}_{\tilde{h}h}(k-q,\omega -\Omega )C_0(q,\Omega ).$$

$$-\Lambda \frac{d\nu (\Lambda )}{d\Lambda }$$

$$-\Lambda \frac{dK(\Lambda )}{d\Lambda }$$

$$-\Lambda \frac{dD(\Lambda )}{d\Lambda }$$

$$-\Lambda \frac{d{D}_d(\Lambda )}{d\Lambda }$$

$$F$$

$$G$$

$$H$$

$$\displaystyle-\Lambda\frac{dF}{d\Lambda}=2F+\frac{G}{2(1+F)^{5}}\biggl{[}$$

$$\displaystyle-\Lambda\frac{dG}{d\Lambda}=7G-\frac{G^{2}}{2(1+F)^{5}}\biggl{[}$$

$$\displaystyle-2(1+F)^{2}\frac{H^{2}}{G^{2}}\biggr{]},$$

Full text

1 Introduction

Eddy viscosity in turbulence, which can explain how a vortex pattern emerges in a non-uniform turbulent flow, depends on the observed length scales Frisch . As exemplified by the Richardson law Richardson , there are cases in which a parameter of a macroscopic description is not given as a definite value, but is rather expressed as a function of the length scale. Another example of scale-dependent parameters has been observed in one- or two- dimensional fluid dynamics, where the viscosity is not uniquely defined in the hydrodynamic description review . Here, it seems reasonable to expect that such scale-dependent parameters in a macroscopic description can be reproduced by an effective stochastic system PhysRevA.16.732 ; Dominicis ; Fournier-Frisch1978 ; Fournier-Frisch1983 ; Yakhot-Orszag ; Yakhot-Orszag2 ; Yakhot-Smith ; Eyink . In this paper, we attempt to determine the effective stochastic system theoretically when scale-dependent parameters are observed.

As the simplest example for scale-dependent parameters, we consider the one-dimensional Kardar-Parisi-Zhang (KPZ) equation PhysRevLett.56.889 . It is known that the effective surface tension $\nu(\Lambda)$ at a scale $2\pi/\Lambda$ for the equation is $\nu(\Lambda)=C_{\nu}\Lambda^{-1/2}$ in the limit $\Lambda\to 0$, which is similar to the Richardson law for turbulence. Recently, the KPZ equation was rigorously derived from a stochastic many-particle model Bertini-Giacomin ; PhysRevLett.104.230602 , and the so-called KPZ class has been extensively discussed both theoretically and experimentally Takeuchi-Sano2010 ; Takeuchi-Sano-Sasamoto-Spohn ; Amir-Corwin-Quastel ; Calabrese-Doussal ; Imamura-Sasamoto ; Hairer . However, in general, even if we find systems that may exhibit scale-dependent parameters similar to those for the KPZ class, a method to determine the parameter values of the corresponding KPZ equation has not yet been reported.

Specifically, let us consider a noisy Kuramoto-Sivashinsky (KS) equation, which exhibits spatially extended chaos in the noiseless limit kuramoto1976persistent ; sivashinsky1977nonlinear ; Kuramoto-book . The model describes turbulent chemical waves and unstable interface motions, which are caused by negative surface tension. It has been conjectured that a KPZ equation may be an effective model for describing the long-wavelength behavior of the noisy KS equation; this conjecture is referred to as the Yakhot conjecture PhysRevA.24.642 ; PhysRevLett.60.1840 . Indeed, direct numerical simulations showed that statistical properties of the long wave length modes are similar to those of the KPZ equations zaleski1989stochastic ; PhysRevA.46.R7351 ; PhysRevE.47.911 ; sakaguchi2002effective ; PhysRevE.71.046138 .

Here, one may recall the renormalization group (RG), which is a standard method for studying scale-dependent parameters. For a given noisy KS equation, the RG equation was calculated using a perturbation theory PhysRevE.71.046138 ; Cuerno-Lauritsen . The infrared fixed point of the RG equation determines the scale-dependent behavior $\nu(\Lambda)=C_{\nu}\Lambda^{-1/2}$ in the limit $\Lambda\to 0$, which has the same power-law form as that for the KPZ equations. Nevertheless, as shown below, the analysis at the infrared fixed point of the RG equation cannot determine the parameter values of the corresponding KPZ equation.

In this paper, we present a framework for studying the effective description. We study an RG equation for generalized KPZ equations that include noisy KS equations and KPZ equations. We then consider solution trajectories of the RG equation, in which each point flows to the infrared fixed point of the noisy KS equation we study. The solution trajectories also approach a subspace in the ultraviolet limit, which enables us to define a collection of bare parameters of the generalized KPZ equations. By using the lowest perturbation theory for the RG equation, we uniquely determine the most effective model among such KPZ equations that describes the infrared universal behavior of a noisy KS equation in the most efficient manner.

This paper is organized as follows. In Sect. 2, we introduce a class of models we study, define scale-dependent parameters for the models, and review RG equations for the parameters. We also discuss ultraviolet and infrared behaviors of solution trajectories for the RG equation, and classify universal and non-universal properties of those. After that, we give a definition of “the most effective model”. In Sect. 3, we simplify a representation of the trajectories so as to determine the most effective model. The solution trajectories for the RG equation are expressed as curves in a five-dimensional parameter space. Then, the trajectory for the noisy KS equation is attracted to a two-dimensional subspace, due to emergence of a time-reversal symmetry. We define the most effective model for the noisy KS equation in this subspace. In Sect. 4, we determine parameter values of the most effective model from its definition. In Sect. 5, we provide concluding remarks. We discuss renormalizability of the KPZ equation and its relevant parameters. We also remark an another application of our formalism to turbulence. In Appendix A, we derive Ward-Takahashi identities for scale-dependent parameters from symmetries of our model.

2 Setup

We study models for the stochastic growth of a surface. We assume that the time evolution of the height $h(x,t)$ of the surface is described by a generalized KPZ equation:

[TABLE]

where $\nu$ is the surface tension, $K$ is the surface diffusion constant, $\lambda$ is the strength of the non-linearity, and $\eta(x,t)$ is the white noise. Here, $D$ and $D_{d}$ are the strength of the noise. When $K=D_{d}=0$, (2) is the KPZ equation, while when $D=D_{d}=0$, (2) with $\nu<0$ is the deterministic KS equation. We refer to (2) with $\nu<0$ and $K,D,D_{d}>0$, as the noisy KS equation.

The five parameters in (2) are collectively denoted by $\mathcal{X}\equiv(\nu,K,D,D_{d},\lambda)$. More precisely, these parameters are defined for a field $h(x)$ whose Fourier transform $\hat{h}(k)$ is assumed to be zero for $|k|>\Lambda$. $\Lambda$ is called a cut-off wavenumber. We explicitly express the cutoff dependence of the parameters as $\mathcal{X}(\Lambda)$. Here, for a given model with $\mathcal{X}(\Lambda_{0})$, we define a model with $\mathcal{X}(\Lambda)$ for $\Lambda<\Lambda_{0}$ by eliminating the contribution $\Lambda\leq|k|\leq\Lambda_{0}$ in the dynamics, which may be formally expressed as $\mathcal{X}(\Lambda;\mathcal{X}(\Lambda_{0}))$. Note that we do not employ a rescaling transformation after the coarse-graining. This functional relation trivially satisfies

[TABLE]

From this, we obtain the RG equation

[TABLE]

which determines $\mathcal{X}(\Lambda;\mathcal{X}(\Lambda_{0}))$ under an initial condition $\mathcal{X}(\Lambda_{0})=\mathcal{X}_{0}$. In the next subsection, we review the RG equation for the generalized KPZ equations.

2.1 definition of scale-dependent parameters

We first define the scale-dependent parameters $\nu(\Lambda)$, $D(\Lambda)$, $K(\Lambda)$, $D_{d}(\Lambda)$ and $\lambda(\Lambda)$, and then introduce a perturbation theory leading to the equation for determining them.

We start with the generating functional $Z[J,\tilde{J}]$ by which all statistical quantities of the KPZ equations are determined. Following the Martin-Siggia-Rose-Janssen-deDominicis (MSRJD) formalism MSR ; J ; D1 ; D2 , $Z[J,\tilde{J}]$ is expressed as

[TABLE]

where $i\tilde{h}$ is the auxiliary field, $J$ and $\tilde{J}$ are source fields, and $S[h,i\tilde{h};\Lambda_{0}]$ is the MSRJD action for the generalized KPZ equation. Hereafter, we use the notation $A(k,\omega)$ for the Fourier transform of $A(x,t)$ for any field A. The action $S[h,i\tilde{h};\Lambda_{0}]$ is explicitly written as

[TABLE]

where $G_{0}^{-1}$ is the inverse matrix of the bare propagator

[TABLE]

Here, we consider a coarse-grained description at a cutoff $\Lambda<\Lambda_{0}$. Let us define

[TABLE]

for any quantity $A(k,\omega)$, where $\theta(x)$ is the Heaviside step function. The statistical quantities of $h^{<}$ are described by the generating functional $Z[J^{<},\tilde{J}^{<}]$ with replacement of $(J,\tilde{J})$ by $(J^{<},\tilde{J}^{<})$. We thus define the effective MSRJD action $S[h^{<},i\tilde{h}^{<};\Lambda]$ by the relation

[TABLE]

We can then confirm that $S[h^{<},i\tilde{h}^{<};\Lambda]$ is determined as

[TABLE]

Then, the propagator and the three point vertex function for the effective MSRJD action at $\Lambda$ are defined as

[TABLE]

From these quantities, we define the parameters as

[TABLE]

From a tilt symmetry of the generalized KPZ equation, we can obtain

[TABLE]

In Appendix A, we will provide a non-perturbative proof for (20) based on symmetry properties freyPRE1994 ; Canet2011 . Below, we derive a set of equations that determines $\nu(\Lambda)$, $D(\Lambda)$, $K(\Lambda)$, and $D_{d}(\Lambda)$.

2.2 renormlization group equations

We can calculate $(G^{-1})_{ij}(k,\omega;\Lambda)$ by using the perturbation theory in $\lambda_{0}$. At the second-order level, the propagators are calculated as

[TABLE]

where $C_{0}(k,\omega)$ is the bare correlation function defined by

[TABLE]

In the calculation of (21), one should carefully note the relation PhysRevE.71.046138

[TABLE]

We emphasize that the Feynman rule does not distinguish these.

By setting $(\Lambda-\Lambda_{0})/\Lambda_{0}\ll 1$ for (15) - (18), (21) and (22), we obtain the RG equation as

[TABLE]

where we have introduced the dimensionless parameters $F$, $G$ and $H$ as

[TABLE]

Here, from (25)-(28), we derive the autonomous equation for $(F(\Lambda),G(\Lambda),H(\Lambda))$ as

[TABLE]

2.3 Infrared and ultraviolet behaviors of solution trajectories of the RG equation

The stable fixed point of the equations (32) - (34) is found to be $(F^{*},G,^{*},H^{*})=(10.7593,\;680.652,\;63.2614)$. By substituting the fixed point values to (25)-(28) and solving them, we obtain the scaling laws

[TABLE]

where $C_{\nu}$, $C_{D}$, $C_{K}$, and $C_{D_{d}}$ are constants that depend on the initial condition $\mathcal{X}_{0}$.

We next consider the dimensionless quantities given by

[TABLE]

Substituting the scaling relations (35) - (38) to these equalities, we have

[TABLE]

Since $(F,G,H)$ takes the value $(10.7593,\;680.652,\;63.2614)$ in the limit $\Lambda\rightarrow 0$, we obtain

[TABLE]

which is independent of $\mathcal{X}_{0}$ . The singular behavior $\nu(\Lambda)=C_{\nu}\Lambda^{-1/2}$ implies that the effective surface tension depends on the observed scale $\Lambda$. This is contrasted with cases in which each $\mathcal{X}(\Lambda)$ converges to a finite value in the limit $\Lambda\rightarrow 0$. Then, $\mathcal{X}(\Lambda=0)$ is interpreted as renormalized parameters measured in experiments. Since the exponents characterizing the divergent behaviors are common to all the models given by (2), we refer to the power-law region as the universal range. The smallest characteristic wavenumber scale is also denoted by $\Lambda_{IR}$, the value of which depends on $\mathcal{X}_{0}$. Then, the universal range is defined as $\Lambda\ll\Lambda_{IR}$. As another common aspect of the RG equation (4), we observe that $\mathcal{X}(\Lambda)$ shows a plateau region in the ultraviolet limit when $\Lambda_{0}$ is sufficiently large. This enables us to define a collection of bare parameters, which is denoted by $\mathcal{X}_{B}$.

Here, we focus on a specific model, a noisy KS equation with $\mathcal{X}_{0}^{KS},(\nu_{0}=-1.0,D_{0}=0,K_{0}=D_{d0}=\lambda_{0}=1.0)$, defined at $\Lambda_{0}=2\pi$. In Fig. 1, we display the numerical solution of (4) for this initial condition $\mathcal{X}_{0}^{KS}$. It can be seen that $\Lambda_{0}$ is in the plateau region. Thus, the collection of the bare parameters $\mathcal{X}_{B}^{KS}$ is assumed to be identical to the initial condition $\mathcal{X}_{0}^{KS}$ without loss of accuracy. On the other hand, the numerical solution in the infrared limit obeys $\nu(\Lambda)=C_{\nu}\Lambda^{-0.5}$ and $D(\Lambda)=C_{D}\Lambda^{-0.5}$ in accordance with the analysis of the fixed point.

We note that $D_{d}(\Lambda)$ does not show the plateau region in Fig. 1. However, this graph quickly converges to $D_{d}(\Lambda)$ with $D_{d0}=0$ at $\Lambda_{0}=\infty$. As shown in Fig. 2, the graphs of $D_{d}(\Lambda)$ with $D_{d0}=0$ at $\Lambda_{0}=2\pi$, $10\pi$ and $1000\pi$ do not exhibit the plateau. Instead, the graphs at $\Lambda_{0}=2\pi$ and $10\pi$ quickly approach $D_{d}(\Lambda)$ in the limit $\Lambda_{0}=\infty$ when $\Lambda$ is smaller than $\Lambda_{0}$. Therefore, we define the bare parameter as $D_{dB}=0$ for such cases.

2.4 definition of the most effective model

Now, for the noisy KS equation with $\mathcal{X}_{B}^{KS}$, we consider the set ${\cal B}(\mathcal{X}_{B}^{KS})$ of bare parameters $\mathcal{X}_{B}$, each of which has the same factors $C_{\nu}$, $C_{D}$, $C_{K}$, and $C_{D_{d}}$ in the universal range and the same wavenumber scale $\Lambda_{IR}$ as those for the noisy KS equation. The graph of $\mathcal{X}(\Lambda)$ for a given $\mathcal{X}_{B}\in{\cal B}(\mathcal{X}_{B}^{KS})$ determines the wavenumber scale $\Lambda_{UV}$ that represents the end of the ultraviolet plateau. Note that the value of $\Lambda_{UV}$ depends on $\mathcal{X}_{B}\in{\cal B}(\mathcal{X}_{B}^{KS})$. Then, there is a special model with $\mathcal{X}_{B}\in{\cal B}(\mathcal{X}_{B}^{KS})$ such that $\Lambda_{UV}=\Lambda_{IR}^{KS}$. For this model, as soon as the graph of $\mathcal{X}(\Lambda)$ exits from the ultraviolet plateau region, it enters the infrared universal range. In other words, this special model represents the universal behavior of the noisy KS equation in the most efficient manner. We refer to it as the most effective model for the universal range of the noisy KS equation with $\mathcal{X}_{B}^{KS}$. Below, we determine the most effective model.

3 Representation of the parameter space

The solution trajectories for the RG equation are expressed as curves in the five-dimensional parameter space consisting of $\mathcal{X}$. We attempt to simplify a representation of the trajectories so as to determine the most effective model.

First, recalling $\lambda(\Lambda)=\lambda_{0}$, we may restrict the parameter space into the subspace $\lambda=\lambda_{0}=1$.

Next, as shown in Fig. 3, we find that $D(\Lambda)/\nu(\Lambda)$ and $D_{d}(\Lambda)/K(\Lambda)$ converge to the same value, 2.24, in the universal range for the noisy KS equation. We can explain this phenomenon as follows. First, for the generalized KPZ equations with $\mathcal{X}_{B}$ satisfying $D_{B}/\nu_{B}=D_{dB}/K_{B}\equiv\chi>0$, we can show the fluctuation-dissipation relation with the effective temperature $\chi$ fixed by using a time-reversal symmetry. The time-reversal transformation is given as

[TABLE]

The variation of the action (6) under this transformation is calculated as

[TABLE]

The generalized KPZ equation is invariant when $D_{0}/\nu_{0}=D_{d0}/K_{0}$ or $K_{0}=D_{d0}=0$. This symmetry leads to the invariance property of $D(\Lambda)/\nu(\Lambda)$ and $D_{d}(\Lambda)/K(\Lambda)$ along the solution trajectories of the RG equation. See Appendix A for the time-reversal symmetry of the generalized KPZ equation and the derivation of the fluctuation-dissipation relation. For the other cases where $D_{B}/\nu_{B}\not=D_{dB}/K_{B}$ including for noisy KS equations, $D(\Lambda)/\nu(\Lambda)$ and $D_{d}(\Lambda)/K(\Lambda)$ change in $\Lambda$. However, they satisfy $D(\Lambda)/\nu(\Lambda)=D_{d}(\Lambda)/K(\Lambda)$ in the universal range. Therefore, it is reasonable to conjecture that the time-reversal symmetry emerges in the universal range. Now, since the most effective model represents the universal behavior most efficiently, this special model should be in the subspace satisfying $D_{B}/\nu_{B}=D_{dB}/K_{B}\equiv\chi=2.24$. On the basis of the results, we express the bare-parameter space by $(\nu_{B},K_{B},D_{B}=2.24\nu_{B},D_{dB}=2.24K_{B},\lambda_{B}=1)$, as illustrated in Fig. 4. For each value of $(\nu_{B},K_{B})$, we have a model that exhibits the infrared universal behavior of $\mathcal{X}_{B}^{KS}$.

Finally, for a generalized KPZ equation with $\mathcal{X}_{B}$ at $\Lambda_{0}$ in the ultraviolet plateau region, we consider the following scale transformation:

[TABLE]

which yields another generalized KPZ equation with a different collection of bare parameters $\mathcal{X}_{B}^{\prime}$ at $\Lambda_{0}^{\prime}=b_{x}^{-1}\Lambda_{0}$ in the ultraviolet plateau region. These are the equivalent models in different unit systems. For the cases that $D=\chi\nu$ and $D_{d}=\chi K$, the equation for $H(X,T)$ is written as

[TABLE]

where we have introduced

[TABLE]

By imposing $\chi^{\prime}=\chi$ and $\lambda^{\prime}=\lambda$, we obtain $b_{h}=b_{x}^{1/2}$ and $b_{t}=b_{x}^{3/2}$. Then, we have the relation

[TABLE]

We find that $J\equiv K_{B}/\nu_{B}^{5}$ is invariant under the transformation. Thus, we parameterize $(\nu_{B},K_{B})$ as $(b_{x}^{1/2},b_{x}^{5/2}J)$. The next problem is to determine the values of $b_{x}$ and $J$ of the most effective model for the universal range of the noisy KS equation.

4 the most effective model

Since $J$ is invariant under the scale transformation, the determination of $J$ can be separated from the determination of $b_{x}$. Here, we notice the condition $\Lambda_{UV}=\Lambda_{IR}$ for the most effective model. Because this condition is invariant under the scale transformation, the value of $J$ is uniquely determined. Furthermore, the condition $\Lambda_{IR}=\Lambda_{IR}^{KS}$ fixes the value of $b_{x}$. Below, we explicitly calculate these values.

In order to determine the value of $J$, we study the dimensionless quantity $F(\Lambda)=\nu(\Lambda)/(K(\Lambda)\Lambda^{2})$ as a function of

[TABLE]

where $F$ and $s$ are invariant under the scale transformation. It should be noted that, for any $J$ and $b_{x}$, $F$ approaches

[TABLE]

in the ultraviolet limit $s\to-\infty$, while

[TABLE]

in the infrared limit $s\to\infty$. In Fig. 5, we show graphs of $F$ as functions of $s$ for several values of $J$. In general, there are two characteristic scales of $s$, the departure scale from $e^{2s}$ and the relaxation scale to $F^{*}$, as clearly observed for $J=0.1$. When $J$ increases, the peak of $F$ decreases and eventually vanishes at $J=7.3$. In this case, the transition scale between the infrared universal region and the ultraviolet region is simply given by the cross point $s_{c}$ of the ultraviolet behavior $F=e^{2s}$ and the infrared behavior $F^{*}=10.8$. That is,

[TABLE]

which gives $s_{c}=1.2$. Thus, we conclude that the value of $J$ of the most effective model is $J=7.3$.

Next, we determine the value of $b_{x}$. From the cross point $s_{c}$, we define the transition length scale $\Lambda_{c}^{-1}$ by $s_{c}=-\ln(J^{1/2}b_{x}\Lambda_{c})$, which gives $\Lambda_{c}^{-1}=\sqrt{JF^{*}}b_{x}=8.9b_{x}$. Here, the value of $b_{x}$ is determined by identifying $\Lambda_{c}$ with $\Lambda_{IR}^{KS}$. Then, we estimate $\Lambda_{IR}^{KS}$ from the graph of $\nu(\Lambda)$ for the noisy KS equation under study. In Fig. 6, we show how $\nu(\Lambda)$ approaches $C_{\nu}\Lambda^{-0.5}$. We find that $|\nu(\Lambda)-C_{\nu}\Lambda^{-0.5}|$ is well fitted to a power-law function of $\Lambda^{-1}$, which does not provide any wavenumber scale. Through more detailed analysis, we find a fitting function

[TABLE]

with $A=3.57$, $B=0.431$, $C=1.1\times 10^{-2}$, and $D=195$. From the second term of (62), we obtain the characteristic scale $(\Lambda_{IR}^{KS})^{-1}=D=195$. Now, from the condition

[TABLE]

we obtain $b_{x}=22$. Thus, we have arrived at the most effective model for the universal range of the noisy KS equation with $\mathcal{X}_{B}^{KS}$, where the collection of the bare parameter values of the most effective model, $\mathcal{X}_{B}^{ME}$, is determined as $(\nu_{B}=4.7,D_{B}=10,K_{B}=1.6\times 10^{4},D_{dB}=3.7\times 10^{4},\lambda_{B}=1)$.

Now, the linear decay rate of the disturbance of a wavenumber $k$ in the universal range is expressed as $\nu_{B}k^{2}+K_{B}k^{4}$ at an early time. Here, we notice that $(\nu_{B}/K_{B})^{0.5}$ defines one wavenumber scale. Since the most effective model has only one wavelength scale $\Lambda_{c}$, $(\nu_{B}/K_{B})^{0.5}\simeq\Lambda_{c}$ holds. This implies that the linear decay rate $\nu_{B}k^{2}+K_{B}k^{4}$ is estimated as $\nu_{B}k^{2}$ for $k\ll\Lambda_{c}$. In this manner, $\nu_{B}$ can be measured in experiments. Indeed, by applying this method to the numerical simulation of the noisy KS equation, the result $\nu_{B}^{\rm exp}\simeq 5.5$ was obtained PhysRevE.71.046138 . Thus, our theoretical value $\nu_{B}=4.7$ is in good agreement with the numerical value.

5 Concluding remarks

The main result of this paper is illustrated in Fig. 7. For a given noisy KS equation, we construct the most effective model exhibiting the same infrared universal behavior with just one cross-over wavenumber scale $\Lambda_{IR}^{KS}$ connecting the infrared behavior and the ultraviolet behavior. We emphasize that our theory enables us to calculate the bare surface tension $\nu_{B}$ of the effective model in the universal range, which could not be obtained by previous studies. We conclude this paper by presenting a few remarks.

The first remark is on the relevant parameter space in the universal range. Since $\lambda(\Lambda)$ is a conserved quantity along the solution of the RG equation, it obviously depends on the initial condition $\mathcal{X}_{0}$. Thus, it is relevant in the universal range. Furthermore, $D/\nu-D_{d}/K$ is not relevant because $D(\Lambda)/\nu(\Lambda)-D_{d}(\Lambda)/K(\Lambda)$ approaches zero. At the same time, $\chi=D/\nu$ is a relevant parameter because its value is invariant along the solution trajectory when $D_{0}/\nu_{0}=D_{d0}/K_{0}$. Finally, in the limit $\Lambda\to 0$, $K(\Lambda)\Lambda^{2}/\nu(\Lambda)$ approaches the universal constant value $0.0929$ which is independent of $\mathcal{X}_{0}$. Thus, we can state that $K(\Lambda)\Lambda^{2}/\nu(\Lambda)$ is irrelevant, following the argument in polchinski1984 ; Weingberg1995 . In other words, $\nu(\Lambda)$ and $K(\Lambda)$ are not independent of each other in the universal range, as shown in Fig. 8. In summary, the relevant parameter space in the universal range is spanned by the three parameters $(\nu,\chi,\lambda)$. However, the parameter $K$ cannot be negligible because the irrelevant parameter $K(\Lambda)\Lambda^{2}/\nu(\Lambda)$ is not zero in the universal range. This is different from many standard RG analysis Wilson1974 .

Second, we remark that the original Yakhot conjecture claims a statistical property of the deterministic KS equation PhysRevA.24.642 . Here, we discuss the noiseless limit $D_{0}\to 0$ for the noisy KS equation. In this case, we obtain $\chi\to 0$ which is not consistent with observations. This implies that the lowest order contribution in loop expansions is not sufficient to yield statistical properties for the small $D_{0}$ limit. In order to overcome this situation, we have to formulate a non-perturbative calculation. This is an interesting problem for future work.

Finally, we expect that the concept proposed in this paper will be applied to various systems, although we have studied a specific phenomenon as an example of scale-dependent parameters. The most interesting example may be fluid turbulence. The effective model for the universal range in the turbulence is given by a noisy Navier-Stokes equation Dominicis ; Fournier-Frisch1978 ; Fournier-Frisch1983 ; Yakhot-Orszag ; Yakhot-Orszag2 ; Yakhot-Smith ; Eyink

[TABLE]

where $\bm{v}$ is the fluid velocity, $p$ is the pressure, $\eta_{0}$ is the viscosity and $\bm{f}$ is the noise. Here, $P_{ij}$ and $F$ in the Fourier space are given as

[TABLE]

where $d$ is the space dimension, $D_{0}$ is the noise strength, and $y$ is a positive parameter. When $y=d=3$, this model exhibits the Kolmogorov scaling law

[TABLE]

where $E(k)$ is the energy spectra and $C$ is a universal constant and takes the value $\sim 1.5$. The analysis of solution trajectories of an RG equation for such the noisy Navier-Stokes equation may provide fresh insight into the understanding of the turbulence such as the universal constant $C$. We hope that this paper stimulates the study of whole solutions of RG equations in various research fields.

The authors thank K. A. Takeuchi, M. Itami, and T. Haga for useful discussions. The present study was supported by KAKENHI (Nos. 25103002 and 17H01148).

Appendix A Ward-Takahashi identities

In this section, we prove

[TABLE]

for all generalized KPZ equations, and

[TABLE]

for $K_{0}=D_{d0}=0$ or $K_{0}/D_{d0}=\nu_{0}/D_{0}$, and

[TABLE]

for $K_{0}/D_{d0}=\nu_{0}/D_{0}$. These results are easily obtained from the following Ward-Takahashi identities freyPRE1994 ; Canet2011 :

[TABLE]

[TABLE]

and

[TABLE]

These identities are relaed to invariance properties of the MSRJD action for a shift transformation, a tilt transformation, and a time-reversal transformation, respectively. In the next subsections, we will derive (72)-(74) following the arguments freyPRE1994 ; Canet2011 .

Here, we derive (69)-(71) from (72)-(74). First, by differentiating (73) with respect to $k_{1}$ and taking the limit $k_{1}\rightarrow 0$, we have

[TABLE]

Next, we substitute (72) to (75) and take the limit $\omega_{1}\rightarrow 0$. Then, we obtain

[TABLE]

By recalling the definition (19), we find that this equality is (69). Second, we differentiate (74) twice with respect to $k_{1}$. Then, we have

[TABLE]

By taking the limit $\omega_{1},k_{1}\rightarrow 0$ and using (15) and (17), we obtain (70). Finally, by differentiating (74) four times with respect to $k_{1}$, we arrive at (71).

A.1 Proof of (72)

We consider a shift transformation

[TABLE]

where $c(t)$ is an infinitesimal parameter that depends on time. The variation of the MSRJD action for the transformation is calculated as

[TABLE]

It should be noted that this simple form comes from the invariance property of the MSRJD action for the time-independent $c$ 111In general, by assuming time dependence of the infinitesimal parameter for a continuous symmetry transformation, we can obtain non-trivial identities such as (72). This technique, which has been referred to as “gauging a global symmetry”, is standard when we derive identities from a continuous global symmetry Weingberg1995 . For such a case, the variation of an action under a time-gauged transformation is expressed as $\delta S=\int dtQ(t)\partial_{t}\epsilon(t)$, where $Q(t)$ is a Noether charge of the corresponding global symmetry, and $\epsilon(t)$ is the time-gauged infinitesimal parameter. The Noether charge of the shift symmetry is calculated as $Q_{\rm shift}=\int dxi\tilde{h}(x,t)$, which is consistent with (79).. Then, the variation of the effective MSRJD action is derived as

[TABLE]

When we obtain the fourth line in (80) from the third line, we have used

[TABLE]

Here, noting the trivial relation

[TABLE]

we rewrite (80) as

[TABLE]

which is further expressed as

[TABLE]

Since this equality holds for any $c(t)$, we obtain

[TABLE]

The differentiation of (85) with respect to $i\tilde{h}^{<}(t^{{}^{\prime}},x^{{}^{\prime}})$ leads to

[TABLE]

By performing the Fourier transformation, we arrive at (72).

A.2 Proof of (73)

We consider a tilt transformation

[TABLE]

where $v$ is an infinitesimal parameter. The tilt transformation for their Fourier transforms is expressed as

[TABLE]

We then find the symmetry property

[TABLE]

from which we obtain

[TABLE]

where $\mathcal{J}=1+va$ is the Jacobian for the tilt transformation, and $a$ is a field independent quantity. The expansion of (92) in $v$ leads to the identity

[TABLE]

We differentiate this identity with respect to $i\tilde{h}^{<}(k_{1},t_{1})$ and $h^{<}(k_{2},t_{2})$. Then, we have

[TABLE]

By taking the limit $i\tilde{h}^{<},h^{<}\rightarrow 0$ and recalling the definitions given in (12) - (14), we obtain

[TABLE]

The Fourier transform of this equality is (73).

A.3 Proof of (74)

We consider a time-reversal transformation

[TABLE]

The variation of the action (6) under this transformation is calculated as

[TABLE]

The generalized KPZ equation is invariant when $D_{0}/\nu_{0}=D_{d0}/K_{0}$ or $K_{0}=D_{d0}=0$.

Here, we focus on the case $D_{0}/\nu_{0}=D_{d0}/K_{0}$. Then, we obtain

[TABLE]

where $\mathcal{J}$ is the Jacobian of the time-reversal transformation. By differentiating this equality with respect to $i\tilde{h}^{<}(k_{1},\omega_{1})$ and $h^{<}(k_{2},\omega_{2})$, we have

[TABLE]

where we have used the relation

[TABLE]

By recalling the definition given in (12) - (14), we obtain

[TABLE]

By rearranging (103), we arrive at the identities (74).