The one-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: limit distributions

TL;DR

This paper demonstrates that the one-dimensional Kuramoto-Sivashinsky (KS) PDE exhibits the same universal limit distributions as the KPZ equation in a nonequilibrium steady state, confirmed through extensive numerical simulations.

Contribution

It establishes that the 1D KS PDE belongs to the KPZ universality class by numerically showing identical limit distributions in a steady state.

Findings

KS height fluctuations follow Tracy-Widom and Baik-Rains distributions

Numerical simulations confirm universality class membership

Statistical properties are consistent across different initial conditions

Abstract

Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) \textit{stochastic} partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state (NESS) of the one-dimensional Kuramoto-Sivashinsky (KS) \textit{deterministic} PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile $h(x,t)$ for different initial conditions. We establish, therefore, that the statistical properties of the 1D KS PDE in this state are in the 1D KPZ universality class.

Equations

$$h(x,t)-h(x,0)\approx v_{\infty }t+(\Gamma t{)}^{{\text{upbeta}}_{\text{KPZ}}}{\text{upchi}}_{\beta }+o(t^{{\text{upbeta}}_{\text{KPZ}}}),\text{for}t\to \infty ,$$

$${\partial }_{t}h(\mathbf{x},t)+\Delta h(\mathbf{x},t)+{\Delta }^{2}h(\mathbf{x},t)+\frac{1}{2}(\nabla h(\mathbf{x},t){)}^2=0,$$

$${\partial }_{t}h(x,t)$$

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

$$w(l,t)={\left(⟨[{\Delta }_{l}h(x,t){]}^2{⟩}_{x,l}\right)}^{1/2},$$

$${\mu }_n=⟨{\left({\Delta }_{L}h(x,t)\right)}^n⟩/⟨{\left({\Delta }_{L}h(x,t)\right)}^2{⟩}^{n/2}-3{\delta }_{n,4};$$

$$\Lambda =\sqrt{2}+\frac{1}{\sqrt{2}}{N}^{-2/3}{\text{upchi}}_{\beta },$$

$$\text{P}(\Lambda ,N)\approx \{\begin{array}{ll}\exp [-\beta N^2{\varphi }_{-}(\Lambda )],& \Lambda <\sqrt{2},|\Lambda -\sqrt{2}|\sim \mathcal{O}(1),\\ \sqrt{2}{N}^{2/3}{\text{P}}_{\text{TW},\beta }({\text{upchi}}_{\beta }),& |\Lambda -\sqrt{2}|\sim \mathcal{O}(N^{-2/3}),\\ \exp [-\beta N{\varphi }_{+}(\Lambda )],& \Lambda >\sqrt{2},|\Lambda -\sqrt{2}|\sim \mathcal{O}(1),\end{array}$$

$${\varphi }_{-}(\Lambda )$$

$${\varphi }_{+}(\Lambda )$$

$$\underset{N\to \infty }{\lim }-\frac{1}{N^2}\ln F_{\beta }(\Lambda ,N)=\{\begin{array}{ll}{\varphi }_{-}(\Lambda ),& \Lambda <\sqrt{2},\\ 0,& \Lambda >\sqrt{2}.\end{array}$$

$$\mathcal{F}(\overline{h})=\underset{t,L\to \infty }{\lim }-\frac{1}{t^2}\ln F(\text{upchi},t),$$

Full text

Kuramoto-Sivashinsky equation, Kardar-Parisi-Zhang equation, Tracy-Widom distribution, Baik-Rains distribution.

Fundamental investigations of the statistical properties of hydrodynamical turbulence often use randomly forced versions of the deterministic Navier-Stokes (NS) equations (3D NSE, in three dimensions); the latter use a non-random forcing term to produce a turbulent, but nonequilibrium, statistically steady state (NESS). A randomly forced 3D, incompressible NS equation (3D RFNSE), proposed first by Edwards Edwards (1964) in 1964, has been studied extensively, via renormalization-group (RG) and other theoretical Forster et al. (1977); DeDominicis and Martin (1979); Fournier and Frisch (1983); Yakhot and Orszag (1986); Mou and Weichman (1995); Bhattacharjee (1988); Adzhemyan et al. (1996, 1999) and numerical Sain et al. (1998); Biferale et al. (2004) methods; these studies have shown that many statistical properties of turbulence in the 3D RFNSE are akin to their 3D NSE counterparts. In particular, the wave-number $k$ dependence of the energy spectrum Kolmogorov (1941a, b); Frisch (1995) $E(k)$, and even the mutiscaling corrections Frisch (1995); pf1 (1985); Benzi et al. (1984); Benzi and Frisch (2010); Meneveau and Sreenivasan (1991) to the Kolmogorov phenomenology Kolmogorov (1941a, b); Frisch (1995) of 1941 are similar in both these models.

Can we find such similarity between the statistical properties of NESSs in deterministic and related stochastic partial differential equations (PDEs) that are simpler than their 3D hydrodynamical counterparts? It has been suggested, since the 1980s, that the Kuramoto-Sivashinsky (KS) PDE, a deterministic interface-growth model for a height field $h({\bf x},t)$, which is used in studies of chemical waves, flame fronts, and the surfaces of thin films flowing under gravity Kuramoto and Tsuzuki (1976); Sivashinsky (1977); Sivashinsky and Michelson (1980); Ruyer-Quil and Manneville (1998); pz1 (1985); Chen and Chang (1986); Grinstein et al. (1996), is a simplified model for turbulence pz1 (1985). It has been conjectured Yakhot (1981), and subsequently shown by compelling numerical studies Hyman et al. (1986); Sneppen et al. (1992); Hayot et al. (1993); Jayaprakash et al. (1993); Boghosian et al. (1999); Kalogirou et al. (2015), in both one dimension (1D) and two dimensions (2D), that the long-distance and long-time behaviors of correlation functions, in the spatiotemporally chaotic NESS of the KS PDE, exhibit the same power-law scaling as their couterparts in the the Kardar-Parisi-Zhang (KPZ) equation Kardar et al. (1986); Halpin-Healy and Zhang (1995); Halpin-Healy and Takeuchi (2015); Quastel and Spohn (2015), a stochastic PDE (SPDE), in which the height field $h({\bf x},t)$ is kinetically roughened. The elucidation of the statistics of $h({\bf x},t)$ in the KPZ SPDE has played a central role in nonequilibrium statistical mechanics, in general, and interface-growth phenomena, in particular. Early KPZ studies Kardar et al. (1986); Halpin-Healy and Zhang (1995) have concentrated on height-field correlations, the width $w(L,t)$ of the fluctuating KPZ interface, and their power-law dependences on the linear system size $L$ and time $t$, for large $L$ and $t$ (see below); especially for the 1D case, several results can be obtained analytically. The universality of the power-law exponents has been demonstrated by explicit numerical calculations, e.g., in the poly-nuclear growth (PNG) model, directed polymers in random media (DPRM), and the asymmetric simple exclusion process (ASEP), and by experiments in turbulent liquid crystals Takeuchi et al. (2011); Takeuchi and Sano (2012); Takeuchi (2013), all of which lie (in suitable parameter regimes) in the KPZ universality class. The seminal work of Prähofer and Spohn work (recently referred to as “the $2^{nd}$ KPZ Revolution” Halpin-Healy and Takeuchi (2015)) on the PNG model Prähofer and Spohn (2000) has led to a new set of studies of the 1D KPZ universality class Sasamoto and Spohn (2010); Calabrese and Le Doussal (2011); Imamura and Sasamoto (2012); Corwin (2012); Halpin-Healy and Lin (2014); Quastel and Spohn (2015); Saberi et al. (2019), which have led to the remarkable result that, at a point $x$ and at large times $t$,

[TABLE]

where $v_{\infty}$ and $\Gamma$ are model-dependent constants (Supplemental Material sup ), the exponent $\upbeta_{\text{KPZ}}=1/3$, and $\upchi_{\beta}$ is a random variable distributed according to the Tracy-Widom (TW) distribution for the Gaussian Orthogonal Ensemble (GOE) ($\beta=1$) and for the Gaussian Unitary Ensemble (GUE) ($\beta=2$), familiar from the theory of random matrices Tracy and Widom (1994), or the Baik-Rains (BR $F_{0}$) distribution Baik and Rains (2000) ($\beta=0$); the value of $\beta$ depends on the initial condition. We show, by extensive direct numerical simulations (DNSs), that the result (1) holds for the NESS of the 1D KS PDE. Thus, the correspondence between the statistical properties of these states, in the 1D KS (PDE) and their counterparts in the 1D KPZ (SPDE), does not stop at the simple correlation functions, investigated so far Hyman et al. (1986); Sneppen et al. (1992); Hayot et al. (1993); Jayaprakash et al. (1993); we demonstrate that this correspondence includes the universal limit distributions obtained in “the $2^{nd}$ KPZ Revolution” Halpin-Healy and Takeuchi (2015). Such a result has not been obtained hitherto for a spatiotemporally chaotic NESS of a deterministic PDE.

The KS PDE, which predates the KPZ SPDE, is

[TABLE]

where $\nabla\equiv\partial/\partial\mathbf{x}$, $\partial_{t}\equiv\partial/\partial t$, $\Delta\equiv\nabla^{2}$, and $h,\,\mathbf{x}$, and $t$ have been scaled such that the linear system size $L$ is the only control parameter. The dynamical and long-wavelength properties of the 1D KS PDE have been explored via DNSs in Refs. Hyman et al. (1986); Sneppen et al. (1992); Hayot et al. (1993); Hyman and Nicolaenko (1986); Kevrekidis et al. (1990); several mathematical results have been obtained in Refs. Collet et al. (1993); Jolly et al. (1990); Conte and Musette (1989).

The 1D KPZ SPDE is

[TABLE]

where $\nu$, the diffusivity, and $\lambda$, the strength of the nonlinearity, are real parameters, and $\eta$ is a zero-mean Gaussian white noise, with variance $D$.

We solve the 1D KS PDE (2), with periodic boundary conditions on a domain of size $L$, by using the pseudospectral method Canuto and Quarteroni (1981); Canuto et al. (2006); Trefethen (2000) and the $2/3$ dealiasing rule. For time marching we use the fourth-order, exponential time-differencing Runge-Kutta scheme ETDRK4 Kassam and Trefethen (2005); Cox and Matthews (2002). For reliable statistics, it is important to carry out long simulations with large values of $L$; we report results with $L=2^{20}$, by far the highest spatial resolution that has been used for a DNS of the 1D KS PDE (2); for this we have developed a CUDA C code that runs very efficiently on a GPU cluster with NVIDIA Tesla K80 accelerators.

From our DNSs we compute $h(x,t)$ for six different kinds of initial conditions, IC1-IC6, which we depict by plots of $h(x,0)$ versus $x$ in Figs. 1 (a), (e), (i), (m), (q), and (u); we show the short-time spatiotemporal evolution of $h(x,t)$, in the interval $x\in[-100,100]$, in Figs. 1 (b), (f), (j), (n), (r), and (v) (see the videos V1-V6 in the Supplemental Material sup ). We choose these ICs to mimic the effect of wedge, flat, stationary, wedge-to-stationary, wedge-to-flat, and flat-to-stationary geometries in the ASEP model, which are listed in Refs. Corwin (2012); Borodin et al. (2008); Corwin et al. (2010) as initial conditions for six different sub-classes of the 1D KPZ universality class. Previous numerical studies Hayot et al. (1993); Sneppen et al. (1992) of the 1D KS PDE have shown that two-point, equal-time height-field correlations show the scaling behaviors of their 1D KPZ SPDE counterparts for times greater than a crossover time $t_{c}\simeq 18700$ and lengths larger than the crossover size $L_{c}\simeq 3600$. Therefore, we use a very large system size $L=2^{20}$ and very long simulation times $t_{max}\geq 2\times 10^{5}$ (see the Supplemental Material sup ).

Our results for two-point height correlation functions are consistent with those of earlier investigations Hayot et al. (1993); Sneppen et al. (1992) of the statistical properties of the spatiotemporally chaotic state of the 1D KS PDE: We show, e.g., the equal-time compensated spectrum $k^{2}E(k)=\langle L\tilde{h}(k,t)\tilde{h}^{*}(k,t)\rangle_{t}$, where $\langle\cdot\rangle_{t}$ is the time average, $\tilde{h}(k,t)$ is the spatial Fourier transform of $h(x,t)$, and $k$ is the wave number, in Fig. (1) of the Supplemental Material sup . In addition, we calculate the time-dependent, two-point correlation function $S(k,\delta t)=\langle k^{2}\tilde{h}(k,t_{0})\tilde{h}^{*}(k,t_{0}+\delta t)\rangle_{t_{0}}$ in Fig. 2, for the IC3 initial condition. We find that the imaginary part of $S(k,\delta t)$ fluctuates around zero and its magnitude is much smaller than that of its real part, which we plot in Fig. 2. Our data are consistent with the scaling form of $S(k,\delta t)$ (orange curve in Fig. 2), which has been obtained analytically by Prähofer and Spohn Prähofer and Spohn (2004) for the 1D KPZ SPDE; this comparison of $S(k,\delta t)$ for the 1D KS and 1D KPZ equations has not been made hitherto.

The scaling properties of the interface width $w(L,t)$ distinguish different universality classes in interface-growth models;

[TABLE]

with $\Delta_{l}h(x,t)=h(x,t)-h(x,0)-\langle h(x,t)-h(x,0)\rangle_{x,l}$ and $\langle\cdot\rangle_{x,l}$ the spatial average over a region of spatial extent $l$. For $t\gg 1$ in the 1D KPZ equation, $w(L,t)\sim t^{\upbeta}$. Before crossover occurs in systems with $L>L_{c}$, the exponent $\upbeta$ assumes the value $\upbeta_{\text{EW}}=1/4$, which is the Edwards-Wilkinson (EW) result Edwards and Wilkinson (1982); Halpin-Healy and Zhang (1995) for the linear SPDE with $\lambda=0$ in Eq. (3); finally, $\upbeta$ assumes the KPZ value $\upbeta_{\text{KPZ}}=1/3$ in the NESS (for $t>t_{c}$). Moreover, the growing KPZ surface involves the length scale $\mathcal{L}(t)\sim t^{1/z}$, where the dynamic exponent $z=3/2$; and the width $w(l,t)\sim l^{\alpha}$, for $l\ll\mathcal{L}(t)$, with $\alpha=1/2$ Takeuchi et al. (2011). We find from our DNSs of the 1D KS equation that these Family-Vicsek scaling Family and Vicsek (1985) forms are indeed satisfied as we show in Figs. 3 (a), (c), and (e) for IC1-IC3 (see the Supplemental Material sup for IC4-IC6).

We define

[TABLE]

for $n=3$ ($n=4$), $\mu_{n}$ is the skewness (kurtosis); we plot $\mu_{3}$ and $\mu_{4}$ versus time $t$ in the right panel of Fig. 3; for each initial condition, IC1-IC6, we average these quantities for $100$ surfaces, over a time interval of $10^{4}$, and five independent DNS runs; i.e., our overall sample size is $\simeq 5\times 10^{8}$ data points. [For our 1D KS, $\mu_{3}<0$ because of the sign of the nonlinear term in Eq. (2); we ignore the sign of $\mu_{3}$ for it can be reversed by the transformation $h(x,t)\rightarrow-h(x,t)$.] In addition, we calculate the probability distribution function (PDF) $\textrm{P}(\upchi)$ of the shifted and rescaled fluctuations, namely, $\upchi=(h(x,t)-v_{\infty}t)/(\Gamma t)^{1/3}$, when both $\mu_{3}$ and $\mu_{4}$ are close to their standard values for the relevant TW or BR $F_{0}$ PDFs; for IC2, e.g., we compute $\textrm{P}(\upchi)$ when we have $\mu_{3}\simeq 0.27$ and $\mu_{4}\simeq 0.19$, which are close to the standard values $\mu_{3,\text{GOE}}\simeq 0.29$ and $\mu_{4,\text{GOE}}\simeq 0.16$, respectively.

For IC1, IC2, IC3, and IC4 we compare, on semilog plots, the PDFs with TW-GUE, TW-GOE, BR $F_{0}$, and $(F_{\text{GOE}})^{2}$ Corwin (2012) in Figs. 1 (d), (h), (l), and (p), respectively. For ease of comparison, we show in Fig. 4 that the PDFs we obtain from our DNSs of the 1D KS Eq. (2) are very close to the TW-GUE, TW-GOE, and BR $F_{0}$ PDFs over at least three orders of magnitude. Stricly speaking, we must collect data only from those two points ($x=L/4,3L/4$) at which the two different type of height profiles meet in cases IC4, IC5 and IC6. However, this leads to inadequate statistics. Therefore, the PDFs of $\upchi$ for IC4-6, which we show in Figs. 1 (t) and (x), have been computed by using data from the regions $\left[7L/32,9L/32\right]$ and $\left[23L/32,25L/32\right]$; we see that this averaging procedure already leads to PDFs (Figs. 1 (p), (t) and (x)) that are distinctly different from TW-GUE, TW-GOE, and BR $F_{0}$ distributions.

The TW distributions, for IC1 and IC2 initial conditions in the 1D KPZ equation, have been studied in the context of $N\times N$ GOE ($\beta=1$) and GUE ($\beta=2$) random matrices. The largest eigenvalue (after scaling with $N$) $\Lambda$ of such random matrices is

[TABLE]

where $\upchi_{\beta}$ has the PDF Majumdar and Schehr (2014)

[TABLE]

$\textrm{P}_{\textrm{TW},\beta}(\upchi_{\beta})$ denotes TW distributions, and the right and left large-deviation functions (LDFs) $\phi_{+}(\Lambda)$ and $\phi_{-}(\Lambda)$, respectively, display the following asymptotic behaviors:

[TABLE]

The LDFs, which yield the probabilities of atypically large fluctuations, match smoothly with the tails of $\textrm{P}_{\textrm{TW},\beta}(\upchi_{\beta})$. Because of different behaviors of the tails of $\textrm{P}(\Lambda,N)$, a third-order transition Majumdar and Schehr (2014) can be associated with $\Lambda$ at $\Lambda_{c}=\sqrt{2}$ by defining the free energy $\propto\ln F_{\beta}(\Lambda,N)$, $F_{\beta}(\Lambda,N)$ being the cumulative density function (CDF) for $\Lambda$, for we have Majumdar and Schehr (2014)

[TABLE]

Similarly, we define, for the KS initial conditions IC1 and IC2, the free-energy function $\mathcal{F}(\overline{h})$, for $t,L\rightarrow\infty$, as follows:

[TABLE]

where $\overline{h}=h(x,t)/t$ and $F(\upchi,t)$ is the CDF for $\upchi$ at time $t$. Therefore, for IC1 and IC2, we should obtain a third-order phase transition for $\overline{h}$ at the critical value $\overline{h}_{c}=v_{\infty}$; an explicit demonstration requires much better statistics for $\textrm{P}(\upchi)$ than is possible with our DNS.

We have shown, by extensive pseudospectral DNSs of the 1D KS deterministic PDE, that the statistical properties of its spatiotemporally chaotic NESS are in the 1D KPZ universality class. This is not limited, merely, to the power-law forms of simple correlation functions and the width of the interface. It includes, in addition, (a) the complete scaling form for the two-point time-dependent correlation function $S(k,\delta t)$ (Fig. 2), (b) the skewness and kurtosis shown in Fig. 2, and (c) most important of all, the unversal limit distributions in Fig. 1, obtained in “the $2^{nd}$ KPZ Revolution” Halpin-Healy and Takeuchi (2015). Such results have not been obtained hitherto for a spatiotemporally chaotic NESS of any deterministic PDE. We conjecture that similar conclusions should ensue for the phase-chaos regime of the 1D Complex-Ginzburg-Landau equation Grinstein et al. (1996). Such studies are also being pursued for the 1D Calogero-Moser model Agarwal et al. .