Crossover from Growing to Stationary Interfaces in the Kardar-Parisi-Zhang Class
CCrossover from Growing to Stationary Interfaces in the Kardar-Parisi-Zhang Class
Kazumasa A. Takeuchi ∗ Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan (Dated: October 13, 2018)This Letter reports on how the interfaces in the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ)class undergo, in the course of time, a transition from the flat, growing regime to the stationaryone. Simulations of the polynuclear growth model and experiments on turbulent liquid crystalreveal universal functions of the KPZ class governing this transition, which connect the distributionand correlation functions for the growing and stationary regimes. This in particular shows howinterfaces realized in experiments and simulations actually approach the stationary regime, whichis never attained unless a stationary interface is artificially given as an initial condition.
PACS numbers: 05.40.-a, 64.70.qj, 89.75.Da, 64.70.mj
Aside from their ubiquity in nature, surface growthphenomena constitute an important situation of statisti-cal mechanics out of equilibrium, where scale invarianceand universal scaling laws arise generically [1]. Theseare usually evidenced in the roughness of the interfaces,whose amplitude w ( L, t ) measured at the system (sub-strate) size L and time t obeys the following power laws: w ( L, t ) ∼ (cid:40) L α for L (cid:28) L ∗ , t β for L (cid:29) L ∗ , ( L ∗ ∼ t /z ) , (1)with scaling exponents α, β, z ≡ α/β [1, 2]. At theheart of such growth processes is the Kardar-Parisi-Zhang (KPZ) equation [3] and the corresponding uni-versality class [1, 3], describing the simplest case withoutany conservation laws and long-range interactions. Forone-dimensional interfaces, the KPZ equation reads ∂∂t h ( x, t ) = ν ∂ h∂x + λ (cid:18) ∂h∂x (cid:19) + √ Dη ( x, t ) , (2)where h ( x, t ) denotes the fluctuating height profile and η ( x, t ) white Gaussian noise with (cid:104) η ( x, t ) (cid:105) = 0 and (cid:104) η ( x, t ) η ( x (cid:48) , t (cid:48) ) (cid:105) = δ ( x − x (cid:48) ) δ ( t − t (cid:48) ). The values ofthe scaling exponents are exactly known in this one-dimensional case [1, 3, 4]: the height fluctuation δh ≡ h − (cid:104) h (cid:105) grows as δh ∼ t / ( β = 1 /
3) and the correlationlength ξ as ξ ∼ t / ( z = 3 / h is describedby a rescaled random variable χ ( x (cid:48) , t ) as h ( x, t ) (cid:39) v ∞ t + (Γ t ) / χ ( x (cid:48) , t ) (3)with a rescaled coordinate x (cid:48) ≡ ( Ax/ t ) − / and con-stant parameters A ≡ ν/ D, Γ ≡ A λ/
2, and v ∞ . TheKPZ-class exponents have indeed been reported in vari-ous models and theoretical situations [1, 3–5] as well asby a growing number of experiments [6–12].Studies on the (1+1)-dimensional KPZ class enteredan unprecedented stage in 2000, when Johansson [13] andothers [5] rigorously derived asymptotic distributions ofthe height fluctuations for a few models. Among oth-ers, it has brought about two outstanding outcomes. (i) The KPZ class splits into a few subclasses according tothe global geometry of the interfaces, or, equivalently,to the initial condition. These subclasses are character-ized by different distribution and correlation functions,whereas they share the same scaling exponents. (ii) Anunexpected link to random matrix theory has been re-vealed. In particular, the asymptotic distribution of χ for the flat and curved interfaces is given by the largest-eigenvalue distribution, called the Tracy-Widom (TW)distribution [14, 15], for the Gaussian orthogonal ensem-ble (GOE) and the Gaussian unitary ensemble, respec-tively [16]. The stationary interfaces also form a distinctsubclass. To study it analytically, one usually sets the ini-tial condition h ( x,
0) to be a stationary interface, whichis simply the one-dimensional Brownian motion for theKPZ equation [1]. The height difference h ( x, t ) − h ( x, χ obeys the F distributionintroduced by Baik and Rains [17], as proved for thepolynuclear growth (PNG) model [16, 17], for the to-tally asymmetric simple exclusion process [18, 19], and,very recently, for the KPZ equation [20]. The two-pointcorrelation function being exactly derived as well [18–22],this subclass is now firmly established like the ones forthe flat and curved interfaces.Such a stationary regime is, however, never attainedwithin a finite time in an infinitely large system, unless astationary interface is artificially given as an initial condi-tion. This is readily seen by recalling that the correlationlength grows as ξ ∼ t / , whereas it is infinite for thestationary interfaces. Therefore, in practical situationsstarting from a smooth or uncontrolled initial profile, oneneeds to elucidate how the interfaces approach the sta-tionary regime in the course of time. This is achieved bythe present Letter. Simulations of the PNG model showthat the height difference of the flat interfaces exhibits atransition from the flat, growing regime to the stationaryone. We find scaling functions describing this crossoverand determine their functional forms, which smoothlyconnect the GOE-TW and Baik-Rains F distributions.We also study the two-point correlation function andshow how those for the two subclasses interplay at fi- a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y -2 -1 -2 -1 -3 -2 -1 -3 -4 -2 ∆ t -0.8-0.6-0.4-0.20 ∆ t / t ∆ t ∆ t / t ∆ t ∆ t / t -2 -4 -2 -3 -1 -2 (a) (b) s k e w n e ss ku r t o s i s 〈 ∆ q 〉 c 〈 ∆ q 〉 c 〈 ∆ q 〉 c ∆ t 〈 ∆ q 〉 c ∆ t / t ∆ t / t ∆ t / t -6 -4 -2 (e) ∆ q p r ob a b ilit y d e n s it y Baik-Rains F GOE-TW ∆ t / t t t = 0.1 t ≥ 1 F GOE-TW ∆ t / t 〈 χ 〉 c – 〈 ∆ q 〉 c 〈 ∆ q 〉 c – 〈 χ 〉 c 〈 χ 〉 c – 〈 ∆ q 〉 c 〈 ∆ q 〉 c – 〈 χ 〉 c (c) (d) slope 2/3slope 1/2 slope -1/3slope -2/3 ∆ t / t FIG. 1. (color online). Crossover in the one-point distribution for the PNG model. (a,b) First- to fourth-order cumulants (cid:104) ∆ q n (cid:105) c against ∆ t (a) and ∆ t/t (b), for t = 0 , . , , , , , , F and GOE-TW distributions, (cid:104) χ n (cid:105) c and (cid:104) χ n (cid:105) c , respectively. The black solid lines in (b) show fitting to the collapsed curves (see text). The insets in (b) show theskewness and the kurtosis. (c,d) Asymptotic behavior of the data in (b) for small and large ∆ t/t . The data for t = 0 . q for given pairs of t and ∆ t/t . nite times. These results are quantitatively reproducedby experimental data on turbulent liquid crystal [8, 9],indicating that they are universal characteristics of theKPZ-class interfaces.First we study the PNG model. Starting from a flatsubstrate h ( x,
0) = 0, an interface experiences randomnucleation events at a uniform rate. On each nucleation,the local height h ( x, t ) increases by one, producing aplateau that expands laterally at constant speed. Whentwo plateaux encounter, they simply coalesce. For thesimulations in continuous space and time, we numericallyimplement space-time representation used for analyticalderivation of the distribution function [16, 17], with thenucleation rate 2 per unit space and time and the plateauexpansion speed 1. This choice of the parameters corre-sponds to v ∞ = 2 , A = 2, and Γ = 1. We impose the pe-riodic boundary condition with system size L = 10 andrealize 10 independent interfaces up to time 10 . Thesize is chosen to satisfy L (cid:29) L ∗ until t = 10 [Eq. (1)]so that the system does not reach the saturated regime,which is not the crossover addressed in this Letter.The quantity of interest is the height difference ∆ h ( x, ∆ t, t ) ≡ h ( x, t + ∆ t ) − h ( x, t ), rescaled here as∆ q ( x, ∆ t, t ) ≡ ∆ h − v ∞ ∆ t (Γ∆ t ) / . (4)By construction, ∆ q d → χ for t → t → ∞ ,while ∆ q d → χ for t → ∞ and then ∆ t → ∞ , where χ and χ are random variables obeying the GOE-TW andBaik-Rains F distributions, respectively, with the factor2 − / multiplied with the usual definition for the former[16, 17]. Figure 1(a) shows the first- to fourth-order cu-mulants of ∆ q , (cid:104) ∆ q n (cid:105) c , as functions of ∆ t for different t , displayed with the values for the GOE-TW and Baik-Rains F distributions (dashed and dotted lines, respec-tively). The cumulants agree with those for the GOE-TW distribution as ∆ t tends to infinity, while they in-dicate the values of the Baik-Rains F distribution forlarge t and small enough ∆ t . The transition curvesare found to collapse very well when ∆ t is scaled by t [Fig. 1(b)], except for too small t and ∆ t . In par-ticular, for t → ∞ , the cumulants converge to a sin-gle set of functions, (cid:104) ∆ q n (cid:105) c → ∆ Q n (∆ t/t ), satisfying∆ Q n ( τ ) → (cid:104) χ n (cid:105) c for τ → ∞ and ∆ Q n ( τ ) → (cid:104) χ n (cid:105) c for τ →
0. One can indeed draw the functions ∆ Q n ( τ )by making histograms for (cid:104) ∆ q n (cid:105) c at each ∆ t/t withvarying t and fitting their modes by, e.g., spline func-tions, as shown by the black solid lines in Fig. 1(b).Theoretical expressions of ∆ Q n ( τ ) are unknown, be-cause they involve time correlation which still remainsanalytically unsolved. Asymptotically, the data suggest (cid:104) χ (cid:105) c − ∆ Q ( τ ) ∼ τ / , (cid:104) χ (cid:105) c − ∆ Q ( τ ) ∼ τ / for small τ and ∆ Q ( τ ) −(cid:104) χ (cid:105) c ∼ τ − / , ∆ Q ( τ ) −(cid:104) χ (cid:105) c ∼ τ − / forlarge τ [Fig. 1(c,d)]. While this convergence to the GOE-TW distribution ( τ → ∞ ) is analogous to that of theheight variable h ( x, t ) [9, 23, 24], the power laws towardthe Baik-Rains F distribution ( τ →
0) indicate unusualexponents that need to be explained theoretically. Forhigher orders n ≥
3, one needs better statistical accuracyto determine the asymptotics. In between the two limits,the transition occurs earlier for larger n ( ≤ (cid:104) ∆ q (cid:105) c / (cid:104) ∆ q (cid:105) / and the kurtosis (cid:104) ∆ q (cid:105) c / (cid:104) ∆ q (cid:105) [insets of Fig. 1(b)]. Fi-nally, this crossover can also be checked directly in thedistribution; Fig. 1(e) shows that the probability densityfunctions of ∆ q overlap for fixed ∆ t/t , and that theyshift from the Baik-Rains F to the GOE-TW distribu-tions as ∆ t/t is increased.Now we turn our attention to the two-point correlationfunction, defined here by C ( l, ∆ t, t ) ≡ (cid:10) [ δh ( x + l, t + ∆ t ) − δh ( x, t )] (cid:11) (5)with δh ( x, t ) ≡ h ( x, t ) − (cid:104) h ( x, t ) (cid:105) . If one takes thestationary limit t → ∞ and then considers large ∆ t ,one has C (cid:48) ( ζ, ∆ t, t ) ≡ (Γ∆ t ) − / C ( l, ∆ t, t ) (cid:39) g ( ζ )with rescaled length ζ ≡ ( Al/ t ) − / , where g ( ζ )is the exact solution for the rescaled stationary correla-tion [20, 21]. This is tested in Fig. 2(a) with finite t and∆ t , where ∆ C (cid:48) ( ζ, ∆ t, t ) ≡ C (cid:48) ( ζ, ∆ t, t ) − C (cid:48) (0 , ∆ t, t ) iscompared with g ( ζ ) − g (0) in the main panel. First wenote that the data for fixed ∆ t/t and different t over-lap with each other, confirming that ∆ t/t is the onlytime scale that controls the dynamics. Now, we focuson the data with the smallest ∆ t/t we have, namely∆ t/t = 0 . ζ , with or with-out subtraction of C (cid:48) (0 , ∆ t, t ) (main panel and inset,respectively). By contrast, for large ζ , the correlationis governed by the spatial correlation of the flat in-terfaces, namely the Airy correlation g ( · ), defined by g ( v ) ≡ (cid:104)A ( u + v ) A ( u ) (cid:105)−(cid:104)A ( u ) (cid:105) with the Airy pro-cess A ( u ) [25–27]. To see this, we take ∆ t → t , ∆ C (cid:48) ( ζ, , t ) = C (cid:48) ( ζ, , t ) (cid:39) t/t ) − / (cid:2) g (0) − g (cid:0) (∆ t/t ) / ζ (cid:1)(cid:3) . This functionwith ∆ t/t = 0 .
006 is indicated by the dotted line inFig. 2(a) and accounts for the data with large ζ . In short, -3 -4 -2 -3 -3 -1 -2 -6 -4 -2 ζ C ’( ζ , ∆ t , t ) -2 ζ ’( l , ∆ t , t ) C stationarycorrelation g ( ζ ) – g (0) Airy correlation C ’( ζ , ∆ t , t ) li m ζ → C ’( ζ , ∆ t , t ) ζ ∼ ( ∆ t / t ) -4/3 g ’ (0)/2 ≈ ∆ t / t ∆ t / t li m ζ → ∞ (a) (b) (c) ’ ∼ ( ∆ t / t ) -4/3 ∆∆ ∆ g( ζ )short-time correlation[Eq.(7)] FIG. 2. (color online). Crossover in the correlation func-tion for the PNG model. (a) ∆ C (cid:48) ( ζ, ∆ t, t ) against ζ for∆ t/t = 0 . , . , , ,
100 (different sets of data; increas-ing from top to bottom) and t = 1 , , , , , correlation as described in Eq. (6), respec-tively, with ∆ t/t = 0 .
006 for the latter. The inset shows thedata for ∆ t/t = 0 .
006 without subtraction of C (cid:48) (0 , ∆ t, t ).(b,c) Asymptotics of ∆ C (cid:48) ( ζ, ∆ t, t ) for ζ → ∞ (b) and ζ → t = 10 for (b) and with various t for (c).The gray dots in (b) show the values of the right-hand side ofEq. (9), where (cid:104) ∆ q (cid:105) c for t = 10 and 100 is used as Q ( τ ). when ∆ t/t is small enough, C (cid:48) ( ζ, ∆ t, t ) (cid:39) g ( ζ ) ( ζ (cid:28) ζ c ) , ∆ tt ) − (cid:20) g (0) − g (cid:18) ( ∆ tt ) ζ (cid:19)(cid:21) ( ζ (cid:29) ζ c ) , (6)where the crossover length ζ c is defined by the intersec-tion of the two functions. If ∆ t/t is further decreased inFig. 2(a), the Airy branch moves away as (∆ t/t ) − / along both axes, leaving, asymptotically, only the sta-tionary correlation g ( ζ ) as expected. Alternatively, if C and l are rescaled by t / instead of ∆ t / , what remainsasymptotically is the Airy correlation. For tiny but fi-nite ∆ t/t , the two branches are connected by C (cid:48) (cid:39) ζ .We then study how the correlation function varies forlarge ∆ t/t . The data series in Fig. 2(a) show that∆ C (cid:48) ( ζ, ∆ t, t ) decreases with increasing ∆ t/t . In thelimit ζ → ∞ , since (cid:104) δh ( x + l, t + ∆ t ) δh ( x, t ) (cid:105) → C (cid:48) ( ζ, ∆ t, t ) → t ) − / C t (∆ t, t ) with C t (∆ t, t ) ≡ (cid:104) δh ( x, t + ∆ t ) δh ( x, t ) (cid:105) , i.e., the time cor-relation function. Despite the lack of analytical solution,its short-time behavior (∆ t/t (cid:28)
1) is given by C t (∆ t, t ) (cid:39) (Γ t t r ) (cid:104) χ (cid:105) c (cid:20) − (cid:104) χ (cid:105) c (cid:104) χ (cid:105) c (cid:18) − t t r (cid:19) (cid:21) (7)with t r ≡ t + ∆ t [9, 28, 29]. For ∆ t/t (cid:29)
1, numerical[29] and experimental [9] studies showed C t (∆ t, t ) (cid:39) (Γ t t r ) / F (∆ t/t ) with F ( τ ) ∼ τ − . They indicatelim ζ →∞ ∆ C (cid:48) ( ζ, ∆ t, t ) ∼ (cid:40) (∆ t/t ) − / , (∆ t/t (cid:28) , (∆ t/t ) − / , (∆ t/t (cid:29) , (8)and correctly account for the data [Fig. 2(b)]. Further,since the second-order cumulant of the rescaled heightdifference, Q (∆ t/t ), involves the two-point time corre-lation C t (∆ t, t ), we also obtain for arbitrary ∆ t/t lim ζ →∞ ∆ C (cid:48) ( ζ, ∆ t, t )= (cid:104) χ (cid:105) c (cid:20)(cid:18) t/t (cid:19) + (∆ t/t ) − (cid:21) − Q (∆ t/t ) . (9)This is also confirmed as shown by gray dots in Fig. 2(b).In contrast to the long-length limit, one cannot apriori predict how the short-length limit ζ → C (cid:48) ( ζ, ∆ t, t ) depends on ∆ t/t . The data in Fig. 2(a)suggest ∆ C (cid:48) ( ζ, ∆ t, t ) ∼ ζ for any ∆ t/t . Figure 2(c)shows that the coefficient of this quadratic term varies aslim ζ → ∆ C (cid:48) ( ζ, ∆ t, t ) ζ − = 12 ∂ C (cid:48) ∂ζ (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 (cid:39) (cid:40) g (cid:48)(cid:48) (0) / ≈ .
085 (∆ t/t (cid:28) ,c (∆ t/t ) − / (∆ t/t (cid:29) , (10)with a constant c and the second derivative g (cid:48)(cid:48) (0),which naturally arises since C (cid:48) ( ζ, ∆ t, t ) → g ( ζ ) for∆ t/t →
0. To examine the other limit, let us note ∂ C (cid:48) ∂ζ (cid:12)(cid:12)(cid:12) ζ =0 = ( A/ − (Γ∆ t ) / (cid:104) ∂h∂x ( x, t + ∆ t ) ∂h∂x ( x, t ) (cid:105) ,which is simply time correlation in the slope of the in-terface. It is suggestive that the short- and long-lengthlimits of ∆ C (cid:48) ( ζ, ∆ t, t ) are governed by the slope-slopeand height-height time correlations, respectively, decay-ing with the same power in the rescaled units [Eqs. (8)and (10)]. The results may also remind us of the space-like and time-like paths argued in the literature [30, 31],though precise relation is yet to be clarified.Finally, we test universality of the presented crossover,analyzing experimental data of growing interfaces in tur-bulent liquid crystal. While the readers are referred toRefs. [8, 9] for detailed descriptions, in this series of workthe author and a coworker studied planar evolution ofborders between two distinct regimes of spatiotemporalchaos, called the dynamic scattering modes 1 and 2, inthe electroconvection of nematic liquid crystal. The in-terfaces grow under high applied voltage, clearly exhibit-ing, besides the exponents, the distribution and correla-tion functions for the flat and curved KPZ-class interfaces -0.8-0.40 ∆ t / t -2 -4 -6 -2 ζ ∆ t / t ∆ t / t -2 ∆ t -2 s k e w n e ss 〈 ∆ q 〉 c 〈 ∆ q 〉 c 〈 ∆ q 〉 c 〈 ∆ q 〉 c ∆ t / t ∆ t / t C ’( ζ , ∆ t , t ) ∆ (a) (b)(c) ∆ t / t t -3-1 -1 (s) 〈 ∆ q 〉 c FIG. 3. (color online). Crossover in the liquid-crystal exper-iment. (a,b) Cumulants (a) and skewness (b) against ∆ t/t with t = 2 , , , , , ,
60 s (from right to left). Theinset shows (cid:104) ∆ q (cid:105) c against ∆ t . The dotted and dashed linesindicate the values for the Baik-Rains F and GOE-TW dis-tributions, respectively. The solid curves show the fitting tothe PNG data obtained in Fig. 1(b). (c) Rescaled correla-tion function ∆ C (cid:48) ( ζ, ∆ t, t ) against ζ for given t and ∆ t/t (∆ t/t increases from top to bottom). The red dashed andblack dotted lines indicate the stationary and Airy correla-tion functions as described in Eq. (6), respectively, the latterbeing set with ∆ t/t = 0 . t/t . [8, 9]. Here, we employ the data for 1128 flat interfacesused in Ref. [9] and perform the crossover analyses de-veloped in the present study.Figure 3 shows the results. The n th-order cumulants ofthe rescaled height difference ∆ q [Eq. (4)] with various t , which sufficiently fall apart as functions of ∆ t [see,e.g., inset of Fig. 3(a)], collapse reasonably well whenplotted against ∆ t/t [Fig. 3(a)], despite a rather strongfinite-time effect for n ≥
2. The collapsed data are foundasymptotically on top of the fitting curves obtained forthe PNG model, Q n (∆ t/t ) (black solid lines). This im-plies that Q n ( τ ) are universal functions of the KPZ classdescribing the crossover in question, and so is the dis-tribution function of ∆ q parametrized by ∆ t/t . Theundershoot in the skewness is also confirmed experimen-tally [Fig. 3(b)], while it was not clearly identified for thekurtosis because of larger statistical error (not shown).Moreover, extrapolation of the finite-time corrections inthe cumulants allows us to roughly estimate the timeneeded for direct observation of the Baik-Rains F dis-tribution, longer than 10 s here, which is unfortunatelyunreachable in the current setup [8, 9].The results on the correlation function are also repro-duced experimentally [Fig. 3(c)]. The functional formis parametrized solely by ∆ t/t (see two data sets for∆ t/t = 10 − overlapping with each other) and agreesvery well with the one obtained for the PNG model (blacksolid lines). In particular, the crossover between the sta-tionary and Airy correlations [Eq. (6)] is clearly con-firmed for small enough ∆ t/t (top yellow data set).In summary, we have studied the flat-stationarycrossover in the KPZ class, which takes place graduallyin time. Analyzing numerical and experimental data, wehave found and determined universal functions describ-ing the cumulants and the two-point correlation duringthis crossover. These functions show multifaceted rela-tions to the analytically unsolved time correlation, andhence may provide an important clue toward its solution.Seeking a possible connection to analogous, mathemati-cally tractable crossover in space [5] is another interestingissue left for future studies. Besides such fundamentalimportance, our results also answer a practical questionof how interfaces realized in experiments and simulationsapproach the stationary regime, which is never attainedwithout full control on the initial condition.The author acknowledges enlightening suggestions byT. Sasamoto and H. Spohn during the MSRI workshop in2010 “Random Matrix Theory and its Applications II,”which gave birth to the present work. Fruitful discus-sions with them and T. Imamura are also appreciated,as well as a remark by Y. Nakayama on numerical im-plementation of the PNG model. Further, the authorthanks M. Pr¨ahofer for providing the theoretical curve ofthe GOE-TW distribution, T. Imamura for those of theBaik-Rains F distribution and the stationary correlationfunction g ( ζ ), and F. Bornemann for that of the Airy correlation function g ( ζ ) [32]. This work is supported inpart by Grant for Basic Science Research Projects fromThe Sumitomo Foundation. ∗ [email protected][1] A.-L. 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