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Comment on “Parameter-free scaling for nonequilibrium growth processes” A. Kolakowska Department of Physics, The University of Memphis, Memphis, TN 38152 (Dated: January 23, 2015) 5 In the paper [Phys. Rev. E 79, 051605 (2009)] by Chou and Pleimling a claim is made that a 1 parameter-free scaling that gives data collapse for some simulation models would replace universal 0 Family-Vicsek(FV)scaling. Here,bygivingtheexplicitform ofthisscalingforcompetitivegrowth 2 models, it is shown that data collapse in this procedure is obtained by a shift-and-scale operator n that gives no information about stochastic dynamicsand has norelation with FV function. a J PACSnumbers: 81.15.Aa,64.60.Ht,05.20.-y,05.70.Np 2 2 Nonequilibrium sufrace-growth processes of Ref. [1] are In the language of ‘data collapse,’ Eq.(6) says that for ] SOSmodelsin(1+1)dimensionswithperiodicboundary t [0;t (p)]all thedataforall Landforall parealready h ∈ 1 and time t is a number of deposited monolayers to a in one curve w =c √t. In order to collapse the data for c 1 e substrate of L sites. At t = 0 the substrate is flat. The t (t1(p);+ ) it is required to simultaneously multiply m growth rule is a competitive growth model: “either RD t∈and w(t) b∞y different scale factors a(p,L) and b(p,L): - (active with probability q) or X (active with probability t t/a(p,L), and w(t) w(t)/b(p,L). (7) at p = 1 q),” where RD is random deposition and X is a → → − t process that builds correlations. For these models, the But,ifaffinetransformationsinEq.(7)givedatacollapse s . initial time evolution of the surface width w(t) has two for t (t1(p);+ ), when they are applied to w(t) = t ∈ ∞ a growthregimesbeforecross-overtimetosaturation[1,2]. c1√t for t [0;t1(p)] they will produce ‘data scatter.’ m Inprocesses“RDorX”foranyLandp,w(t)hasthree This is beca∈use a(p,L) = b(p,L) and the data for t 6 ∈ - evolution regimes, as shown in Fig.1 of Ref.[1]: [0;t1(p)] already follow one curve for any L and p. That d is, Family-Vicsek (FV) scaling produces data collapse in on t∈[0;t1]∪(t1;t2)∪[t2;+∞)=[0;+∞). (1) EW-scaling regime for t ∈ (t1;+∞) and destroys data coalescence in RD-growth regime for t [0;t ]. c First, w(t) obeys the RD universal power law: ∈ 1 [ Because of this, the questions are about the rationale t [0;t ]:w(t) tβ1, β =1/2. (2) for the data collapse shown in Ref.[1] and about a rele- 1 1 1 ∀ ∈ ∝ vance of the proposed parameter-free scaling to the uni- v Note, β isnot ascalingexponent[3]. Later,w(t)obeys: versal dynamic scaling. The answers are given below. 5 1 In Ref.[1], data are collapsed in two-step transforma- 1 5 t (t1;t2) : w(t) tβ2, (3) tionsperformedindividuallyforeachcurveofEq.(6),i.e., ∀ ∈ ∝ 5 t [t ;+ ) : w(t) Lα2, (4) for each set of points (t,w) representing one curve in- 2 0 ∀ ∈ ∞ ∝ dexed by p and L. In the first step, t is divided by t , 1 1. where β2 and α2 are scaling exponents (growth and and w(t) is divided by w1, and the log is taken of all 0 roughness, respectively) of the universality class of X. numbers. In this way intervals in Eq.(1) are mapped 5 For fixed L and p, values of w(t) are in the interval ontot t′′ ( ;0] (0;τ′′) [τ′′;+ )=( ;+ ), Xiv:1 [tww0h(;itsw1is2)in]l∝,atwer√rghvetea1rbleubistwy2fi[E0n=;qiwt.(ew22;]()ht.=2eI)nnc∝[0et;,hLweiαn1l2]igm∪beyint(ewEorq1af;l.l:(wa4r2)g.],eAwbfuhteterrfieRniwetfe1.[1L=],, mwbwehhareeprrλpeeei→dswτ′2′so′′enl==teo∈ctlweloo−dgg→∞(s(wut2c2wh//′tw′∪1th∈)1.)a.(t−TλI∞nwhe′;∪′t0h=i]en∪t1se,(er0cva;oa∞nwnldsd2′′]ti′ns′=tea−Ep(n−,q∞d.∞a(w5)n;′′wu∞aam2′′rr]ee-, 2 2 r multiplied by λ. This gives a w(t) [0;w ] (w ;w ]=[0;w ]. (5) ∈ 1 ∪ 1 2 2 w′′ w′ ( ;0] (0;1]=( ;1], (8) → ∈ −∞ ∪ −∞ In Eqs.(1) and (5), t and w are functions of L. But, ′′ ′ 2 2 t t ( ;0] (0;τ) [τ;+ ), (9) w(t) in Eq.(2) does not depend on L [3]. In Ref.[1] this → ∈ −∞ ∪ ∪ ∞ ′′ ′′ is seen in Fig.2a, where w(t) is for the model “RD or where τ = τ /w2 = 1/β. In effect, original w val- X”andXis‘randomdepositionwithsurfacerelaxation,’ ues in Eq.(5) are mapped onto Eq.(8), and t values are i.e., in Edwards-Wilkinson (EW) universality class. For mapped from Eq.(1) onto Eq.(9). The outcome of this the model “RD or EW,” the data can be summarized as mapping is one curve, regardless of the pair p and L. the family of curves parameterized by p and L: After transformation, RD-growth phase is in ( ;0] −∞ × ( ;0],correlation-growthphaseisin(0;τ) (0;1),and −∞ × c √t , t [0;t (p)] saturation-growthphase is in [τ;+ ) 1 .  1 ∈ 1 ∞ ×{ } w(t;L;p)= c tβ2 , t (t (p);t (L,p)) (6) For curves in Eq.(6) (shown here in the left Fig.1) the  2 1 2 c Lα2 , t∈[t (L,p);+ ], procedure of Ref.[1] is described by the operator Gˆ , 3 ∈ 2 ∞ (p,L)  where c , c , and c are constants. Gˆ :(x,y) (x′,y′), (10) 1 2 3 (p,L) −→ 2 7 Eq.(7) reflects universal dynamics of correlations. FV 1 6 β function summarizes a relation between scaling proper- β 2 2 0.5 tiesofgrowingsurfacesandsymmetrypropertiesofequa- 5 tionsthatdescribegrowthdynamics. InFVfunction,ar- w)4 w’) 0 β gumentandprefactorcontainexplicit informationabout og( β og( 1 the way the dynamics is affected by growth parameters. y=l3 1 y’=l-0.5 x-x 1 Physics wise, FV function provides explanationnot only 2 L=500, p=15/16 x x’= x -x1 β fordatacollapsebutfirstofallforthe universalshapeof L=500, p=1/2 -1 2 1 2 L=500, p=1/16 y-y w(t), in contrast to the scaling of Ref.[1]. 1 L=2000, p=15/16 y y’= y -y1 Proofthat parameter-freescaling is a dynamic scaling L=2000, p=1/2 -1.5 2 1 0 callsforshowingthatitconnectswithstochasticdynam- 0 5 10 15 20 -4 -2 0 2 4 6 8 x=log(t) x’=log(t’) ics, i.e., it must be shownthat Eqs.(11) give t1 and t2 as functions of L and p. But Eqs.(11) and (6) are equiva- FIG.1: (Coloronline)Domain(left)andimage(right)ofthe lent,hence,Eqs.(11)donotcontainanynewinformation shift-and-scale operator Gˆ . In the image, cross-over to abovethatinEq.(6): Parameter-freescalingdoesnotex- (p,L) saturationisat(τ =1/β,1). Thecurvesarefor“RDorEW” plainthe universalshape ofw(t). In orderto find t and 1 competitivegrowth processinEq.(6). Here,log(·)≡log10(·). t2 one must analyze the scale invariance of stochastic growth equation, as universal scaling functions express this invariance: Parameter-free scaling does not express ′ ′ ′ ′ where x = logt, y = logw, x = logt, and y = logw . dynamical scale invariance. Denotingx =logt andy =logw fori=1,2,standard i i i i algebra methods give the explicit form of Gˆ(p,L): Evenformodelswithp=1parameter-freescalingdoes not connect with dynamics. When in Eq.(7) of Ref.[1] a x x 1 y y ′ 1 ′ 1 heuristicparameterλissetλ=1,inorderto findw (L) x x = − , y y = − . (11) 2 → x2−x1β2 → y2−y1 and t2(L) one must perform finite-size scaling analysis. On the other hand, when w (L) and t (L) are known 2 2 Explicitly, Gˆ(p,L) = Sˆ(p,L) Tˆ(p) is the composition of beforehand, it is possible to verify if Eq.(7) of Ref.[1] ◦ the translation operator, Tˆ : (x,y) (x′′,y′′) = (x represents dynamic scaling, which shows that there are (p) → − x ,y y ), and the scale operator, Sˆ : (x′′,y′′) some examples of dynamics where the scaling of Ref.[1] 1 1 (p,L) (x′,y′−) = (x′′/β2, y′′ ), in (x,y)-plane. Thus, Gˆ →is maymapdirectlyonFVfunctionwhenheuristicallyλ= x2−x1 y2−y1 (p,L) 1. However, there is no physical reason to set λ = 1. a shift-and-scale operator that translates the curves to When λ = 1, then Eq.(7) of Ref.[1] does not give FV one position and adjusts the length of the correlation- 6 scaling function even if w (L) and t (L) are known. growth phase to √τ2+1, as shown here in Fig.1. Such 2 2 shift-and-scale operation is possible because each curve Perhaps Eq.(7) in Ref.[1] should have been supple- in this family carries one universal footprint: the initial mented by an explanation that we must always have RDtransient—whereeachcurvehasthesameoneslope λ=1andexplicitly knoww , w , t , andt asfunctions 1 2 1 2 of1/2butitslengthdependsonp—thatisfollowedbya ofLandpforamodel,inordertobeabletoidentifythe specificuniversalcorrelationphase,whereeachcurvehas parameter-freescalingofRef.[1]. Ofcoursethent andt 1 2 the same one slope of β2 and ends at saturation phase. must come from dynamical scale-invarianceanalysis and However,asimilarpicturemayormaynotoccurinother from FV function, because there is no other way to ob- competitive growthphenomena, so Gˆ(p,L) is not general. taincrossovertimes. ThereforetheclaimmadeinRef.[1] Moreover,as this explicit derivationof Gˆ shows,the that parameter-freescaling is ‘something more’ than FV (p,L) shift-and-scale operator in (x,y)-plane is not a dynamic scaling does not hold: To the contrary, parameter-free scaling. scalingis‘somethingless.’ Thisisbecausewhenweknow Data collapse by shift-and-scale operation is a nice il- FV function we can construct parameter-free scaling in lustration of the known fact that all systems in one uni- suchawaythatitreflectsthescaleinvarianceofstochas- versality class follow one universal curve. But it has no tic dynamics. But if we do not know the symmetries connection with finite-size dynamic scaling and with FV of the stochastic growth equation the data collapse via function. Affine scaling of interfaces such as the one in parameter-free scaling is meaningless. [1] Y.-L Chou and M. Pleimling, Phys. Rev. E 79, 051605 M. Horowitz et al, Phys. Rev.E 63, 066132 (2001). (2009). [3] In RD universality class, all curves w(t) for any L follow [2] A.Kolakowska,M.A.Novotny,andP.Verma,Phys.Rev. onlyonecurvew(t)=c√t,wherecisnonuniversal,i.e., c E 73, 011603 (2006); A. Kolakowska and M. A. Novotny, is model-dependent and does not depend on L. arXiv:cond-mat/0511688v3, (2006); C. M. Horowitz and E. V. Albano, J. Phys. A: Math. Gen. 34, 357 (2001); C.