Maximal and minimal height distributions of fluctuating interfaces
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Maximal and minimal height distributions of fluctuatinginterfaces
T. J. Oliveira a ) and F. D. A. Aar˜ao Reis b ) ∗ Instituto de F´ısica, Universidade Federal Fluminense,Avenida Litorˆanea s/n, 24210-340 Niter´oi RJ, Brazil (Dated: November 1, 2018)
Abstract
We study numerically the maximal and minimal height distributions (MAHD, MIHD) of thenonlinear interface growth equations of second and fourth order and of related lattice modelsin two dimensions. MAHD and MIHD are different due to the asymmetry of the local heightdistribution, so that, in each class, the sign of the relevant nonlinear term determines which oneof two universal curves is the MAHD and the MIHD. The average maximal and minimal heightsscale as the average roughness, in contrast to Edwards-Wilkinson (EW) growth. All extremeheight distributions, including the EW ones, have tails that cannot be fit by generalized Gumbeldistributions.
PACS numbers: 68.35.Ct, 68.55.Jk, 81.15.Aa, 05.40.-a ∗ a) Email address: [email protected]ff.brb) Email address: [email protected]ff.br n th extrema is described bythe Gumbel’s first asymptotic distribution [1, 5] if the probability density functions (PDF)of those sets decrease faster than a power law. However, deviations from this statistics areexpected in fluctuating interfaces if there are strong correlation of local heights. In onedimension, this is the case of Edwards-Wilkinson (EW) interfaces (Brownian curves) [6, 7]and other Gaussian interface models [8]. On the other hand, fluctuations of other globalquantities in various physical systems follow generalized Gumbel distributions [9, 10, 11],i. e. the first asymptotic distribution with noninteger n values. This is explained by theconnections between the EVS of correlated variables and sums of independent variablesdrawn from exponential PDF [12], which shows that Gumbel statistics goes far beyond thedescription of uncorrelated variables sets.The interface models where maximal height distributions (MAHD) were previously cal-culated [6, 7, 8] are symmetric with respect to the average height, consequently MAHD andminimum height distributions (MIHD) are the same. However, there is a large number ofreal interfaces where the up-down symmetry is broken [13], such as those described by thenonlinear growth models of Kardar-Parisi-Zhang (KPZ) [14] and of Villain-Lai-Das Sarma(VLDS) for molecular beam epitaxy [15], which raises the question whether MAHD andMIHD are the same in those systems. This is particularly important for two-dimensionalinterfaces due to the variety of real growth processes which show KPZ [16, 17] and VLDSscaling [18]. Recent works on persistence in VLDS growth also motivate such study, sincedifferent exponents for positive and negative height persistence were obtained [19]. More-over, the above scenario raises additional (and not less important) questions for the nonlinearmodels. The first one is connected to the possibility of fitting their extreme height distri-butions (EHD) by generalized Gumbel distributions, similarly to other correlated systems.The second one is the scaling of the average maximal height, since EW interfaces showedan unanticipated scaling as the square of the average roughness [11], in contrast to severalone-dimensional interfaces. This is essential to correlate surface roughness with the extremeevents.The aim of this letter is to address those questions by performing a numerical study of2he MAHD and MIHD in the steady states of the KPZ and VLDS equations and of variouslattice models belonging to those classes in 2 + 1 dimensions. We will show that, for eachgrowth class, two universal distributions are obtained, which may be a MAHD or a MIHDof a given model depending on the sign of the coefficient of the relevant nonlinear term.Combination of data collapse and extrapolation of amplitude ratios (e. g. skewness andkurtosis) of those distributions are used to separate systems with coefficients of differentsigns. In order to illustrate the drastic effects that asymmetric PDF (i. e. distributions oflocal heights) may have on MAHD and MIHD, we will discuss their differences in a randomdeposition-erosion model on an inert flat substrate. We will also show that average maximaland minimal heights is all those models scale as the average roughness, as usually expected,which shows that the EW scaling is an exception [11]. Finally, we will show that KPZ,VLDS and EW distributions cannot be fit by generalized Gumbel distributions.MAHD and MIHD were calculated for the KPZ equation ∂h∂t = ν ∇ h + λ ( ∇ h ) + η ( ~x, t )[ h η ( ~x, t ) η ( ~x ′ , t ′ ) i = Dδ d ( ~x − ~x ′ ) δ ( t − t ′ )] in dimension d = 2, with ν = 0 . D = 5 × − and λ = √
75 ( g ≡ λ D/ν = 24), in discretized boxes with spatial step ∆ x = 1, timeincrement ∆ t = 0 .
04 and linear sizes 8 ≤ L ≤
64. A simple Euler integration method [20]and a scheme for suppression of instabilities [21] were adopted. We also simulated threediscrete KPZ models in sizes 32 ≤ L ≤ λ > λ < ∂h∂t = − ν ∇ h + λ ∇ ( ∇ h ) + η ( ~x, t )was integrated with ν = 1, λ = 1, D = 1 /
2, and ∆ t = 0 .
01, using the same methods, insizes 8 ≤ L ≤
32. We also simulated a generalized conserved RSOS model (CRSOS) [26],whose original version was proposed in Ref. [27]) and which belongs to the VLDS class, insizes 16 ≤ L ≤ λ = 0) was integrated with ν = 1 . D = 1 / t = 0 .
01 in box sizes 8 ≤ L ≤
64. For each model and each lattice size,distributions with at least 10 different configurations were obtained to ensure high accuracy,which is particularly important at their tails. The extremes were calculated relatively tothe average height of each configuration, the minima being absolute values of the differencesfrom the average.In Fig. 1a we show the scaled MAHD and MIHD of the KPZ equation in box size L = 64.In these plots, P ( m ) dm is the probability that the extreme lies in the range [ m, m + dm ],3 ≡ ( m − h m i ) /σ and σ ≡ (cid:16) h m i − h m i (cid:17) / . The high accuracy in Fig. 1a allows us todistinguish those curves (log-linear plots also show discrepancies in the right tails). Resultsfor smaller box sizes show that the finite-size effects are negligible, confirming that MAHDand MIDH are actually different.Gumbel’s first asymptotic distribution used to compare our data is g ( x ; n ) = ω exp (cid:16) − n h e − b ( x + s ) + b ( x + s ) i(cid:17) , where b = q ψ ′ ( n ), s = [ln n − ψ ( n )] /b and ω = n n b/ Γ ( n ), with Γ ( x ) the Gamma function and ψ ( x ) = ∂ ln Γ ( x ) /∂x [9, 11]. In Fig. 1b,we show the MAHD of the KPZ equation and the Gumbel curve with the same skewness0 .
79, which has n = 1 .
95. Although the fit near the peak is reasonable, there are significantdifferences in the tails. Data for different lattice sizes in Fig. 1b show that discrepanciesare not consequence of finite-size effects. Similar disagreement is observed when we try tofit the MIHD with a Gumbel curve of skewness 0 .
65 ( n = 2 . m , similarlyto the Gumbel curves.In Fig. 2a we show the scaled MAHD of the integrated KPZ equation and of BD, andthe MIHD of the RSOS model. In Fig. 2b we show the MIHD of the KPZ equation and ofthe etching model, and the MAHD of the RSOS model. There is excellent data collapse inboth plots, which confirm that MAHD (MIHD) of models with λ > λ <
0. This illustrates the possibility of using the MAHD andMIHD to identify the sign of the coefficient of the nonlinear term in cases where it is notknown a priori. At this point, EVS is superior to the scaling of the local height distributions(the PDF), which might also reveal the sign of the nonlinear terms if distortion by hugefinite-size effects were not so frequent [28].The visual agreement between those distributions and the small finite-size effects arequantitatively confirmed by estimates of their skewnesses and kurtosis in various systemsizes. Figs. 3a and 3b show the skewness of the same models of Figs. 2a and 2b, respectively,as a function of 1 /L / (BD data were not shown in Figs. 3a and 3b because they superimposethe etching model data). The small finite-size dependence of the data for the KPZ equation,BD and the etching models leads to S ≈ .
79 for ( λ > λ < S ≈ .
65 for ( λ > λ < L = 32 are plotted in Fig. 4a.Differences in the peaks are tiny, but discrepancies in the tails are clearly observed. Finite-size effects are also negligible in this case. The MAHD has skewness S ≈ .
63 and the MIHDhas S ≈ .
55. In contrast to the KPZ models, those distributions have Gaussian-shapedright tails [exp ( − m )]. For this reason, fits with generalized Gumbel distributions (whichtend to simple exponentials as m → ∞ ) are not possible. This is confirmed for MAHD inFig. 4a by comparison with Gumbel’s curve with n = 2 .
90, which has skewness 0 . λ > q > / − q of deposition, and assume thaterosion is possible only if h >
0. A steady state is attained with average height (relativelyto the substrate) h h i = q/ (2 q −
1) and PDF P ( h ) ∝ exp ( − h/ h h i ) (for q close to 1 / L (independent) columns. Withthat PDF, MAHD is given by Gumbel’s first asymptotic distribution with n = 1. Forlarge L , the minimum absolute height is typically at the substrate, thus fluctuations of therelative minima are dominated by fluctuations of the average height, which are Gaussian[ ∝ exp (cid:16) − m /a √ L (cid:17) , a ≡ q − √ q (1 − q ) ], and so it is the MIHD. The difference between MAHDand MIHD is easily confirmed by visual inspection of the plots of these functions. It isrelated to the highly asymmetric local height distribution of this model (skewness of PDFis S P DF = 2), in contrast to the slight asymmetry of KPZ ( S P DF ≈ .
26 [28]) and VLDS( S P DF ≈ .
20 [26]).Now we analyze the average values of extremes of KPZ and VLDS interfaces. They areassumed to scale as h m i ∼ L α m , while the average roughness scales with the roughness expo-nent α . We estimate α m by extrapolation of effective exponents α m ( L ) ≡ ln [ h m i ( L ) / h m i ( L/ ,as shown in Figs. 5a (KPZ) and 5b (VLDS). The estimates of α m are consistent with the bestknown estimates of the roughness exponents α ≈ .
39 (KPZ) [28] and α ≈ .
67 (VLDS)526]. Consequently, the average values of extremes scale as the average roughness in theKPZ and VLDS classes in 2 + 1 dimensions. This contrasts with the scaling as the squaredroughness in the EW class [11], which we also confirmed by simulation.Finally, in Fig. 6 we show the MAHD of the EW equation and the generalized Gumbeldistribution with n = 2 .
6, which has the same skewness. Despite the good agreement inalmost three decades of the scaled P ( m ), the discrepancy in the tails is clear. Again, datafor two box sizes ( L = 64 and L = 32) show that this is not caused by finite-size effectsnor to low accuracy of the data. On the other hand, the analytical prediction by Lee [11]of a Gaussian-shaped tail [ ∼ exp ( − m )] of the MAHD is confirmed by the trend of ourdata for large m . Together with the above results for KPZ and VLDS classes, it shows thatEVS of important interface growth models in two dimensions are not connected to the EVSof independent variables, despite the wide applicability of Gumbel statistics to correlatedsystems.In summary, we showed that interface growth models with asymmetric local height dis-tributions have different maximal and minimal height distributions, the most importantexamples being the KPZ and the VLDS classes in two dimensions. In each class, a pair ofuniversal curves may be maximal of minimal height distributions depending on the sign ofthe relevant nonlinear term. The average maximal and minimal heights of KPZ and VLDSmodels scale as the average roughness, in contrast to the EW class. All extreme heightdistributions, including the EW ones, cannot be fit by generalized Gumbel distributions.Although most works on statistical properties of interfaces focus on features of height dis-tributions and/or roughness scaling [13], recent studies show that the statistics of globalquantities are very useful to characterize real growth processes [2, 30]. The EVS has thesame advantages of roughness distribution scaling for this task, such as weak finite-size ef-fects, and also reveals the sign of the nonlinear terms. Information on rare events is alsoessential in systems where drastic changes in the dynamics occur if the global minima ormaxima attain certain values, such as in corrosion damage. On the other hand, for someapplications (from friction to parallel computing) the distributions of local extremes may beimportant, and the present study certainly motivates additional studies of those quantities[31, 32]. 6 cknowledgments TJO acknowledges support from CNPq and FDAAR acknowledges support from CNPqand FAPERJ (Brazilian agencies). [1] E. J. Gumbel,
Statistics of Extremes (Columbia University, New York, 1958).[2] S. T. Bramwell et al, Nature (London) , 552 (1998).[3] J. P. Bouchaud and M. M´ezard, J. Phys. A , 7997 (1997); R. W. Katz et al, Adv. WaterResour. , 1287 (2002); S. T. Bramwell et al, Europhys. Lett. , 310 (2002); J. F. Eichneret al, Phys. Rev. E , 016130 (2006); S. Redner and M. R. Petersen, Phys. Rev. E , 061114(2006).[4] G. Engelhardt and D. D. Macdonald, Corros. Sci. , 2755 (2004).[5] R. A. Fisher and L. A. Tippett, Proc. Cambridge Philos. Soc. , 180 (1928).[6] S. Raychaudhuri et al, Phys. Rev. Lett. , 136101 (2001).[7] S. N. Majumdar and A. Comtet, Phys. Rev. Lett. , 225501 (2004); J. Stat. Phys. , 777(2005).[8] G. Gy¨orgyi et al, Phys. Rev. E , 056116 (2003); G. Schehr and S. N. Majumdar, Phys. Rev.E , 056103 (2006); G. Gy¨orgyi et al, Phys. Rev. E , 021123 (2007).[9] S. T. Bramwell et al, Phys. Rev. Lett. , 3744 (2000).[10] T. Antal et al, Phys. Rev. Lett. , 240601 (2001).[11] D.-S. Lee, Phys. Rev. Lett. , 150601 (2005).[12] E. Bertin, Phys. Rev. Lett. , 170601 (2005); E. Bertin and M. Clusel, J. Phys. A , 7607(2006).[13] A.L. Barab´asi and H.E. Stanley, Fractal concepts in surface growth (Cambridge UniversityPress, Cambribge, England, 1995).[14] M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. , 889 (1986).[15] J. Villain, J. Phys. I , 19 (1991); Z.-W. Lai and S. Das Sarma, Phys. Rev. Lett. , 2348(1991).[16] J. Krim and G. Palasantzas, Int. J. Mod. Phys. B , 599 (1995).[17] R. Paniago et al, Phys. Rev. B , 13442 (1997); M. C. Salvadori et al, Phys. Rev. E , 6814 , 4638 (1999); A. E. Lita and J. E. Sanchez, Jr., Phys. Rev. B , 1801 (2001).[18] Y.-L. He et al, Phys. Rev. Lett. , 3770 (1992); C. Thompson et al, Phys. Rev. B , 4902(1994); D. C. Law et al, J. Appl. Phys. , 508 (2000).[19] M. Constantin et al, Phys. Rev. E , 061608 (2004).[20] K. Moser et al, Physica A , 215 (1991).[21] C. Dasgupta et al, Phys. Rev. E , 2235 (1997).[22] J. M. Kim and J. M. Kosterlitz, Phys. Rev. Lett. , 2289 (1989).[23] M. J. Vold, J. Coll. Sci. (1959) 168; J. Phys. Chem. , 1608 (1959).[24] B. A. Mello et al, Phys. Rev. E , 041113 (2001).[25] W. E. Hagston and H. Ketterl, Phys. Rev. E , 2699 (1999).[26] F. D. A. Aar˜ao Reis, Phys. Rev. E , 031607 (2004); F. D. A. Aar˜ao Reis and D. F. Frances-chini, Phys. Rev. E , 3417 (2000).[27] Y. Kim et al, J. Phys. A: Math. Gen. , L533 (1994).[28] E. Marinari et al, J. Phys. A , 8181 (2000); F. D. A. Aar˜ao Reis, Phys. Rev. E , 021610(2004).[29] S.-C. Park et al, Phys. Rev. E , 015102(R) (2002).[30] S. Moulinet et al, Phys. Rev. E , 035103(R) (2004).[31] Z. Toroczkai et al, Phys. Rev. E , 276 (2000).[32] F. Hivert et al, J. Stat. Phys. , 243 (2007). x σ P ( m ) -4 -2 0 2 4 6 8 x -6 σ P ( m ) a)b) FIG. 1: (Color online) a) Scaled MAHD (solid curve) and MIHD (dashed curve) of the KPZequation in box size L = 64. b) Scaled MAHD of the KPZ equation (solid curve for L = 64,squares for L = 32) and generalized Gumbel distribution with n = 1 .
95 (dashed curve). x -6 -4 -2 σ P ( m ) -4 -2 0 2 4 6 8 x -6 -4 -2 σ P ( m ) b)a) FIG. 2: (Color online) a) Scaled MAHD of the KPZ equation (solid curve) and BD (triangles),and MIHD of the RSOS model (squares). b) Scaled MIHD of the KPZ equation (solid curve) andetching model (triangles), and MAHD of the RSOS model (squares). For KPZ equation, box sizeis L = 64, and for discrete models L = 256. .0 0.1 0.2 0.3 0.4 L S S a)b) FIG. 3: a) Finite-size dependence of the skewness S of MAHD of the KPZ equation (squares) andetching model (triangles), and of MIHD of the RSOS model (crosses). b) Finite-size dependenceof S of MIHD of the KPZ equation (squares) and etching model (triangles), and of MAHD of theRSOS model (crosses). In both plots, the variable in the abscissa was chosen to make clearer theevolution of the data as L → ∞ . .0 0.1 0.2 0.3 L S -4 -2 0 2 4 6 x -6 -4 -2 σ P ( m ) a)b) FIG. 4: (Color online) a) Scaled MAHD (solid curve) and MIHD (dashed curve) of the VLDSequation in L = 32 and generalized Gumbel distribution with n = 2 .
90 (dotted curve). b) Finite-size dependence of the skewness S of EHD of the VLDS equation (crosses for MAHD, circles forMIHD) and of the CRSOS model (squares for MAHD, triangles for MIHD). The variable 1 /L / provides the best linear fits of the data (dashed lines). L ∆ α m ( L ) α m ( L ) a)b) FIG. 5: Finite-size dependence of effective exponents α m ( L ) of discrete models: a) KPZ class (BD:crosses; etching: triangles; RSOS: squares) and b) VLDS class (CRSOS model). L is the averagesize among L and L/
2. The variables in the abscissa provide the best linear fits (dashed lines) withexponents ∆ = 1 / / -4 -2 0 2 4 6 8 10 x -6 -4 -2 σ P ( m ) FIG. 6: (Color online) Scaled MAHD of the EW equation in L = 64 (solid curve) and L = 32(squares) and the generalized Gumbel distribution with n = 2 .6 (dashed curve).