Universal fluctuations in KPZ growth on one-dimensional flat substrates
We present a numerical study of the evolution of height distributions (HDs) obtained in interface growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. The growth is done on an initially flat substrate. The HDs obtained for all investigated models are very well fitted by the theoretically predicted Gaussian Orthogonal Ensemble (GOE) distribution. The first cumulant has a shift that vanishes as , while the cumulants of order converge to GOE as or faster, behaviors previously observed in other KPZ systems. These results yield a new evidence for the universality of the GOE distribution in KPZ growth on flat substrates. Finally, we further show that the surfaces are described by the Airy process.
pacs:68.43.Hn, 68.35.Fx, 81.15.Aa, 05.40.-a
The last decade witnessed a great advance in the understanding of paradigmatic nonequilibrium models for interface growth by means of rigorous analytical results Kriecherbauer and Krug (2010); Sasamoto and Spohn (2010a) as well as their experimental realizations Takeuchi and Sano (2010); Takeuchi et al. (2011). Among the most prominent examples are several models belonging to the Kardar-Parisi-Zhang (KPZ) universality class introduced by the stochastic equation Kardar et al. (1986)
that describes the evolution of an interface . In this equation, is a white noise defined by and . In , the KPZ equation yields self-affine surfaces with an interface width in a scale of length , defined as , given by the Family-Vicsek scaling ansatz Family and Vicsek (1985): , where for and for . In , the exponents of the KPZ class are and Kardar et al. (1986). Several models with the scaling exponents of the KPZ universality class have been reported Barabasi and Stanley (1995); Krug (1997).
For system belonging to the KPZ class in , the Family-Vicsek scaling suggests an interface height evolving as Kriecherbauer and Krug (2010)
where and are non-universal (model dependent) parameters, and is a time independent random variable. The height distributions (HDs) and consequently were computed exactly for some models in the KPZ class (see Refs. Sasamoto and Spohn (2010a); Kriecherbauer and Krug (2010) for recent reviews), strongly suggesting that is a universal feature of the KPZ class. In a pioneer work, Johansson Johansson (2000) established a link between random matrix theory and KPZ class by determining the exact asymptotic HDs of the totally asymmetric exclusion process (single step model) as the Tracy-Widom (TW) distribution of the Gaussian unitary ensemble (GUE) Tracy and Widom (1994). In the same year, Prähofer and Spohn Prähofer and Spohn (2000a, b) determined the HDs of the radial polynuclear growth (PNG) model as a GUE distribution whereas the Gaussian orthogonal ensemble (GOE) distribution was obtained for the growth on a flat substrate. Quite recently, exact solutions of the KPZ equation in corroborated GOE distributions for flat Calabrese and Le Doussal (2011) and GUE for radial Sasamoto and Spohn (2010b); Amir et al. (2011) geometries, respectively.
Besides TW distributions for the heights, the limiting processes describing the surfaces were identified as the so-called Airy processes Kriecherbauer and Krug (2010). For circular growth, analytic solutions Prähofer and Spohn (2002), experiments Takeuchi and Sano (2010), and simulations Alves et al. (2011) strongly suggest that the limiting process ruling the KPZ universality class is given by the dynamics of the largest eigenvalue in Dyson’s Brownian motion of GUE matrices, the Airy process. Analogously, a limiting process for the flat case, named Airy, was found for PNG Borodin et al. (2008) and a similar model Sasamoto (2005), both belonging to the KPZ class, indicating that it may be a universal feature of the KPZ class. It is worth mentioning that Airy process does not correspond to the dynamics of largest eigenvalue in Dyson’s Brownian motion of GOE matrices Bornemann et al. (2008).
KPZ universality class in 1+1 dimensions was convincingly verified in a meticulous experiment involving two turbulent phases in the electroconvection of nematic liquid crystal films Takeuchi and Sano (2010); Takeuchi et al. (2011). Using an unprecedented statistics for experimental systems ( realizations), it was possible to observed not only the KPZ scaling exponents and , but also the GUE and GOE distributions for radial and flat geometries, respectively. In both cases, cumulants of order and 4 converge fast to the corresponding GOE and GUE cumulants, but the mean () approaches the theoretical value as a power law . This decay was also found in an analytical solution of the KPZ equation in with an edge initial condition Sasamoto and Spohn (2010b) and in simulations of radial Eden model Alves et al. (2011). However, Ferrari and Frings Ferrari and Frings (2011) have shown the non-universality of the amplitudes of this correction and that higher order cumulants have no correction up to order ).
The analysis of height distributions in computer simulations of non-solvable models, supposedly belonging to the KPZ class, are relatively less frequent than the analytical counterpart of solvable models. Indeed, some reports for flat geometries dating from the beginning of nineties, when computer resources were quite limited, demonstrate a surprisingly good agreement with the lately found out theoretical GOE distribution Krug et al. (1992); Amar and Family (1992). For a radial geometry, we have recently observed asymptotic GUE distributions in computer simulations of distinct Eden models grown from a single seed Alves et al. (2011).
In the present work, we revisit the computer simulations of non-solvable models having the KPZ exponents in dimensions. A flat initial condition is used. We also integrated the KPZ equation numerically. In all cases, we confirmed that HDs and cumulants agree with the GOE distribution. As recently observed for other systems Takeuchi and Sano (2010); Takeuchi et al. (2011); Sasamoto and Spohn (2010b), a correction in the mean approaching zero as was found while corrections in higher order cumulants consistent with were obtained for three of four investigated models. These results are in agreement with the corrections obtained analytically for solvable KPZ models Ferrari and Frings (2011). Finally, the covariance of all investigated models agrees with the Airy process, as verified for some solvable models Sasamoto (2005); Borodin et al. (2008).
We study the restricted solid-on-solid (RSOS) and ballistic deposition (BD) models Barabasi and Stanley (1995) as well as the numerical integration of Eq. (1) for two sets of parameters. In all cases, we use one-dimensional lattices of size and periodic boundary conditions. Statistical averages were performed over at least 400 independent samples. The RSOS model consists in the deposition of a particle in a randomly chosen site. The deposition is rejected everytime the height difference between nearest neighbors is greater than one lattice unity. In the BD model, the deposition site is randomly chosen and the particle falls normally to the substrate. When the first contact with the deposit happens, it irreversibly sticks in this position.
We investigated a discrete one-dimensional KPZ equation given by Moser et al. (1991)
where and is a random number uniformly distributed in the interval . Integration instabilities are prevented by replacing the gradient term in KPZ equation by where is a parameter to be adjusted to control the instability Dasgupta et al. (1996). Integration was performed for fixed parameters , and . Two sets of parameters were studied. In the first one, named as KPZ I, we used , the coupling constant , and . In the second one, the KPZ II, we used , and .
Comparison between HDs and random matrix distributions needs highly accurate estimates of the parameters and given by Eq. (2). In BD and RSOS models studied here, estimates are reported in literature Krug et al. (1992), but the accuracy is not sufficient for the aims of the present work. Thus, we determined the parameters for all studied models. Using Eq. (2), the asymptotic growth velocity can be obtained from the extrapolation of plots against when , as illustrated in Fig. 1 for the BD model. The asymptotic velocities for all investigated models are presented in Table 1. Notice that our estimates for RSOS and BD models are consistent with those formerly reported in Ref. Krug et al. (1992): and . The parameter was obtained from the second order cumulant of the HDs. Accordingly Eq. (2), it reads as , where is the th cumulant of . Since we expect that fluctuations are given by the GOE distribution, we used Prähofer and Spohn (2000a) in our calculations. The insertion to Fig. 1 shows as a function of time for the BD model. The estimated values for all investigated models are presented in Table 1. Plots similar to Fig. 1 were obtained for all models.
We study the rescaled HDs following the KPZ ansatz given by Eq. (2). If this equation holds, the fluctuations of would be given by the GOE distribution. Fig. 2 shows the rescaled distributions for all investigated models. As expected, the agreement with GOE is noticeable, but the distributions are slightly shifted in relation to GOE, as highlighted in the inset of Fig. 2.
The absolute differences between and GOE cumulants, , are shown as functions of time in Fig. 3. The first cumulant difference decays closely to a power law for all investigated models, except for KPZ I. In the latter, the initial decay is faster but seems to converge to . Ferrari et al. Ferrari and Frings (2011) have shown that the corrections in the first cumulant for a number of solvable models belonging to the KPZ class read as
where is a model dependent constant that can even be null for some models. Numerically, if the factor is much smaller than the factor of higher order correction, one must observe a fast initial decay  followed by a decay. This is observed in the KPZ I results shown in Fig. 3(e). For KPZ I, the higher order cumulants reach GOE very fast and then the differences fluctuate around zero. Furthermore, the non-universality of the constant is verified in our simulation since is negative for BD and RSOS and positive for the KPZ equation integration. Ferrari et al Ferrari and Frings (2011) have also shown that the higher order cumulants have no correction up to order . Our simulations are in agreement with this claim as one can see in Figs. 3(b)-3(e), where some curves are consistent with a decay while others decay faster, which is particularly evident in KPZ I case. The conclusion of Ref. Ferrari and Frings (2011) stating the non-universality of the correction refers only to the amplitudes since the scenario suggested by Eq. (4) is quite general: any correction to the first moment, if it exists, should decay with an exponent , while corrections in higher order cumulants, if they exist, must decay as fast as or faster than .
Besides the universality of the height fluctuations, the limiting process describing the interface is an important issue on surface growth. To check if our simulations yield surfaces given by an Airy processes, we studied the two-point correlation function
that is expected to scale as with , where is the covariance of the Airy process and Prähofer and Spohn (2002). The amplitude can be obtained from the squared local roughness
where means averages over windows of size Krug et al. (1992); Takeuchi and Sano (2010). Fig. 4(a) shows the plots used to determine the amplitudes shown in Table 1. The short plateau for the BD model is due to strong finite time effects of this model. Notice that the amplitude is given by for the KPZ equation, implying that and are expected for KPZ I and II, respectively. The slightly different amplitudes observed are due to the approximated integration method Lam and Shin (1998). Figure 4(b) shows the results for the rescaled two-point correlation function exhibiting a very satisfactory agreement among all models and the covariance of the Airy process (). This confirms that all models are described by the Airy process and gives one more evidence of this universal feature in KPZ universality class. Similar results were obtained at different times, indicating small finite-time effects in the two-point correlation function.
In conclusion, we investigated non-solvable models in the KPZ universality class in one-dimensional flat substrates as well as a numerical integration of the KPZ equation. We analyzed the height distributions and the two-point correlation function. Height fluctuations of all investigated models are described by GOE distributions as conjectured for the KPZ class for flat substrates, and previously observed in analytical Calabrese and Le Doussal (2011) and experimental Takeuchi et al. (2011) works. The cumulant analysis yields a correction vanishing as for the first cumulant whereas the corrections for higher order cumulants go to zero as or faster. The latter result is in agreement with theoretical analysis of Ferrari et al Ferrari and Frings (2011). Finally, we have found that the limiting processes describing the surface is the Airy process in agreement with previous results for solvable models Borodin et al. (2008); Sasamoto (2005).
Acknowledgements.This work was partially supported by the Brazilian agencies CNPq, FAPEMIG, and CAPES. Authors thank F. Bornemann by kindly providing the covariance of the Airy process.
- Kriecherbauer and Krug (2010) T. Kriecherbauer and J. Krug, J. Phys. A 43, 403001 (2010).
- Sasamoto and Spohn (2010a) T. Sasamoto and H. Spohn, J. Stat. Mech. 2010, P11013 (2010a).
- Takeuchi and Sano (2010) K. A. Takeuchi and M. Sano, Phys. Rev. Lett. 104, 230601 (2010).
- Takeuchi et al. (2011) K. A. Takeuchi, M. Sano, T. Sasamoto, and H. Spohn, Sci. Rep. (Nature) 1, 34 (2011).
- Kardar et al. (1986) M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56, 889 (1986).
- Family and Vicsek (1985) F. Family and T. Vicsek, J. Phys. A 18, L75 (1985).
- Barabasi and Stanley (1995) A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, England, 1995).
- Krug (1997) J. Krug, Advances in Physics 46, 139 (1997).
- Johansson (2000) K. Johansson, Commun. Math. Phys. 209, 437 (2000).
- Tracy and Widom (1994) C. Tracy and H. Widom, Commun. Math. Phys. 159, 151 (1994).
- Prähofer and Spohn (2000a) M. Prähofer and H. Spohn, Phys. Rev. Lett. 84, 4882 (2000a).
- Prähofer and Spohn (2000b) M. Prähofer and H. Spohn, Physica A 279, 342 (2000b).
- Calabrese and Le Doussal (2011) P. Calabrese and P. Le Doussal, Phys. Rev. Lett. 106, 250603 (2011).
- Sasamoto and Spohn (2010b) T. Sasamoto and H. Spohn, Phys. Rev. Lett. 104, 230602 (2010b).
- Amir et al. (2011) G. Amir, I. Corwin, and J. Quastel, Commun. Pure Appl. Math. 64, 466 (2011).
- Prähofer and Spohn (2002) M. Prähofer and H. Spohn, J. Stat. Phys. 108, 1071 (2002).
- Alves et al. (2011) S. G. Alves, T. J. Oliveira, and S. C. Ferreira, Europhys. Lett. 96, 48003 (2011).
- Borodin et al. (2008) A. Borodin, P. Ferrari, and T. Sasamoto, Commun. Math. Phys. 283, 417 (2008).
- Sasamoto (2005) T. Sasamoto, J. Phys. A 38, L549 (2005).
- Bornemann et al. (2008) F. Bornemann, P. Ferrari, and M. Prähofer, J. Stat. Phys. 133, 405 (2008).
- Ferrari and Frings (2011) P. Ferrari and R. Frings, J. Stat. Phys. 144, 1 (2011).
- Krug et al. (1992) J. Krug, P. Meakin, and T. Halpin-Healy, Phys. Rev. A 45, 638 (1992).
- Amar and Family (1992) J. G. Amar and F. Family, Phys. Rev. A 45, 5378 (1992).
- Moser et al. (1991) K. Moser, J. Kertész, and D. E. Wolf, Physica A 178, 215 (1991).
- Dasgupta et al. (1996) C. Dasgupta, S. Das Sarma, and J. M. Kim, Phys. Rev. E 54, R4552 (1996).
- Lam and Shin (1998) C.-H. Lam and F. G. Shin, Phys. Rev. E 57, 6506 (1998).