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Miranda VG, Aarão Reis FDA. Numerical study of the Kardar-Parisi-Zhang equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:031134. [PMID: 18517356 DOI: 10.1103/physreve.77.031134] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2007] [Revised: 01/10/2008] [Indexed: 05/26/2023]
Abstract
We integrate numerically the Kardar-Parisi-Zhang (KPZ) equation in 1+1 and 2+1 dimensions using a Euler discretization scheme and the replacement of (nablah)(2) by exponentially decreasing functions of that quantity to suppress instabilities. When applied to the equation in 1+1 dimensions, the method of instability control provides values of scaling amplitudes consistent with exactly known results, in contrast to the deviations generated by the original scheme. In 2+1 dimensions, we spanned a range of the model parameters where transients with Edwards-Wilkinson or random growth are not observed, in box sizes 8< or =L< or =128 . We obtain a roughness exponent of 0.37< or =alpha< or =0.40 and steady state height distributions with skewness S=0.25+/-0.01 and kurtosis Q=0.15+/-0.1 . These estimates are obtained after extrapolations to the large L limit, which is necessary due to significant finite-size effects in the estimates of effective exponents and height distributions. On the other hand, the steady state roughness distributions show weak scaling corrections and evidence of stretched exponential tails. These results confirm previous estimates from lattice models, showing their reliability as representatives of the KPZ class.
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Affiliation(s)
- Vladimir G Miranda
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil.
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Katzav E, Schwartz M. What is the connection between ballistic deposition and the Kardar-Parisi-Zhang equation? PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:061608. [PMID: 15697382 DOI: 10.1103/physreve.70.061608] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2003] [Revised: 06/03/2004] [Indexed: 05/24/2023]
Abstract
Ballistic deposition (BD) is believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. In this paper we study the validity of this belief by rigorously deriving a continuum equation from the BD microscopic rules, which deviates from the KPZ equation. We show that in one dimension and in the presence of noise the deviation is not important. This is not the case in the absence of noise. In more than one dimension and in the presence of noise we obtain an equation that superficially seems to be a continuum equation but in which the symmetry under rotations around the growth direction is broken.
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Affiliation(s)
- Eytan Katzav
- School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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Buendia GM, Rikvold PA, Park K, Novotny MA. Low-temperature nucleation in a kinetic Ising model under different stochastic dynamics with local energy barriers. J Chem Phys 2004; 121:4193-202. [PMID: 15332967 DOI: 10.1063/1.1772358] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Using both analytical and simulational methods, we study low-temperature nucleation rates in kinetic Ising lattice-gas models that evolve under two different Arrhenius dynamics that interpose between the Ising states a transition state representing a local energy barrier. The two dynamics are the transition-state approximation [T. Ala-Nissila, J. Kjoll, and S. C. Ying, Phys. Rev. B 46, 846 (1992)] and the one-step dynamic [H. C. Kang and W. H. Weinberg, J. Chem. Phys. 90, 2824 (1989)]. Even though they both obey detailed balance and are here applied to a situation that does not conserve the order parameter, we find significant differences between the nucleation rates observed with the two dynamics, and between them and the standard Glauber dynamic [R. J. Glauber, J. Math. Phys. 4, 294 (1963)], which does not contain transition states. Our results show that great care must be exercised when devising kinetic Monte Carlo transition rates for specific physical or chemical systems.
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Affiliation(s)
- Gloria M Buendia
- Department of Physics, Universidad Simon Bolivar, Caracas 1080, Venezuela.
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Pal S, Landau DP, Binder K. Dynamical scaling of surface growth in simple lattice models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:021601. [PMID: 14524980 DOI: 10.1103/physreve.68.021601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2002] [Indexed: 05/24/2023]
Abstract
We present extensive simulations of the atomistic Edwards-Wilkinson (EW) and Restricted Edwards-Wilkinson (REW) models in 2+1 dimensions. Dynamic finite-size scaling analyses of the interfacial width and structure factor provide the estimates for the dynamic exponent z=1.65+/-0.05 for the EW model and z=2.0+/-0.1 for the REW model. The stochastic contribution to the interface velocity U due to the deposition and diffusion of particles is characterized for both the models using a blocking procedure. For the EW model the time-displaced temporal correlations in U show nonexponential decay, while the temporal correlations decay exponentially for the REW model. Dynamical scaling of the temporal correlation function for the EW model yields a value of z, which is consistent with the estimate obtained from finite-size scaling of the interfacial width and structure factor.
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Affiliation(s)
- S Pal
- Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802, USA
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Rikvold PA, Kolesik M. Microstructure and velocity of field-driven Ising interfaces moving under a soft stochastic dynamic. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:066113. [PMID: 16241310 DOI: 10.1103/physreve.67.066113] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2003] [Indexed: 05/04/2023]
Abstract
We present theoretical and dynamic Monte Carlo simulation results for the mobility and microscopic structure of (1+1)-dimensional Ising interfaces moving far from equilibrium in an applied field under a single-spin-flip "soft" stochastic dynamic. The soft dynamic is characterized by the property that the effects of changes in field energy and interaction energy factorize in the transition rate, in contrast to the nonfactorizing nature of the traditional Glauber and Metropolis rates "hard" dynamics). This work extends our previous studies of the Ising model with a hard dynamic and the unrestricted solid-on-solid (SOS) model with soft and hard dynamics. [P. A. Rikvold and M. Kolesik, J. Stat. Phys. 100, 377 (2000); J. Phys. A 35, L117 (2002); Phys. Rev. E 66, 066116 (2002).] The Ising model with soft dynamics is found to have closely similar properties to the SOS model with the same dynamic. In particular, the local interface width does not diverge with increasing field as it does for hard dynamics. The skewness of the interface at nonzero field is very weak and has the opposite sign of that obtained with hard dynamics.
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Affiliation(s)
- Per Arne Rikvold
- Center for Materials Research and Technology, School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4350, USA.
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Rikvold PA, Kolesik M. Microstructure and velocity of field-driven solid-on-solid interfaces: analytic approximations and numerical results. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066116. [PMID: 12513356 DOI: 10.1103/physreve.66.066116] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2002] [Indexed: 05/24/2023]
Abstract
The local structure of a solid-on-solid interface in a two-dimensional kinetic Ising ferromagnet or attractive lattice-gas model with single-spin-flip Glauber dynamics, which is driven far from equilibrium by an applied field or chemical potential, is studied by an analytic mean-field, nonlinear-response theory [P. A. Rikvold and M. Kolesik, J. Stat. Phys. 100, 377 (2000)], and by dynamic Monte Carlo simulations. The probability density of the height of an individual step in the surface is obtained, both analytically and by simulation. The width of the probability density is found to increase dramatically with the magnitude of the applied field, with close agreement between the theoretical predictions and the simulation results. Excellent agreement between theory and simulations is also found for the field dependence and anisotropy of the interface velocity. The joint distribution of nearest-neighbor step heights is obtained by simulation. It shows increasing correlations with increasing field, similar to the skewness observed in other examples of growing surfaces.
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Affiliation(s)
- Per Arne Rikvold
- Center for Materials Research and Technology, School of Computational Science and Information Technology, Florida State University, Tallahassee 32306-4350, USA.
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Giada L, Giacometti A, Rossi M. Pseudospectral method for the Kardar-Parisi-Zhang equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:036134. [PMID: 11909192 DOI: 10.1103/physreve.65.036134] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2001] [Indexed: 05/23/2023]
Abstract
We discuss a numerical scheme to solve the continuum Kardar-Parisi-Zhang equation in generic spatial dimensions. It is based on a momentum-space discretization of the continuum equation and on a pseudospectral approximation of the nonlinear term. The method is tested in (1+1) and (2+1) dimensions, where it is shown to reproduce the current most reliable estimates of the critical exponents based on restricted solid-on-solid simulations. In particular, it allows the computations of various correlation and structure functions with high degree of numerical accuracy. Some deficiencies that are common to all previously used finite-difference schemes are pointed out and the usefulness of the present approach in this respect is discussed.
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Affiliation(s)
- Lorenzo Giada
- International School for Advanced Studies (SISSA) and INFM Unità di Trieste, Via Beirut 2-4, Trieste I-34014, Italy.
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Wang XH, Havlin S, Schwartz M. Entropy fluctuations for directed polymers in 2+1 dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:032601. [PMID: 11308690 DOI: 10.1103/physreve.63.032601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/1999] [Revised: 07/17/2000] [Indexed: 05/23/2023]
Abstract
We find numerically that the sample to sample fluctuation of the entropy DeltaS is a more sensitive tool in distinguishing low from high temperature behaviors than the common corresponding fluctuation in the free energy. In 1+1 dimensions we find a single phase for all temperatures, since (DeltaS)(2) is always extensive. In 2+1 dimensions we find a behavior that at first sight might appear to be a transition from a low temperature phase where (DeltaS)(2) is extensive to a high temperature phase where it is subextensive. This is observed in spite of the relatively large system we use. The observed behavior is explained not as a phase transition but as a strong crossover behavior. We use an analytical argument to obtain (DeltaS)(2) for high temperature, and find that while it is always extensive it is also extremely small, and that the leading extensive part decays very quickly to zero as the temperature is increased.
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Affiliation(s)
- X H Wang
- Minerva Center and Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
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Kim JM, Kosterlitz JM, Ala-Nissila T. Surface growth and crossover behaviour in a restricted solid-on-solid model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/24/23/022] [Citation(s) in RCA: 72] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Chakrabarti A. Computer simulation of stochastically growing interfaces with a conservation law. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/23/17/013] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Sánchez A, Cai D, Gronbech-Jensen N, Bishop AR, Wang ZJ. Roughening transitions of driven surface growth. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:14664-14668. [PMID: 9978401 DOI: 10.1103/physrevb.51.14664] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Wu M, Billah KY, Shinozuka M. Numerical solution of the Kardar-Parisi-Zhang equation with a long-range spatially correlated noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:995-998. [PMID: 9962738 DOI: 10.1103/physreve.51.995] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Tsai YC, Shapir Y. Dynamics of particle deposition on a disordered substrate. II. Far-from-equilibrium behavior. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:4445-4469. [PMID: 9962525 DOI: 10.1103/physreve.50.4445] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Beccaria M, Curci G. Numerical simulation of the Kardar-Parisi-Zhang equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:4560-4563. [PMID: 9962534 DOI: 10.1103/physreve.50.4560] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Dickman R. Numerical study of a field theory for directed percolation. ACTA ACUST UNITED AC 1994; 50:4404-4409. [PMID: 9962521 DOI: 10.1103/physreve.50.4404] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Sun T, Plischke M. Field-theory renormalization approach to the Kardar-Parisi-Zhang equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:5046-5057. [PMID: 9961826 DOI: 10.1103/physreve.49.5046] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Stewart J, Goldenfeld N. Spinodal decomposition of a crystal surface. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:6505-6512. [PMID: 9907960 DOI: 10.1103/physreva.46.6505] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Tsai YC, Shapir Y. Kinetic roughening in surfaces of crystals growing on disordered substrates. PHYSICAL REVIEW LETTERS 1992; 69:1773-1776. [PMID: 10046310 DOI: 10.1103/physrevlett.69.1773] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Kim JM, Bray AJ, Moore MA. Domain growth, directed polymers, and self-organized criticality. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:8546-8550. [PMID: 9906953 DOI: 10.1103/physreva.45.8546] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Tang LH, Forrest BM, Wolf DE. Kinetic surface roughening. II. Hypercube-stacking models. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:7162-7179. [PMID: 9906789 DOI: 10.1103/physreva.45.7162] [Citation(s) in RCA: 95] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Yao JH, Roland C, Guo H. Interfacial dynamics with long-range screening. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:3903-3912. [PMID: 9907439 DOI: 10.1103/physreva.45.3903] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kim JM, Bray AJ, Moore MA. Finite-temperature directed polymers in a random potential. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:R4782-R4785. [PMID: 9906590 DOI: 10.1103/physreva.44.r4782] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kim JM, Moore MA, Bray AJ. Zero-temperature directed polymers in a random potential. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:2345-2351. [PMID: 9906215 DOI: 10.1103/physreva.44.2345] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Rácz Z, Siegert M, Liu D, Plischke M. Scaling properties of driven interfaces: Symmetries, conservation laws, and the role of constraints. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:5275-5283. [PMID: 9904839 DOI: 10.1103/physreva.43.5275] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lai Z. Kinetic growth with surface relaxation: Continuum versus atomistic models. PHYSICAL REVIEW LETTERS 1991; 66:2348-2351. [PMID: 10043462 DOI: 10.1103/physrevlett.66.2348] [Citation(s) in RCA: 230] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Roland C, Guo H. Interface growth with a shadow instability. PHYSICAL REVIEW LETTERS 1991; 66:2104-2107. [PMID: 10043392 DOI: 10.1103/physrevlett.66.2104] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Paczuski M. Divergence of energy fluctuations in a growing interface. PHYSICAL REVIEW LETTERS 1991; 66:1545. [PMID: 10043236 DOI: 10.1103/physrevlett.66.1545] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Richards PM. Equilibrium interface width for epitaxial growth with step-height-independent hopping of adatoms. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:6750-6753. [PMID: 9998121 DOI: 10.1103/physrevb.43.6750] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Grossmann B, Guo H, Grant M. Kinetic roughening of interfaces in driven systems. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:1727-1743. [PMID: 9905213 DOI: 10.1103/physreva.43.1727] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Hwa T, Kardar M, Paczuski M. Growth-induced roughening of crystalline facets. PHYSICAL REVIEW LETTERS 1991; 66:441-444. [PMID: 10043808 DOI: 10.1103/physrevlett.66.441] [Citation(s) in RCA: 26] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Tamborenea P. A new universality class for kinetic growth: One-dimensional molecular-beam epitaxy. PHYSICAL REVIEW LETTERS 1991; 66:325-328. [PMID: 10043777 DOI: 10.1103/physrevlett.66.325] [Citation(s) in RCA: 89] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Willett E, Pandey R. Power-law exponents for a spreading front and growing interface in an irreversible wetting. PHYSICAL REVIEW LETTERS 1990; 65:3413-3416. [PMID: 10042865 DOI: 10.1103/physrevlett.65.3413] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Tang LH, Nattermann T, Forrest BM. Multicritical and crossover phenomena in surface growth. PHYSICAL REVIEW LETTERS 1990; 65:2422-2425. [PMID: 10042544 DOI: 10.1103/physrevlett.65.2422] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Yan H, Kessler DA. Comment on "Phase transition in a restricted solid-on-solid surface-growth model in 2+1 dimensions". PHYSICAL REVIEW LETTERS 1990; 65:661. [PMID: 10042980 DOI: 10.1103/physrevlett.65.661] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Guo H, Grossmann B, Grant M. Crossover scaling in the dynamics of driven systems. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:7082-7085. [PMID: 9903136 DOI: 10.1103/physreva.41.7082] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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