1
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Roy D, Huse DA, Kulkarni M. Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations. Phys Rev E 2023; 108:054112. [PMID: 38115452 DOI: 10.1103/physreve.108.054112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2023] [Accepted: 10/11/2023] [Indexed: 12/21/2023]
Abstract
The out-of-time-ordered correlator (OTOC) has emerged as an interesting object in both classical and quantum systems for probing the spatial spread and temporal growth of initially local perturbations in spatially extended chaotic systems. Here, we study the (classical) OTOC and its "light cone" in the nonlinear Kuramoto-Sivashinsky (KS) equation, using extensive numerical simulations. We also show that the linearized KS equation exhibits a qualitatively similar OTOC and light cone, which can be understood via a saddle-point analysis of the linearly unstable modes. Given the deep connection between the KS (deterministic) and the Kardar-Parisi-Zhang (KPZ, which is stochastic) equations, we also explore the OTOC in the KPZ equation. While our numerical results in the KS case are expected to hold in the continuum limit, for the KPZ case it is valid in a discretized version of the KPZ equation. More broadly, our work unravels the intrinsic interplay between noise/instability, nonlinearity, and dissipation in partial differential equations (deterministic or stochastic) through the lens of OTOC.
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Affiliation(s)
- Dipankar Roy
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
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2
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Minoguchi M, Sasa SI. Divergent Stiffness of One-Dimensional Growing Interfaces. PHYSICAL REVIEW LETTERS 2023; 130:197101. [PMID: 37243631 DOI: 10.1103/physrevlett.130.197101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Revised: 01/12/2023] [Accepted: 04/13/2023] [Indexed: 05/29/2023]
Abstract
When a spatially localized stress is applied to a growing one-dimensional interface, the interface deforms. This deformation is described by the effective surface tension representing the stiffness of the interface. We present that the stiffness exhibits divergent behavior in the large system size limit for a growing interface with thermal noise, which has never been observed for equilibrium interfaces. Furthermore, by connecting the effective surface tension with a space-time correlation function, we elucidate the mechanism that anomalous dynamical fluctuations lead to divergent stiffness.
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Affiliation(s)
| | - Shin-Ichi Sasa
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
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3
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Golden A, Dukovski I, Segrè D, Korolev KS. Growth instabilities shape morphology and genetic diversity of microbial colonies. Phys Biol 2022; 19:056005. [PMID: 35901792 DOI: 10.1088/1478-3975/ac8514] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2022] [Accepted: 07/28/2022] [Indexed: 11/11/2022]
Abstract
Cellular populations assume an incredible variety of shapes ranging from circular molds to irregular tumors. While we understand many of the mechanisms responsible for these spatial patterns, little is known about how the shape of a population influences its ecology and evolution. Here, we investigate this relationship in the context of microbial colonies grown on hard agar plates. This a well-studied system that exhibits a transition from smooth circular disks to more irregular and rugged shapes as either the nutrient concentration or cellular motility is decreased. Starting from a mechanistic model of colony growth, we identify two dimensionless quantities that determine how morphology and genetic diversity of the population depend on the model parameters. Our simulations further reveal that population dynamics cannot be accurately described by the commonly-used surface growth models. Instead, one has to explicitly account for the emergent growth instabilities and demographic fluctuations. Overall, our work links together environmental conditions, colony morphology, and evolution. This link is essential for a rational design of concrete, biophysical perturbations to steer evolution in the desired direction.
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Affiliation(s)
- Alexander Golden
- Department of Physics, Graduate Program in Bioinformatics, and Biological Design Center, Boston University, Boston, MA 02215, United States of America
| | - Ilija Dukovski
- Graduate Program in Bioinformatics, and Biological Design Center, Boston University, Boston, MA 02215, United States of America
| | - Daniel Segrè
- Department of Physics, Department of Biology, Department of Biomedical Engineering, Graduate Program in Bioinformatics, and Biological Design Center, Boston University, Boston, MA 02215, United States of America
| | - Kirill S Korolev
- Department of Physics, Graduate Program in Bioinformatics, and Biological Design Center, Boston University, Boston, MA 02215, United States of America
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4
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Daryaei E. Universality and crossover behavior of single-step growth models in 1+1 and 2+1 dimensions. Phys Rev E 2020; 101:062108. [PMID: 32688564 DOI: 10.1103/physreve.101.062108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2020] [Accepted: 05/15/2020] [Indexed: 06/11/2023]
Abstract
We study the kinetic roughening of the single-step (SS) growth model with a tunable parameter p in 1+1 and 2+1 dimensions by performing extensive numerical simulations. We show that there exists a very slow crossover from an intermediate regime dominated by the Edwards-Wilkinson class to an asymptotic regime dominated by the Kardar-Parisi-Zhang (KPZ) class for any p<1/2. We also identify the crossover time, the nonlinear coupling constant, and some nonuniversal parameters in the KPZ equation as a function p. The effective nonuniversal parameters are continuously decreasing with p but not in a linear fashion. Our results provide complete and conclusive evidence that the SS model for p≠1/2 belongs to the KPZ universality class in 2+1 dimensions.
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Affiliation(s)
- E Daryaei
- Department of Physics, Faculty of Basic Sciences, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran
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5
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Roy D, Pandit R. One-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: Limit distributions. Phys Rev E 2020; 101:030103. [PMID: 32289936 DOI: 10.1103/physreve.101.030103] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2019] [Accepted: 03/10/2020] [Indexed: 11/07/2022]
Abstract
Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state of the one-dimensional Kuramoto-Sivashinsky (KS) deterministic PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile h(x,t) for different initial conditions. We establish, therefore, that the statistical properties of the one-dimensional (1D) KS PDE in this state are in the 1D KPZ universality class.
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Affiliation(s)
- Dipankar Roy
- Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
| | - Rahul Pandit
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India
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6
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Headrick RL, Ulbrandt JG, Myint P, Wan J, Li Y, Fluerasu A, Zhang Y, Wiegart L, Ludwig KF. Coherent X-ray measurement of step-flow propagation during growth on polycrystalline thin film surfaces. Nat Commun 2019; 10:2638. [PMID: 31201329 PMCID: PMC6570654 DOI: 10.1038/s41467-019-10629-8] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2018] [Accepted: 05/14/2019] [Indexed: 11/09/2022] Open
Abstract
The properties of artificially grown thin films are strongly affected by surface processes during growth. Coherent X-rays provide an approach to better understand such processes and fluctuations far from equilibrium. Here we report results for vacuum deposition of C60 on a graphene-coated surface investigated with X-ray Photon Correlation Spectroscopy in surface-sensitive conditions. Step-flow is observed through measurement of the step-edge velocity in the late stages of growth after crystalline mounds have formed. We show that the step-edge velocity is coupled to the terrace length, and that there is a variation in the velocity from larger step spacing at the center of crystalline mounds to closely-spaced, more slowly propagating steps at their edges. The results extend theories of surface growth, since the behavior is consistent with surface evolution driven by processes that include surface diffusion, the motion of step-edges, and attachment at step edges with significant step-edge barriers.
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Affiliation(s)
- Randall L Headrick
- Department of Physics and Materials Science Program, University of Vermont, Burlington, VT, 05405, USA.
| | - Jeffrey G Ulbrandt
- Department of Physics and Materials Science Program, University of Vermont, Burlington, VT, 05405, USA
| | - Peco Myint
- Division of Materials Science and Engineering, Boston University, Boston, MA, 02215, USA
| | - Jing Wan
- Department of Physics and Materials Science Program, University of Vermont, Burlington, VT, 05405, USA
| | - Yang Li
- Department of Physics and Materials Science Program, University of Vermont, Burlington, VT, 05405, USA
| | | | - Yugang Zhang
- National Synchrotron Light Source II, Upton, NY, 11967, USA
| | - Lutz Wiegart
- National Synchrotron Light Source II, Upton, NY, 11967, USA
| | - Karl F Ludwig
- Division of Materials Science and Engineering, Boston University, Boston, MA, 02215, USA.,Department of Physics, Boston University, Boston, MA, 02215, USA
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7
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Tomlin RJ, Kalogirou A, Papageorgiou DT. Nonlinear dynamics of a dispersive anisotropic Kuramoto-Sivashinsky equation in two space dimensions. Proc Math Phys Eng Sci 2018; 474:20170687. [PMID: 29662339 PMCID: PMC5897757 DOI: 10.1098/rspa.2017.0687] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2017] [Accepted: 02/27/2018] [Indexed: 11/12/2022] Open
Abstract
A Kuramoto-Sivashinsky equation in two space dimensions arising in thin film flows is considered on doubly periodic domains. In the absence of dispersive effects, this anisotropic equation admits chaotic solutions for sufficiently large length scales with fully two-dimensional profiles; the one-dimensional dynamics observed for thin domains are structurally unstable as the transverse length increases. We find that, independent of the domain size, the characteristic length scale of the profiles in the streamwise direction is about 10 space units, with that in the transverse direction being approximately three times larger. Numerical computations in the chaotic regime provide an estimate for the radius of the absorbing ball in [Formula: see text] in terms of the length scales, from which we conclude that the system possesses a finite energy density. We show the property of equipartition of energy among the low Fourier modes, and report the disappearance of the inertial range when solution profiles are two-dimensional. Consideration of the high-frequency modes allows us to compute an estimate for the analytic extensibility of solutions in [Formula: see text]. We also examine the addition of a physically derived third-order dispersion to the problem; this has a destabilizing effect, in the sense of reducing analyticity and increasing amplitude of solutions. However, sufficiently large dispersion may regularize the spatio-temporal chaos to travelling waves. We focus on dispersion where chaotic dynamics persist, and study its effect on the interfacial structures, absorbing ball and properties of the power spectrum.
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Affiliation(s)
- R J Tomlin
- Department of Mathematics, Imperial College London, SW7 2AZ, London, UK
| | - A Kalogirou
- School of Mathematics, University of East Anglia, NR4 7TJ, Norwich, UK
| | - D T Papageorgiou
- Department of Mathematics, Imperial College London, SW7 2AZ, London, UK
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8
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Mokhtarzadeh M, Ludwig KF. Simulations of Co-GISAXS during kinetic roughening of growth surfaces. JOURNAL OF SYNCHROTRON RADIATION 2017; 24:1187-1194. [PMID: 29091062 DOI: 10.1107/s1600577517011924] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2017] [Accepted: 08/16/2017] [Indexed: 06/07/2023]
Abstract
The recent development of surface growth studies using X-ray photon correlation spectroscopy in a grazing-incidence small-angle X-ray scattering (Co-GISAXS) geometry enables the investigation of dynamical processes during kinetic roughening in greater detail than was previously possible. In order to investigate the Co-GISAXS behavior expected from existing growth models, calculations and (2+1)-dimension simulations of linear Kuramoto-Sivashinsky and non-linear Kardar-Parisi-Zhang surface growth equations are presented which analyze the temporal correlation functions of the height-height structure factor. Calculations of the GISAXS intensity auto-correlation functions are also performed within the Born/distorted-wave Born approximation for comparison with the scaling behavior of the height-height structure factor and its correlation functions.
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Affiliation(s)
| | - Karl F Ludwig
- Division of Materials Science and Engineering, Boston University, Boston, MA 02215, USA
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9
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Muñoz-García J, Gago R, Vázquez L, Sánchez-García JA, Cuerno R. Observation and modeling of interrupted pattern coarsening: surface nanostructuring by ion erosion. PHYSICAL REVIEW LETTERS 2010; 104:026101. [PMID: 20366611 DOI: 10.1103/physrevlett.104.026101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2009] [Indexed: 05/29/2023]
Abstract
We report the experimental observation of interrupted coarsening for surface self-organized nanostructuring by ion erosion. Analysis of the target surface by atomic force microscopy allows us to describe quantitatively this intriguing type of pattern dynamics through a continuum equation put forward in different contexts across a wide range of length scales. The ensuing predictions can thus be consistently extended to other experimental conditions in our system. Our results illustrate the occurrence of nonequilibrium systems in which pattern formation, coarsening, and kinetic roughening appear, each of these behaviors being associated with its own spatiotemporal range.
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Affiliation(s)
- Javier Muñoz-García
- Systems Biology Ireland and Grupo Interdisciplinar de Sistemas Complejos (GISC), University College Dublin, Belfield, Dublin 4, Ireland
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10
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Mori H, Okamura M. Dual structures of chaos and turbulence, and their dynamic scaling laws. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:051124. [PMID: 20364964 DOI: 10.1103/physreve.80.051124] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2009] [Indexed: 05/29/2023]
Abstract
The decay form of the time correlation function Un(t) of a state variable un(t) with a small wave number kn has been shown to take the algebraic decay 1/{1+(gammanat)2} in the initial regime t<taun(gamma) and the exponential decay alphane exp(-gammanet) in the final regime t>taun(gamma), where taun(gamma) denotes the decay time of the memory function Gamman(t). This dual structure of Un(t) is generated by the deterministic short orbits in the initial regime and the stochastic long orbits in the final regime, thus giving the outstanding features of chaos and turbulence. The kn dependence of gammana, alphane, and gammane is obtained for the chaotic Kuramoto-Sivashinsky equation, and it is shown that if kn is sufficiently small, then the dual structure of Un(t) obeys a hydrodynamic scaling law in the final regime t>tau(gamma) with scaling exponent z=2 and a dynamic scaling law in the initial regime t<taun(gamma) with scaling exponent z=1. If kn is increased so that the decay time taun(u) of Un(t) becomes equal to the decay time taun(gamma), then the decay form of Un(t) becomes the power-law decay t-3/2 in the final regime.
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Affiliation(s)
- Hazime Mori
- Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
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11
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Castro M, Hernández-Machado A, Cuerno R. Single-phase-field model of stepped surfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:021601. [PMID: 19391755 DOI: 10.1103/physreve.79.021601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2008] [Revised: 11/28/2008] [Indexed: 05/27/2023]
Abstract
We formulate a phase-field description of step dynamics on vicinal surfaces that makes use of a single dynamical field, at variance with previous analogous works in which two coupled fields are employed, namely, a phase-field proper plus the physical adatom concentration. Within an asymptotic sharp interface limit, our formulation is shown to retrieve the standard Burton-Cabrera-Frank model in the general case of asymmetric attachment coefficients (Ehrlich-Schwoebel effect). We confirm our analytical results by means of numerical simulations of our phase-field model. Our present formulation seems particularly well adapted to generalization when additional physical fields are required.
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Affiliation(s)
- M Castro
- Grupo Interdisciplinar de Sistemas Complejos (GISC) and Grupo de Dinámica No Lineal (DNL), Escuela Técnica Superior de Ingeniería (ICAI), Universidad Pontificia Comillas, E-28015 Madrid, Spain.
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12
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Sakaguchi H. Soliton turbulence in the complex Ginzburg-Landau equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:017205. [PMID: 17677602 DOI: 10.1103/physreve.76.017205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2007] [Indexed: 05/16/2023]
Abstract
We study spatiotemporal chaos in the complex Ginzburg-Landau equation in parameter regions of weak amplification and viscosity. Turbulent states involving many solitonlike pulses appear in the parameter range, because the complex Ginzburg-Landau equation is close to the nonlinear Schrödinger equation. We find that the distributions of amplitude and wave number of pulses depend only on the ratio of the two parameters of the amplification and the viscosity. This implies that a one-parameter family of soliton turbulence states characterized by different distributions of the soliton parameters exists continuously around the completely integrable system.
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Affiliation(s)
- Hidetsugu Sakaguchi
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
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13
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Ong SW, Tok ES, Kang HC. Sputter roughening of inhomogeneous surfaces: impurity pinning and nanostructure shape selection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:061607. [PMID: 17677273 DOI: 10.1103/physreve.75.061607] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2006] [Revised: 02/12/2007] [Indexed: 05/16/2023]
Abstract
A model is proposed for sputter roughening of inhomogeneous systems with slowly sputtered impurity particles randomly distributed in the bulk. Surface inhomogeneity, which develops as a result of coupling between the time evolution of the local surface impurity concentration and the local surface shape, is tuned by changing the dependence of the sputtering probability upon impurity concentration. In 1+1 dimensions, we find long-time scaling exponents that are consistent with Kardar-Parisi-Zhang (KPZ) values. However, for a range of surface inhomogeneity, impurity pinning results in a persistent growth regime where the surface roughens rapidly. We correlate this rapid roughening to fluctuations of the impurity concentration at the surface. Roughening in this regime leads to the formation of cones whose shape is determined by material property and sputtering flux, suggesting a unique method of nanostructure fabrication. In 2+1 dimensions, a similar variation of the roughening behavior with surface inhomogeneity is observed. For small surface inhomogeneity, there is an initial exponential roughening followed by power-law roughening with an effective growth exponent much smaller than KPZ. For larger surface inhomogeneity two power-law roughening regimes are observed, with an initial rapid roughening that crosses over to slower roughening; the effective exponent in each of these regimes increases with surface inhomogeneity. The surface morphology observed in the simulations is considerably noisier than experimental data for InP and GaSb. Our model shows noisy nonlinear pattern formation in contrast to the marked long-range hexagonal ordering seen in experiments. However, the scaling behavior is robust enough that roughening kinetics similar to that observed experimentally can be obtained depending upon the values of inhomogeneity and the strength of the nonlinear term in the model.
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Affiliation(s)
- S W Ong
- Department of Chemistry, National University of Singapore, 10 Kent Ridge Crescent, Singapore
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14
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Muñoz-García J, Cuerno R, Castro M. Short-range stationary patterns and long-range disorder in an evolution equation for one-dimensional interfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:050103. [PMID: 17279865 DOI: 10.1103/physreve.74.050103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2006] [Indexed: 05/13/2023]
Abstract
A local evolution equation for one-dimensional interfaces is derived in the context of erosion by ion beam sputtering. We present numerical simulations of this equation which show interrupted coarsening in which an ordered cell pattern develops with constant wavelength and amplitude at intermediate distances, while the profile is disordered and rough at larger distances. Moreover, for a wide range of parameters the lateral extent of ordered domains ranges up to tens of cells. We also provide analytical estimates for the stationary pattern wavelength and mean growth velocity.
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Affiliation(s)
- Javier Muñoz-García
- Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Carlos III de Madrid, Avenida de la Universidad 30, E-28911 Leganés, Spain
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15
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Okamura M. Validity of the essential assumption in a projection operator method. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:046210. [PMID: 17155156 DOI: 10.1103/physreve.74.046210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2006] [Revised: 08/18/2006] [Indexed: 05/12/2023]
Abstract
The projection operator method developed by Mori involves the essential assumption that chaotic motion is successfully divided into a coherent motion and a fluctuating one. We investigate the validity of the assumption using the Kuramoto-Sivashinsky equation as a model equation of chaotic systems. It has been found that the assumption is reasonable for both long wave modes and short wave modes. We have also evaluated a value of the eddy viscosity as 9.0 by extracting the nonlinear term from the coherent part. This value is consistent with the former estimates with other methods.
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Affiliation(s)
- Makoto Okamura
- Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan.
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16
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Ueno K, Sakaguchi H, Okamura M. Renormalization-group and numerical analysis of a noisy Kuramoto-Sivashinsky equation in 1 + 1 dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:046138. [PMID: 15903757 DOI: 10.1103/physreve.71.046138] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2004] [Indexed: 05/02/2023]
Abstract
The long-wavelength properties of a noisy Kuramoto-Sivashinsky (KS) equation in 1 + 1 dimensions are investigated by use of the dynamic renormalization group (RG) and direct numerical simulations. It is shown that the noisy KS equation is in the same universality class as the Kardar-Parisi-Zhang (KPZ) equation in the sense that they have scale invariant solutions with the same scaling exponents in the long-wavelength limit. The RG analysis reveals that the RG flow for the parameters of the noisy KS equation rapidly approach the KPZ fixed point with increasing strength of the noise. This is supplemented by numerical simulations of the KS equation with a stochastic noise, in which scaling behavior close to the KPZ scaling can be observed even in a moderate system size and time.
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Affiliation(s)
- K Ueno
- Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580, Japan.
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17
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Ginelli F, Hinrichsen H, Livi R, Mukamel D, Politi A. Directed percolation with long-range interactions: Modeling nonequilibrium wetting. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:026121. [PMID: 15783392 DOI: 10.1103/physreve.71.026121] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2004] [Indexed: 05/24/2023]
Abstract
It is argued that some phase transitions observed in models of nonequilibrium wetting phenomena are related to contact processes with long-range interactions. This is investigated by introducing a model where the activation rate of a site at the edge of an inactive island of length l is 1+a l(-sigma) . Mean-field analysis and numerical simulations indicate that for sigma>1 the transition is continuous and belongs to the universality class of directed percolation, while for 0<sigma<1 , the transition becomes first order. This criterion is then applied to discuss critical properties of various models of nonequilibrium wetting.
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Affiliation(s)
- F Ginelli
- Institut für Theoretische Physik und Astrophysik, University of Würzburg, D-97974 Würzburg, Germany
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18
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Kitahara Y, Okamura M. Mean solutions for the Kuramoto-Sivashinsky equation with incoming boundary conditions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:056210. [PMID: 15600731 DOI: 10.1103/physreve.70.056210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/25/2003] [Indexed: 05/24/2023]
Abstract
We consider herein the Kuramoto-Sivashinsky (KS) equation with incoming boundary conditions. Using a projection operator method, we have derived a set of closed equations for the mean quantities, called a model equation, from the KS equation. One of the characteristics of the model equation is that it does not include any empirical parameters. The adequacy of the model equation is verified by comparing solutions of the model equation with time-averaged solutions obtained from the numerical simulation of the KS equation.
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Affiliation(s)
- Youichi Kitahara
- Department of Earth System Science and Technology, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan
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19
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Tok ES, Ong SW, Kang HC. Dynamical scaling of sputter-roughened surfaces in 2+1 dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:011604. [PMID: 15324062 DOI: 10.1103/physreve.70.011604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2003] [Revised: 04/26/2004] [Indexed: 05/24/2023]
Abstract
The asymptotic scaling behavior of sputter-roughened surfaces is of great current interest. In particular, the disparately wide-ranging values of the growth exponent found experimentally, and whether sputter-roughening belongs to the Kardar-Parisi-Zhang universality class in 2+1 dimensions, are two interesting issues. We address these issues using simulations of an atomistic model. The asymptotic scaling appears to be Edwards-Wilkinson. Crossover behavior in the model leads to effective growth exponents that vary widely depending upon the regime of observation.
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Affiliation(s)
- E S Tok
- Department of Materials Science, National University of Singapore, 3 Science Drive 3, National University of Singapore, Singapore 117543
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Provata A, Tsekouras GA. Spontaneous formation of dynamical patterns with fractal fronts in the cyclic lattice Lotka-Volterra model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:056602. [PMID: 12786291 DOI: 10.1103/physreve.67.056602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2002] [Indexed: 05/24/2023]
Abstract
Dynamical patterns, in the form of consecutive moving stripes or rings, are shown to develop spontaneously in the cyclic lattice Lotka-Volterra model, when realized on square lattice, at the reaction limited regime. Each stripe consists of different particles (species) and the borderlines between consecutive stripes are fractal. The interface width w between the different species scales as w(L,t) approximately L(alpha)f(t/L(z)), where L is the linear size of the interface, t is the time, and alpha and z are the static and dynamical critical exponents, respectively. The critical exponents were computed as alpha=0.49+/-0.03 and z=1.53+/-0.13 and the propagating fronts show dynamical characteristics similar to those of the Eden growth models.
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Affiliation(s)
- A Provata
- Institute of Physical Chemistry, National Research Center Demokritos, 15310 Athens, Greece.
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21
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Merikoski J, Maunuksela J, Myllys M, Timonen J, Alava MJ. Temporal and spatial persistence of combustion fronts in paper. PHYSICAL REVIEW LETTERS 2003; 90:024501. [PMID: 12570549 DOI: 10.1103/physrevlett.90.024501] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2002] [Indexed: 05/24/2023]
Abstract
The spatial and temporal persistence, or first-return distributions are measured for slow-combustion fronts in paper. The stationary temporal and (perhaps less convincingly) spatial persistence exponents agree with the predictions based on the front dynamics, which asymptotically belongs to the Kardar-Parisi-Zhang universality class. The stationary short-range and the transient behavior of the fronts are non-Markovian, and the observed persistence properties thus do not agree with the predictions based on Markovian theory. This deviation is a consequence of additional time and length scales, related to the crossovers to the asymptotic coarse-grained behavior.
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Affiliation(s)
- J Merikoski
- Department of Physics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
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22
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Cuerno R, Castro M. Possible origin for the experimental scarcity of KPZ scaling in non-conserved surface growth. PHYSICA A: STATISTICAL MECHANICS AND ITS APPLICATIONS 2002; 314:192-199. [DOI: 10.1016/s0378-4371(02)01148-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/02/2023]
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23
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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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24
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Cai D, McLaughlin DW, McLaughlin KT. The Nonlinear Schrödinger Equation as Both a PDE and a Dynamical System. HANDBOOK OF DYNAMICAL SYSTEMS 2002. [DOI: 10.1016/s1874-575x(02)80033-9] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/22/2023]
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25
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Cuerno R, Castro M. Transients due to instabilities hinder Kardar-Parisi-Zhang scaling: a unified derivation for surface growth by electrochemical and chemical vapor deposition. PHYSICAL REVIEW LETTERS 2001; 87:236103. [PMID: 11736462 DOI: 10.1103/physrevlett.87.236103] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2001] [Indexed: 05/23/2023]
Abstract
We propose a unified moving boundary problem for surface growth by electrochemical and chemical vapor deposition, which is derived from constitutive equations into which stochastic forces are incorporated. We compute the coefficients in the interface equation of motion as functions of phenomenological parameters. The equation features the Kardar-Parisi-Zhang (KPZ) nonlinearity and instabilities which, depending on surface kinetics, can hinder the asymptotic KPZ scaling. Our results account for the universality and the experimental scarcity of KPZ scaling in the growth processes considered.
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Affiliation(s)
- R Cuerno
- Departamento de Matemáticas & GISC, Universidad Carlos III de Madrid, Avenida Universidad 30, 28911 Leganés, Spain
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26
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Medeiros JT, Simões AS. Heuristic model for the energy spectrum of phase turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:057301. [PMID: 11736148 DOI: 10.1103/physreve.64.057301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2001] [Indexed: 05/23/2023]
Abstract
We present a heuristic model for the energy spectrum of the one-dimensional phase turbulence in the steady state of the Kuramoto-Sivashinsky equation. Our model contains an energy transfer mechanism from low- to high-wave-vector modes. The energy transfer is written as the sum of local and nonlocal interactions. Our analytical results show good agreement with numerical simulations, particularly for the hump in the energy spectrum, which is mainly due to the local interactions.
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Affiliation(s)
- J T Medeiros
- Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, RS, Brazil
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27
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Myllys M, Maunuksela J, Alava M, Ala-Nissila T, Merikoski J, Timonen J. Kinetic roughening in slow combustion of paper. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:036101. [PMID: 11580388 DOI: 10.1103/physreve.64.036101] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2001] [Indexed: 05/23/2023]
Abstract
Results of experiments on the dynamics and kinetic roughening of one-dimensional slow-combustion fronts in three grades of paper are reported. Extensive averaging of the data allows a detailed analysis of the spatial and temporal development of the interface fluctuations. The asymptotic scaling properties, on long length and time scales, are well described by the Kardar-Parisi-Zhang (KPZ) equation with short-range, uncorrelated noise. To obtain a more detailed picture of the strong-coupling fixed point, characteristic of the KPZ universality class, universal amplitude ratios, and the universal coupling constant are computed from the data and found to be in good agreement with theory. Below the spatial and temporal scales at which a crossover takes place to the standard KPZ behavior, the fronts display higher apparent exponents and apparent multiscaling. In this regime the interface velocities are spatially and temporally correlated, and the distribution of the magnitudes of the effective noise has a power-law tail. The relation of the observed short-range behavior and the noise as determined from the local velocity fluctuations is discussed.
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Affiliation(s)
- M Myllys
- Department of Physics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
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28
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Sakaguchi H. Shock structures in time-averaged patterns for the kuramoto-sivashinsky equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:8817-9. [PMID: 11138191 DOI: 10.1103/physreve.62.8817] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2000] [Indexed: 11/07/2022]
Abstract
The Kuramoto-Sivashinsky equation with fixed boundary conditions is numerically studied. Shocklike structures appear in the time-averaged patterns for some parameter range of the boundary values. Effective diffusion constant is estimated from the relation of the width and the height of the shock structures.
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Affiliation(s)
- H Sakaguchi
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
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29
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Myllys M, Maunuksela J, Alava MJ, Ala-Nissila T, Timonen J. Scaling and noise in slow combustion of paper. PHYSICAL REVIEW LETTERS 2000; 84:1946-1949. [PMID: 11017667 DOI: 10.1103/physrevlett.84.1946] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/1999] [Indexed: 05/23/2023]
Abstract
We present results of high resolution experiments on kinetic roughening of slow combustion fronts in paper, focusing on short length and time scales. Using three different grades of paper, we find that the combustion fronts show apparent spatial and temporal multiscaling at short scales. The scaling exponents decrease as a function of the order of the corresponding correlation functions. The noise affecting the fronts reveals short range temporal and spatial correlations, and non-Gaussian noise amplitudes. Our results imply that the overall behavior of slow combustion fronts cannot be explained by standard theories of kinetic roughening.
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Affiliation(s)
- M Myllys
- Department of Physics, University of Jyvaskyla, P.O. Box 35, FIN-40351 Jyvaskyla, Finland
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30
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Wittenberg RW, Holmes P. Scale and space localization in the Kuramoto-Sivashinsky equation. CHAOS (WOODBURY, N.Y.) 1999; 9:452-465. [PMID: 12779842 DOI: 10.1063/1.166419] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We describe a wavelet-based approach to the investigation of spatiotemporally complex dynamics, and show through extensive numerical studies that the dynamics of the Kuramoto-Sivashinsky equation in the spatiotemporally chaotic regime may be understood in terms of localized dynamics in both space and scale (wave number). A projection onto a spline wavelet basis enables good separation of scales, each with characteristic dynamics. At the large scales, one observes essentially slow Gaussian dynamics; at the active scales, structured "events" reminiscent of traveling waves and heteroclinic cycles appear to dominate; while the strongly damped small scales display intermittent behavior. The separation of scales and their dynamics is invariant as the length of the system increases, providing additional support for the extensivity of the spatiotemporally complex dynamics claimed in earlier works. We show also that the dynamics are spatially localized, discuss various correlation lengths, and demonstrate the existence of a characteristic interaction length for instantaneous influences. Our results motivate and advance the search for localized, low-dimensional models that capture the full behavior of spatially extended chaotic partial differential equations. (c) 1999 American Institute of Physics.
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Affiliation(s)
- Ralf W. Wittenberg
- Program in Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544
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31
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Bradley RM. Dynamic scaling of ion-sputtered rotating surfaces. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:6149-6152. [PMID: 9965834 DOI: 10.1103/physreve.54.6149] [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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32
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Lanczycki CJ, Jejjala V. Far from equilibrium nonconserved growth under a surface diffusion bias. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4755-4759. [PMID: 9965654 DOI: 10.1103/physreve.54.4755] [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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33
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Lauritsen KB, Cuerno R, Makse HA. Noisy Kuramoto-Sivashinsky equation for an erosion model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:3577-3580. [PMID: 9965504 DOI: 10.1103/physreve.54.3577] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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34
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Elezgaray J, Berkooz G, Holmes P. Large-scale statistics of the Kuramoto-Sivashinsky equation: A wavelet-based approach. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:224-230. [PMID: 9965064 DOI: 10.1103/physreve.54.224] [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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35
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Sasa SI. Collective patterns arising out of spatio-temporal chaos. CHAOS (WOODBURY, N.Y.) 1996; 6:238-242. [PMID: 12780252 DOI: 10.1063/1.166169] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We present a simple mathematical model in which a time averaged pattern emerges out of spatio-temporal chaos as a result of the collective action of chaotic fluctuations. Our evolution equation possesses spatial translational symmetry under periodic boundary conditions. Thus the spatial inhomogeneity of the statistical state arises through spontaneous symmetry breaking. The transition from a state of homogeneous spatio-temporal chaos to one exhibiting spatial order is explained by introducing a collective viscosity which relates the averaged pattern with a correlation of the fluctuations. (c) 1996 American Institute of Physics.
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Affiliation(s)
- Shin-Ichi Sasa
- Department of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
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36
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Frey E, Täuber UC, Hwa T. Mode-coupling and renormalization group results for the noisy Burgers equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:4424-4438. [PMID: 9964774 DOI: 10.1103/physreve.53.4424] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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37
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Cuerno R, Makse HA, Tomassone S, Harrington ST, Stanley HE. Stochastic model for surface erosion via ion sputtering: Dynamical evolution from ripple morphology to rough morphology. PHYSICAL REVIEW LETTERS 1995; 75:4464-4467. [PMID: 10059915 DOI: 10.1103/physrevlett.75.4464] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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38
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Rost M, Krug J. Anisotropic Kuramoto-Sivashinsky equation for surface growth and erosion. PHYSICAL REVIEW LETTERS 1995; 75:3894-3897. [PMID: 10059758 DOI: 10.1103/physrevlett.75.3894] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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39
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Cuerno R, Lauritsen KB. Renormalization-group analysis of a noisy Kuramoto-Sivashinsky equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:4853-4859. [PMID: 9963982 DOI: 10.1103/physreve.52.4853] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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40
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Malevanets A, Careta A, Kapral R. Biscale chaos in propagating fronts. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:4724-4735. [PMID: 9963967 DOI: 10.1103/physreve.52.4724] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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41
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Schimschak M, Krug J. Bulk defects and surface roughening in epitaxial growth. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:8550-8563. [PMID: 9979860 DOI: 10.1103/physrevb.52.8550] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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42
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Cuerno R, Barabási AL. Dynamic scaling of ion-sputtered surfaces. PHYSICAL REVIEW LETTERS 1995; 74:4746-4749. [PMID: 10058588 DOI: 10.1103/physrevlett.74.4746] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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43
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Statistical physics of growth processes. ACTA ACUST UNITED AC 1995. [DOI: 10.1007/978-1-4899-1421-7_1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register]
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44
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Grinstein G. Generic scale invariance and self-organized criticality. SCALE INVARIANCE, INTERFACES, AND NON-EQUILIBRIUM DYNAMICS 1995. [DOI: 10.1007/978-1-4899-1421-7_11] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/02/2022]
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45
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Kassner K, Misbah C, Müller-Krumbhaar H, Valance A. Directional solidification at high speed. II. Transition to chaos. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:5495-5516. [PMID: 9961876 DOI: 10.1103/physreve.49.5495] [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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46
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Wang XH, Wang KL. Analysis of anomalous diffusion in the Kuramoto-Sivashinsky equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:5853-5854. [PMID: 9961914 DOI: 10.1103/physreve.49.5853] [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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47
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Conrado CV, Bohr T. Singular growth shapes in turbulent field theories. PHYSICAL REVIEW LETTERS 1994; 72:3522-3525. [PMID: 10056221 DOI: 10.1103/physrevlett.72.3522] [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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48
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Kapral R, Livi R, Oppo GL, Politi A. Dynamics of complex interfaces. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:2009-2022. [PMID: 9961441 DOI: 10.1103/physreve.49.2009] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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49
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Sneppen K, Jensen MH. Multidiffusion in critical dynamics of strings and membranes. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:919-922. [PMID: 9961289 DOI: 10.1103/physreve.49.919] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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50
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Pikovsky AS, Kurths J. Roughening interfaces in the dynamics of perturbations of spatiotemporal chaos. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:898-901. [PMID: 9961283 DOI: 10.1103/physreve.49.898] [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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