Growing fractal interfaces in the presence of self-similar hopping surface diffusion

Mann, J. Adin; Woyczyński, Wojbor A.

doi:10.1016/s0378-4371(00)00467-2

Cited by 69 publications

(58 citation statements)

References 30 publications

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“…(1) Fig. 1(b) and [19] for ¼ 1=2 < 1, that corresponds to an unexplored (to our knowledge) instance of unstable superballistic (Lévy flight type) interface relaxation [12]. As we see in the figures, the qualitative behavior is similar to the SMS ( ¼ 1) case, especially at large length scales.…”

Section: Prl 102 256102 (2009) P H Y S I C a L R E V I E W L E T T Esupporting

confidence: 52%

“…1 (see insets of Fig. 1 and [19]) [2,12,15]. Moreover, these results for the morphologically stable condition suggest that Eq.…”

Section: Prl 102 256102 (2009) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 68%

“…In general, the unstable ( < 0) nonlocal Eq. (1) has been proposed as a universal description of systems in which long-range interactions persist at the level of amplitude descriptions [7], while its stable ( > 0) counterpart, studied in the context of kinetic roughening for the ¼ 0 case [3], has been proposed [12] as a description of surface growth mediated by long-range surface hopping mechanisms like Lévy flights. Indeed, k h k ðtÞ is a fractional power of the Laplacian operator, employed to describe anomalous diffusion [13].…”

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confidence: 99%

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Unstable Nonlocal Interface Dynamics

2009

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Nonlocal effects occur in many nonequilibrium interfaces, due to diverse physical mechanisms like diffusive, ballistic, or anomalous transport, with examples from flame fronts to thin films. While dimen sional analysis describes stable nonlocal interfaces, we show the morphologically unstable condition to be nontrivial. This is the case for a family of stochastic equations of experimental relevance, paradigmatically including the Michelson Sivashinsky system. For a whole parameter range, the asymptotic dynamics is scale invariant with dimension independent exponents reflecting a hidden Galilean symmetry. The usual Kardar Parisi Zhang nonlinearity, albeit irrelevant in that parameter range, plays a key role in this behavior. DOI: 10.1103/PhysRevLett.102.256102 PACS numbers: 68.35.Ct, 05.45. a, 47.54. r The interface dynamics of many nonequilibrium systems arises from the interplay between nonlocal interactions and morphological instabilities. Examples range from flame front propagation to thin film growth [1]. Often, although the basic physical interactions are short-ranged and local, the evolution is driven by nonlocal effects implicitly (via projection of the overall dynamics on the interface) or explicitly (as in elastic media or viscous flow) [1]. These nonlocalities appear in many fields, diffusion-limited growth being a prominent example. Here, although diffusion events of aggregating units are local, the morphology of a growing cluster is dominated by shadowing of the most prominent surface features over less exposed ones. Hence, the local growth velocity depends on the global surface shape. This moreover leads to the classic Mullins-Sekerka (MS) morphological instability [1], whereby prominent features grow faster. Still, nonlocality and morphological stability are independent properties. Thus, we may consider diffusion-limited erosion (DLE) [2,3], qualitatively relevant to experiments on, e.g., ion irradiation smoothing [4]. Here the most exposed surface features are eroded faster, leading to nonlocal stable interface evolution in which differences in surface height are smoothed out.Often [3], the nonequilibrium dynamics of these interfaces can be cast into a stochastic equation for the surface height hðr; tÞ at time t and point r on a d-dimensional substrate. Assuming translational and rotational symmetry, we propose the following equation (after space Fourier transform F )Here, , m, n, K, and N are positive constants (0 < 2, m ! 2, and n > m; see below), the linear dispersionproviding the amplification rate for periodic disturbances (with wave vector k) of a planar interface. The term proportional to is the Kardar-Parisi-Zhang (KPZ) nonlinearity, generically expected whenever the interface evolves in the absence of conservation laws [3], while k ðtÞ is Gaussian uncorrelated noise. Indeed, equations like (1) have been derived from constitutive laws, and from symmetry arguments, as weakly nonlinear, long wavelength (k ¼ jkj ( 1) descriptions of a variety of systems [1,3,6]. Locality holds whenever ! ...

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Section: Prl 102 256102 (2009) P H Y S I C a L R E V I E W L E T T Esupporting

confidence: 52%

“…1 (see insets of Fig. 1 and [19]) [2,12,15]. Moreover, these results for the morphologically stable condition suggest that Eq.…”

Section: Prl 102 256102 (2009) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 68%

mentioning

confidence: 99%

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Unstable Nonlocal Interface Dynamics

2009

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“…Both EW and KPZ equations have been generalized by replacing the usual diffusion term with the fractional one [25,43]. This gives the fractional EW equation…”

Section: Dynamical Scalingmentioning

confidence: 99%

Scaling properties of the asymmetric exclusion process with long-range hopping

Szavits-Nossan

Uzelac

2008

Phys. Rev. E

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The exclusion process in which particles may jump any distance l ≥ 1 with the probability that decays as l −(1+σ) is studied from coarse-grained equation for density profile in the limit when the lattice spacing goes to zero. For 1 < σ < 2, the usual diffusion term of this equation is replaced by the fractional one, which affects dynamical-scaling properties of the late-time approach to the stationary state. When applied to an open system with totally asymmetric hopping, this approach gives two results: first, it accounts for the σ-dependent exponent that characterizes the algebraic decay of density profile in the maximum-current phase for 1 < σ < 2, and second, it shows that in this region of σ the exponent is of the mean-field type.

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“…The present work is motivated by various physical applications of nonlinear equations with nonlocal integro-differential or pseudodifferential diffusive terms which include, e.g., anomalous growth models of molecular interfaces involving hopping and trapping phenomena [21] and hydrodynamic models with modified diffusivity [1]. Various linear differential equations involving fractional derivatives, and their applications to statistical physics, hydrodynamics, molecular biology etc., have been discussed in, e.g., [23].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for conservation laws involving Lévy diffusion generators

2001

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Abstract. Let −L be the generator of a Lévy semigroup on L 1 (R n ) and f : R → R n be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations u t + Lu + ∇ · f (u) = 0, analyzing their L p -decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.

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Growing fractal interfaces in the presence of self-similar hopping surface diffusion

Cited by 69 publications

References 30 publications

Unstable Nonlocal Interface Dynamics

Unstable Nonlocal Interface Dynamics

Scaling properties of the asymmetric exclusion process with long-range hopping

Asymptotics for conservation laws involving Lévy diffusion generators

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