17 research outputs found
Quantized Scaling of Growing Surfaces
The Kardar-Parisi-Zhang universality class of stochastic surface growth is
studied by exact field-theoretic methods. From previous numerical results, a
few qualitative assumptions are inferred. In particular, height correlations
should satisfy an operator product expansion and, unlike the correlations in a
turbulent fluid, exhibit no multiscaling. These properties impose a
quantization condition on the roughness exponent and the dynamic
exponent . Hence the exact values for two-dimensional
and for three-dimensional surfaces are derived.Comment: 4 pages, revtex, no figure
Static avalanches and Giant stress fluctuations in Silos
We propose a simple model for arch formation in silos. We show that small
pertubations (such as the thermal expansion of the beads) may lead to giant
stress fluctuations on the bottom plate of the silo. The relative amplitude
of these fluctuations are found to be power-law distributed, as
, . These fluctuations are related to large
scale `static avalanches', which correspond to long-range redistributions of
stress paths within the silo.Comment: 10 pages, 4 figures.p
Super-roughening versus intrinsic anomalous scaling of surfaces
In this paper we study kinetically rough surfaces which display anomalous
scaling in their local properties such as roughness, or height-height
correlation function. By studying the power spectrum of the surface and its
relation to the height-height correlation, we distinguish two independent
causes for anomalous scaling. One is super-roughening (global roughness
exponent larger than or equal to one), even if the spectrum behaves non
anomalously. Another cause is what we term an intrinsically anomalous spectrum,
in whose scaling an independent exponent exists, which induces different
scaling properties for small and large length scales (that is, the surface is
not self-affine). In this case, the surface does not need to be super-rough in
order to display anomalous scaling. In both cases, we show how to extract the
independent exponents and scaling relations from the correlation functions, and
we illustrate our analysis with two exactly solvable examples. One is the
simplest linear equation for molecular beam epitaxy , well known to display
anomalous scaling due to super-roughening. The second example is a random
diffusion equation, which features anomalous scaling independent of the value
of the global roughness exponent below or above one.Comment: 9 pages, 6 figures, Revtex (uses epsfig), Phys. Rev. E, submitte
Velocity fluctuations in forced Burgers turbulence
We propose a simple method to compute the velocity difference statistics in
forced Burgers turbulence in any dimension. Within a reasonnable assumption
concerning the nucleation and coalescence of shocks, we find in particular that
the `left' tail of the distribution decays as an inverse square power, which is
compatible with numerical data. Our results are compared to those of various
recent approaches: instantons, operator product expansion, replicas.Comment: 10 pages latex, one postcript figur
Singularities and Avalanches in Interface Growth with Quenched Disorder
A simple model for an interface moving in a disordered medium is presented.
The model exhibits a transition between the two universality classes of
interface growth phenomena. Using this model, it is shown that the application
of constraints to the local slopes of the interface produces avalanches of
growth, that become relevant in the vicinity of the depinning transition. The
study of these avalanches reveals a singular behavior that explains a recently
observed singularity in the equation of motion of the interface.Comment: 4 pages. REVTEX. 4 figs available on request from [email protected]
Derivation of continuum stochastic equations for discrete growth models
We present a formalism to derive the stochastic differential equations (SDEs)
for several solid-on-solid growth models. Our formalism begins with a mapping
of the microscopic dynamics of growth models onto the particle systems with
reactions and diffusion. We then write the master equations for these
corresponding particle systems and find the SDEs for the particle densities.
Finally, by connecting the particle densities with the growth heights, we
derive the SDEs for the height variables. Applying this formalism to discrete
growth models, we find the Edwards-Wilkinson equation for the symmetric
body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation
for the asymmetric BCSOS model and the generalized restricted solid-on-solid
(RSOS) model, and the Villain--Lai--Das Sarma equation for the conserved RSOS
model. In addition to the consistent forms of equations for growth models, we
also obtain the coefficients associated with the SDEs.Comment: 5 pages, no figur
Noisy Kuramoto-Sivashinsky equation for an erosion model
We derive the continuum equation for a discrete model for ion sputtering. We
follow an approach based on the master equation, and discuss how it can be
truncated to a Fokker-Planck equation and mapped to a discrete Langevin
equation. By taking the continuum limit, we arrive at the Kuramoto-Sivashinsky
equation with a stochastic noise term.Comment: latex (w/ multicol.sty), 4 pages; to appear in Physical Review E (Oct
1996
Stochastic Model for Surface Erosion Via Ion-Sputtering: Dynamical Evolution from Ripple Morphology to Rough Morphology
Surfaces eroded by ion-sputtering are sometimes observed to develop
morphologies which are either ripple (periodic), or rough (non-periodic). We
introduce a discrete stochastic model that allows us to interpret these
experimental observations within a unified framework. We find that a periodic
ripple morphology characterizes the initial stages of the evolution, whereas
the surface displays self-affine scaling in the later time regime. Further, we
argue that the stochastic continuum equation describing the surface height is a
noisy version of the Kuramoto-Sivashinsky equation.Comment: 4 pages, 7 postscript figs., Revtex, to appear in Phys. Rev. Let
Flame front propagation I: The Geometry of Developing Flame Fronts: Analysis with Pole Decomposition
The roughening of expanding flame fronts by the accretion of cusp-like
singularities is a fascinating example of the interplay between instability,
noise and nonlinear dynamics that is reminiscent of self-fractalization in
Laplacian growth patterns. The nonlinear integro-differential equation that
describes the dynamics of expanding flame fronts is amenable to analytic
investigations using pole decomposition. This powerful technique allows the
development of a satisfactory understanding of the qualitative and some
quantitative aspects of the complex geometry that develops in expanding flame
fronts.Comment: 4 pages, 2 figure
Renormalization Group Analysis of a Noisy Kuramoto-Sivashinsky Equation
We have analyzed the Kuramoto-Sivashinsky equation with a stochastic noise
term through a dynamic renormalization group calculation. For a system in which
the lattice spacing is smaller than the typical wavelength of the linear
instability occurring in the system, the large-distance and long-time behavior
of this equation is the same as for the Kardar-Parisi-Zhang equation in one and
two spatial dimensions. For the case the agreement is only qualitative.
On the other hand, when coarse-graining on larger scales the asymptotic flow
depends on the initial values of the parameters.Comment: 8 pages, 5 figures, revte