151 research outputs found
Invading interfaces and blocking surfaces in high dimensional disordered systems
We study the high-dimensional properties of an invading front in a disordered
medium with random pinning forces. We concentrate on interfaces described by
bounded slope models belonging to the quenched KPZ universality class. We find
a number of qualitative transitions in the behavior of the invasion process as
dimensionality increases. In low dimensions the system is characterized
by two different roughness exponents, the roughness of individual avalanches
and the overall interface roughness. We use the similarity of the dynamics of
an avalanche with the dynamics of invasion percolation to show that above
avalanches become flat and the invasion is well described as an annealed
process with correlated noise. In fact, for the overall roughness is
the same as the annealed roughness. In very large dimensions, strong
fluctuations begin to dominate the size distribution of avalanches, and this
phenomenon is studied on the Cayley tree, which serves as an infinite
dimensional limit. We present numerical simulations in which we measured the
values of the critical exponents of the depinning transition, both in finite
dimensional lattices with and on the Cayley tree, which support our
qualitative predictions. We find that the critical exponents in are very
close to their values on the Cayley tree, and we conjecture on this basis the
existence of a further dimension, where mean field behavior is obtained.Comment: 12 pages, REVTeX with 2 postscript figure
On the random neighbor Olami-Feder-Christensen slip-stick model
We reconsider the treatment of Lise and Jensen (Phys. Rev. Lett. 76, 2326
(1996)) on the random neighbor Olami-Feder-Christensen stik-slip model, and
examine the strong dependence of the results on the approximations used for the
distribution of states p(E).Comment: 6pages, 3 figures. To be published in PRE as a brief repor
Exact Results for the One-Dimensional Self-Organized Critical Forest-Fire Model
We present the analytic solution of the self-organized critical (SOC)
forest-fire model in one dimension proving SOC in systems without conservation
laws by analytic means. Under the condition that the system is in the steady
state and very close to the critical point, we calculate the probability that a
string of neighboring sites is occupied by a given configuration of trees.
The critical exponent describing the size distribution of forest clusters is
exactly and does not change under certain changes of the model
rules. Computer simulations confirm the analytic results.Comment: 12 pages REVTEX, 2 figures upon request, dro/93/
Quasi-Static Brittle Fracture in Inhomogeneous Media and Iterated Conformal Maps: Modes I, II and III
The method of iterated conformal maps is developed for quasi-static fracture
of brittle materials, for all modes of fracture. Previous theory, that was
relevant for mode III only, is extended here to mode I and II. The latter
require solution of the bi-Laplace rather than the Laplace equation. For all
cases we can consider quenched randomness in the brittle material itself, as
well as randomness in the succession of fracture events. While mode III calls
for the advance (in time) of one analytic function, mode I and II call for the
advance of two analytic functions. This fundamental difference creates
different stress distribution around the cracks. As a result the geometric
characteristics of the cracks differ, putting mode III in a different class
compared to modes I and II.Comment: submitted to PRE For a version with qualitatively better figures see:
http://www.weizmann.ac.il/chemphys/ander
Flame front propagation V: Stability Analysis of Flame Fronts: Dynamical Systems Approach in the Complex Plane
We consider flame front propagation in channel geometries. The steady state
solution in this problem is space dependent, and therefore the linear stability
analysis is described by a partial integro-differential equation with a space
dependent coefficient. Accordingly it involves complicated eigenfunctions. We
show that the analysis can be performed to required detail using a finite order
dynamical system in terms of the dynamics of singularities in the complex
plane, yielding detailed understanding of the physics of the eigenfunctions and
eigenvalues.Comment: 17 pages 7 figure
Self-organized criticality and synchronization in a lattice model of integrate-and-fire oscillators
We introduce two coupled map lattice models with nonconservative interactions
and a continuous nonlinear driving. Depending on both the degree of
conservation and the convexity of the driving we find different behaviors,
ranging from self-organized criticality, in the sense that the distribution of
events (avalanches) obeys a power law, to a macroscopic synchronization of the
population of oscillators, with avalanches of the size of the system.Comment: 4 pages, Revtex 3.0, 3 PostScript figures available upon request to
[email protected]
Symmetries and Fixed Point Stability of Stochastic Differential Equations Modeling Self-Organized Criticality
A stochastic nonlinear partial differential equation is built for two
different models exhibiting self-organized criticality, the Bak, Tang, and
Wiesenfeld (BTW) sandpile model and the Zhang's model. The dynamic
renormalization group (DRG) enables to compute the critical exponents. However,
the nontrivial stable fixed point of the DRG transformation is unreachable for
the original parameters of the models. We introduce an alternative
regularization of the step function involved in the threshold condition, which
breaks the symmetry of the BTW model. Although the symmetry properties of the
two models are different, it is shown that they both belong to the same
universality class. In this case the DRG procedure leads to a symmetric
behavior for both models, restoring the broken symmetry, and makes accessible
the nontrivial fixed point. This technique could also be applied to other
problems with threshold dynamics.Comment: 19 pages, RevTex, includes 6 PostScript figures, Phys. Rev. E (March
97?
Flame front propagation IV: Random Noise and Pole-Dynamics in Unstable Front Propagation II
The current paper is a corrected version of our previous paper
arXiv:adap-org/9608001. Similarly to previous version we investigate the
problem of flame propagation. This problem is studied as an example of unstable
fronts that wrinkle on many scales. The analytic tool of pole expansion in the
complex plane is employed to address the interaction of the unstable growth
process with random initial conditions and perturbations. We argue that the
effect of random noise is immense and that it can never be neglected in
sufficiently large systems. We present simulations that lead to scaling laws
for the velocity and acceleration of the front as a function of the system size
and the level of noise, and analytic arguments that explain these results in
terms of the noisy pole dynamics.This version corrects some very critical
errors made in arXiv:adap-org/9608001 and makes more detailed description of
excess number of poles in system, number of poles that appear in the system in
unit of time, life time of pole. It allows us to understand more correctly
dependence of the system parameters on noise than in arXiv:adap-org/9608001Comment: 23 pages, 4 figures,revised, version accepted for publication in
journal "Combustion, Explosion and Shock Waves". arXiv admin note:
substantial text overlap with arXiv:nlin/0302021, arXiv:adap-org/9608001,
arXiv:nlin/030201
Anomalous Height Fluctuation Width in Crossover from Random to Coherent Surface Growths
We study an anomalous behavior of the height fluctuation width in the
crossover from random to coherent growths of surface for a stochastic model. In
the model, random numbers are assigned on perimeter sites of surface,
representing pinning strengths of disordered media. At each time, surface is
advanced at the site having minimum pinning strength in a random subset of
system rather than having global minimum. The subset is composed of a randomly
selected site and its neighbors. The height fluctuation width
exhibits the non-monotonic behavior with and it has a
minimum at . It is found numerically that scales as
, and the height fluctuation width at that minimum,
, scales as in 1+1 dimensions. It is found that
the subset-size is the characteristic size of the crossover from
the random surface growth in the KPZ universality, to the coherent surface
growth in the directed percolation universality.Comment: 13 postscript file
Crossover from Percolation to Self-Organized Criticality
We include immunity against fire as a new parameter into the self-organized
critical forest-fire model. When the immunity assumes a critical value,
clusters of burnt trees are identical to percolation clusters of random bond
percolation. As long as the immunity is below its critical value, the
asymptotic critical exponents are those of the original self-organized critical
model, i.e. the system performs a crossover from percolation to self-organized
criticality. We present a scaling theory and computer simulation results.Comment: 4 pages Revtex, two figures included, to be published in PR
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