562 research outputs found
Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model
We study the dynamic behavior of the Ziff-Gulari-Barshad (ZGB) irreversible
surface-reaction model around its kinetic second-order phase transition, using
both epidemic and poisoning-time analyses. We find that the critical point is
given by p_1 = 0.3873682 \pm 0.0000015, which is lower than the previous value.
We also obtain precise values of the dynamical critical exponents z, \delta,
and \eta which provide further numerical evidence that this transition is in
the same universality class as directed percolation.Comment: REVTEX, 4 pages, 5 figures, Submitted to Physical Review
Recent advances and open challenges in percolation
Percolation is the paradigm for random connectivity and has been one of the
most applied statistical models. With simple geometrical rules a transition is
obtained which is related to magnetic models. This transition is, in all
dimensions, one of the most robust continuous transitions known. We present a
very brief overview of more than 60 years of work in this area and discuss
several open questions for a variety of models, including classical, explosive,
invasion, bootstrap, and correlated percolation
Efficient Monte Carlo algorithm and high-precision results for percolation
We present a new Monte Carlo algorithm for studying site or bond percolation
on any lattice. The algorithm allows us to calculate quantities such as the
cluster size distribution or spanning probability over the entire range of site
or bond occupation probabilities from zero to one in a single run which takes
an amount of time scaling linearly with the number of sites on the lattice. We
use our algorithm to determine that the percolation transition occurs at
occupation probability 0.59274621(13) for site percolation on the square
lattice and to provide clear numerical confirmation of the conjectured
4/3-power stretched-exponential tails in the spanning probability functions.Comment: 8 pages, including 3 postscript figures, minor corrections in this
version, plus updated figures for the position of the percolation transitio
Determination of the bond percolation threshold for the Kagome lattice
The hull-gradient method is used to determine the critical threshold for bond
percolation on the two-dimensional Kagome lattice (and its dual, the dice
lattice). For this system, the hull walk is represented as a self-avoiding
trail, or mirror-model trajectory, on the (3,4,6,4)-Archimedean tiling lattice.
The result pc = 0.524 405 3(3) (one standard deviation of error) is not
consistent with the previously conjectured values.Comment: 10 pages, TeX, Style file iopppt.tex, to be published in J. Phys. A.
in August, 199
Kinetics of catalysis with surface disorder
We study the effects of generalised surface disorder on the monomer-monomer
model of heterogeneous catalysis, where disorder is implemented by allowing
different adsorption rates for each lattice site. By mapping the system in the
reaction-controlled limit onto a kinetic Ising model, we derive the rate
equations for the one and two-spin correlation functions. There is good
agreement between these equations and numerical simulations. We then study the
inclusion of desorption of monomers from the substrate, first by both species
and then by just one, and find exact time-dependent solutions for the one-spin
correlation functions.Comment: LaTex, 19 pages, 1 figure included, requires epsf.st
The Largest Cluster in Subcritical Percolation
The statistical behavior of the size (or mass) of the largest cluster in
subcritical percolation on a finite lattice of size is investigated (below
the upper critical dimension, presumably ). It is argued that as the cumulative distribution function converges to the Fisher-Tippett
(or Gumbel) distribution in a certain weak sense (when suitably
normalized). The mean grows like , where is a
``crossover size''. The standard deviation is bounded near with persistent fluctuations due to discreteness. These
predictions are verified by Monte Carlo simulations on square lattices of
up to 30 million sites, which also reveal finite-size scaling. The results are
explained in terms of a flow in the space of probability distributions as . The subcritical segment of the physical manifold ()
approaches a line of limit cycles where the flow is approximately described by
a ``renormalization group'' from the classical theory of extreme order
statistics.Comment: 16 pages, 5 figs, expanded version to appear in Phys Rev
Percolation Threshold, Fisher Exponent, and Shortest Path Exponent for 4 and 5 Dimensions
We develop a method of constructing percolation clusters that allows us to
build very large clusters using very little computer memory by limiting the
maximum number of sites for which we maintain state information to a number of
the order of the number of sites in the largest chemical shell of the cluster
being created. The memory required to grow a cluster of mass s is of the order
of bytes where ranges from 0.4 for 2-dimensional lattices
to 0.5 for 6- (or higher)-dimensional lattices. We use this method to estimate
, the exponent relating the minimum path to the
Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site
and bond percolation, we find (4D) and
(5D). In order to determine
to high precision, and without bias, it was necessary to
first find precise values for the percolation threshold, :
(4D) and (5D) for site and
(4D) and (5D) for bond
percolation. We also calculate the Fisher exponent, , determined in the
course of calculating the values of : (4D) and
(5D)
Catalytic CO Oxidation on Nanoscale Pt Facets: Effect of Inter-Facet CO Diffusion on Bifurcation and Fluctuation Behavior
We present lattice-gas modeling of the steady-state behavior in CO oxidation
on the facets of nanoscale metal clusters, with coupling via inter-facet CO
diffusion. The model incorporates the key aspects of reaction process, such as
rapid CO mobility within each facet, and strong nearest-neighbor repulsion
between adsorbed O. The former justifies our use a "hybrid" simulation approach
treating the CO coverage as a mean-field parameter. For an isolated facet,
there is one bistable region where the system can exist in either a reactive
state (with high oxygen coverage) or a (nearly CO-poisoned) inactive state.
Diffusion between two facets is shown to induce complex multistability in the
steady states of the system. The bifurcation diagram exhibits two regions with
bistabilities due to the difference between adsorption properties of the
facets. We explore the role of enhanced fluctuations in the proximity of a cusp
bifurcation point associated with one facet in producing transitions between
stable states on that facet, as well as their influence on fluctuations on the
other facet. The results are expected to shed more light on the reaction
kinetics for supported catalysts.Comment: 22 pages, RevTeX, to appear in Phys. Rev. E, 6 figures (eps format)
are available at http://www.physik.tu-muenchen.de/~natali
Percolation on two- and three-dimensional lattices
In this work we apply a highly efficient Monte Carlo algorithm recently
proposed by Newman and Ziff to treat percolation problems. The site and bond
percolation are studied on a number of lattices in two and three dimensions.
Quite good results for the wrapping probabilities, correlation length critical
exponent and critical concentration are obtained for the square, simple cubic,
HCP and hexagonal lattices by using relatively small systems. We also confirm
the universal aspect of the wrapping probabilities regarding site and bond
dilution.Comment: 15 pages, 6 figures, 3 table
Self-Organized Dynamical Equilibrium in the Corrosion of Random Solids
Self-organized criticality is characterized by power law correlations in the
non-equilibrium steady state of externally driven systems. A dynamical system
proposed here self-organizes itself to a critical state with no characteristic
size at ``dynamical equilibrium''. The system is a random solid in contact with
an aqueous solution and the dynamics is the chemical reaction of corrosion or
dissolution of the solid in the solution. The initial difference in chemical
potential at the solid-liquid interface provides the driving force. During time
evolution, the system undergoes two transitions, roughening and
anti-percolation. Finally, the system evolves to a dynamical equilibrium state
characterized by constant chemical potential and average cluster size. The
cluster size distribution exhibits power law at the final equilibrium state.Comment: 11 pages, 5 figure
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