240 research outputs found
Gaussian model of explosive percolation in three and higher dimensions
The Gaussian model of discontinuous percolation, recently introduced by
Ara\'ujo and Herrmann [Phys. Rev. Lett., 105, 035701 (2010)], is numerically
investigated in three dimensions, disclosing a discontinuous transition. For
the simple-cubic lattice, in the thermodynamic limit, we report a finite jump
of the order parameter, . The largest cluster at the
threshold is compact, but its external perimeter is fractal with fractal
dimension . The study is extended to hypercubic lattices up
to six dimensions and to the mean-field limit (infinite dimension). We find
that, in all considered dimensions, the percolation transition is
discontinuous. The value of the jump in the order parameter, the maximum of the
second moment, and the percolation threshold are analyzed, revealing
interesting features of the transition and corroborating its discontinuous
nature in all considered dimensions. We also show that the fractal dimension of
the external perimeter, for any dimension, is consistent with the one from
bridge percolation and establish a lower bound for the percolation threshold of
discontinuous models with finite number of clusters at the threshold
Localization of elastic waves in heterogeneous media with off-diagonal disorder and long-range correlations
Using the Martin-Siggia-Rose method, we study propagation of acoustic waves
in strongly heterogeneous media which are characterized by a broad distribution
of the elastic constants. Gaussian-white distributed elastic constants, as well
as those with long-range correlations with non-decaying power-law correlation
functions, are considered. The study is motivated in part by a recent discovery
that the elastic moduli of rock at large length scales may be characterized by
long-range power-law correlation functions. Depending on the disorder, the
renormalization group (RG) flows exhibit a transition to localized regime in
{\it any} dimension. We have numerically checked the RG results using the
transfer-matrix method and direct numerical simulations for one- and
two-dimensional systems, respectively.Comment: 5 pages, 4 figures, to appear in Phys. Rev. Let
Transport on exploding percolation clusters
We propose a simple generalization of the explosive percolation process
[Achlioptas et al., Science 323, 1453 (2009)], and investigate its structural
and transport properties. In this model, at each step, a set of q unoccupied
bonds is randomly chosen. Each of these bonds is then associated with a weight
given by the product of the cluster sizes that they would potentially connect,
and only that bond among the q-set which has the smallest weight becomes
occupied. Our results indicate that, at criticality, all finite-size scaling
exponents for the spanning cluster, the conducting backbone, the cutting bonds,
and the global conductance of the system, change continuously and significantly
with q. Surprisingly, we also observe that systems with intermediate values of
q display the worst conductive performance. This is explained by the strong
inhibition of loops in the spanning cluster, resulting in a substantially
smaller associated conducting backbone.Comment: 4 pages, 4 figure
Fluid flow at the interface between elastic solids with randomly rough surfaces
I study fluid flow at the interface between elastic solids with randomly
rough surfaces. I use the contact mechanics model of Persson to take into
account the elastic interaction between the solid walls and the Bruggeman
effective medium theory to account for the influence of the disorder on the
fluid flow. I calculate the flow tensor which determines the pressure flow
factor and, e.g., the leak-rate of static seals. I show how the perturbation
treatment of Tripp can be extended to arbitrary order in the ratio between the
root-mean-square roughness amplitude and the average interfacial surface
separation. I introduce a matrix D(Zeta), determined by the surface roughness
power spectrum, which can be used to describe the anisotropy of the surface at
any magnification Zeta. I present results for the asymmetry factor Gamma(Zeta)
(generalized Peklenik number) for grinded steel and sandblasted PMMA surfaces.Comment: 16 pages, 14 figure
Anisotropic generalization of Stinchcombe's solution for conductivity of random resistor network on a Bethe lattice
Our study is based on the work of Stinchcombe [1974 \emph{J. Phys. C}
\textbf{7} 179] and is devoted to the calculations of average conductivity of
random resistor networks placed on an anisotropic Bethe lattice. The structure
of the Bethe lattice is assumed to represent the normal directions of the
regular lattice. We calculate the anisotropic conductivity as an expansion in
powers of inverse coordination number of the Bethe lattice. The expansion terms
retained deliver an accurate approximation of the conductivity at resistor
concentrations above the percolation threshold. We make a comparison of our
analytical results with those of Bernasconi [1974 \emph{Phys. Rev. B}
\textbf{9} 4575] for the regular lattice.Comment: 14 pages, 2 figure
Marketing Percolation
A percolation model is presented, with computer simulations for
illustrations, to show how the sales of a new product may penetrate the
consumer market. We review the traditional approach in the marketing
literature, which is based on differential or difference equations similar to
the logistic equation (Bass 1969). This mean field approach is contrasted with
the discrete percolation on a lattice, with simulations of "social percolation"
(Solomon et al 2000) in two to five dimensions giving power laws instead of
exponential growth, and strong fluctuations right at the percolation threshold.Comment: to appear in Physica
Nanopercolation
We investigate through direct molecular mechanics calculations the
geometrical properties of hydrocarbon mantles subjected to percolation
disorder. We show that the structures of mantles generated at the critical
percolation point have a fractal dimension . In addition,
the solvent access surface and volume of these molecules follow
power-law behavior, and ,
where is the system size, and with both critical exponents and
being significantly dependent on the radius of the accessing probing
molecule, . Our results from extensive simulations with two distinct
microscopic topologies (i.e., square and honeycomb) indicate the consistency of
the statistical analysis and confirm the self-similar characteristic of the
percolating hydrocarbons. Due to their highly branched topology, some of the
potential applications for this new class of disordered molecules include drug
delivery, catalysis, and supramolecular structures.Comment: 4 pages, 5 figure
Macroscopic Equations of Motion for Two Phase Flow in Porous Media
The established macroscopic equations of motion for two phase immiscible
displacement in porous media are known to be physically incomplete because they
do not contain the surface tension and surface areas governing capillary
phenomena. Therefore a more general system of macroscopic equations is derived
here which incorporates the spatiotemporal variation of interfacial energies.
These equations are based on the theory of mixtures in macroscopic continuum
mechanics. They include wetting phenomena through surface tensions instead of
the traditional use of capillary pressure functions. Relative permeabilities
can be identified in this approach which exhibit a complex dependence on the
state variables. A capillary pressure function can be identified in equilibrium
which shows the qualitative saturation dependence known from experiment. In
addition the new equations allow to describe the spatiotemporal changes of
residual saturations during immiscible displacement.Comment: 15 pages, Phys. Rev. E (1998), in prin
Driven interfaces in disordered media: determination of universality classes from experimental data
While there have been important theoretical advances in understanding the
universality classes of interfaces moving in porous media, the developed tools
cannot be directly applied to experiments. Here we introduce a method that can
identify the universality class from snapshots of the interface profile. We
test the method on discrete models whose universality class is well known, and
use it to identify the universality class of interfaces obtained in experiments
on fluid flow in porous media.Comment: 4 pages, 5 figure
Optimal path cracks in correlated and uncorrelated lattices
The optimal path crack model on uncorrelated surfaces, recently introduced by
Andrade et al. (Phys. Rev. Lett. 103, 225503, 2009), is studied in detail and
its main percolation exponents computed. In addition to beta/nu = 0.46 \pm 0.03
we report, for the first time, gamma/nu = 1.3 \pm 0.2 and tau = 2.3 \pm 0.2.
The analysis is extended to surfaces with spatial long-range power-law
correlations, where non-universal fractal dimensions are obtained when the
degree of correlation is varied. The model is also considered on a
three-dimensional lattice, where the main crack is found to be a surface with a
fractal dimension of 2.46 \pm 0.05.Comment: 10 pages, 10 figure
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