517 research outputs found
Non-Local Product Rules for Percolation
Despite original claims of a first-order transition in the product rule model
proposed by Achlioptas et al. [Science 323, 1453 (2009)], recent studies
indicate that this percolation model, in fact, displays a continuous
transition. The distinctive scaling properties of the model at criticality,
however, strongly suggest that it should belong to a different universality
class than ordinary percolation. Here we introduce a generalization of the
product rule that reveals the effect of non-locality on the critical behavior
of the percolation process. Precisely, pairs of unoccupied bonds are chosen
according to a probability that decays as a power-law of their Manhattan
distance, and only that bond connecting clusters whose product of their sizes
is the smallest, becomes occupied. Interestingly, our results for
two-dimensional lattices at criticality shows that the power-law exponent of
the product rule has a significant influence on the finite-size scaling
exponents for the spanning cluster, the conducting backbone, and the cutting
bonds of the system. In all three cases, we observe a continuous variation from
ordinary to (non-local) explosive percolation exponents.Comment: 5 pages, 4 figure
Non-Newtonian fluid flow through three-dimensional disordered porous media
We investigate the flow of various non-Newtonian fluids through
three-dimensional disordered porous media by direct numerical simulation of
momentum transport and continuity equations. Remarkably, our results for
power-law (PL) fluids indicate that the flow, when quantified in terms of a
properly modified permeability-like index and Reynolds number, can be
successfully described by a single (universal) curve over a broad range of
Reynolds conditions and power-law exponents. We also study the flow behavior of
Bingham fluids described in terms of the Herschel-Bulkley model. In this case,
our simulations reveal that the interplay of ({\it i}) the disordered geometry
of the pore space, ({\it ii}) the fluid rheological properties, and ({\it iii})
the inertial effects on the flow is responsible for a substantial enhancement
of the macroscopic hydraulic conductance of the system at intermediate Reynolds
conditions. This anomalous condition of ``enhanced transport'' represents a
novel feature for flow in porous materials.Comment: 5 pages, 5 figures. This article appears also in Physical Review
Letters 103 194502 (2009
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
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
Non-universality of elastic exponents in random bond-bending networks
We numerically investigate the rigidity percolation transition in
two-dimensional flexible, random rod networks with freely rotating cross-links.
Near the transition, networks are dominated by bending modes and the elastic
modulii vanish with an exponent f=3.0\pm0.2, in contrast with central force
percolation which shares the same geometric exponents. This indicates that
universality for geometric quantities does not imply universality for elastic
ones. The implications of this result for actin-fiber networks is discussed.Comment: 4 pages, 3 figures, minor clarifications and amendments. To appear in
PRE Rap. Com
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
A mean field description of jamming in non-cohesive frictionless particulate systems
A theory for kinetic arrest in isotropic systems of repulsive,
radially-interacting particles is presented that predicts exponents for the
scaling of various macroscopic quantities near the rigidity transition that are
in agreement with simulations, including the non-trivial shear exponent. Both
statics and dynamics are treated in a simplified, one-particle level
description, and coupled via the assumption that kinetic arrest occurs on the
boundary between mechanically stable and unstable regions of the static
parameter diagram. This suggests the arrested states observed in simulations
are at (or near) an elastic buckling transition. Some additional numerical
evidence to confirm the scaling of microscopic quantities is also provided.Comment: 9 pages, 3 figs; additional clarification of different elastic moduli
exponents, plus typo fix. To appear in PR
Universal Formulae for Percolation Thresholds
A power law is postulated for both site and bond percolation thresholds. The
formula writes , where is the space
dimension and the coordination number. All thresholds up to are found to belong to only three universality classes. For first two
classes for site dilution while for bond dilution. The last one
associated to high dimensions is characterized by for both sites and
bonds. Classes are defined by a set of value for . Deviations
from available numerical estimates at are within and
for high dimensional hypercubic expansions at . The
formula is found to be also valid for Ising critical temperatures.Comment: 11 pages, latex, 3 figures not include
Fracture of disordered solids in compression as a critical phenomenon: I. Statistical mechanics formalism
This is the first of a series of three articles that treats fracture
localization as a critical phenomenon. This first article establishes a
statistical mechanics based on ensemble averages when fluctuations through time
play no role in defining the ensemble. Ensembles are obtained by dividing a
huge rock sample into many mesoscopic volumes. Because rocks are a disordered
collection of grains in cohesive contact, we expect that once shear strain is
applied and cracks begin to arrive in the system, the mesoscopic volumes will
have a wide distribution of different crack states. These mesoscopic volumes
are the members of our ensembles. We determine the probability of observing a
mesoscopic volume to be in a given crack state by maximizing Shannon's measure
of the emergent crack disorder subject to constraints coming from the
energy-balance of brittle fracture. The laws of thermodynamics, the partition
function, and the quantification of temperature are obtained for such cracking
systems.Comment: 11 pages, 2 figure
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