418 research outputs found
Alternative criterion for two-dimensional wrapping percolation
Based on the differences between a spanning cluster and a wrapping cluster,
an alternative criterion for testing wrapping percolation is provided for
two-dimensional lattices. By following the Newman-Ziff method, the finite size
scaling of estimates for percolation thresholds are given. The results are
consistent with those from Machta's method.Comment: 4 pages, 2 figure
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
A random fiber bundle with many discontinuities in the threshold distribution
We study the breakdown of a random fiber bundle model (RFBM) with
-discontinuities in the threshold distribution using the global load sharing
scheme. In other words, different classes of fibers identified on the
basis of their threshold strengths are mixed such that the strengths of the
fibers in the class are uniformly distributed between the values
and where . Moreover, there
is a gap in the threshold distribution between and class. We
show that although the critical stress depends on the parameter values of the
system, the critical exponents are identical to that obtained in the recursive
dynamics of a RFBM with a uniform distribution and global load sharing. The
avalanche size distribution (ASD), on the other hand, shows a non-universal,
non-power law behavior for smaller values of avalanche sizes which becomes
prominent only when a critical distribution is approached. We establish that
the behavior of the avalanche size distribution for an arbitrary is
qualitatively similar to a RFBM with a single discontinuity in the threshold
distribution (), especially when the density and the range of threshold
values of fibers belonging to strongest ()-th class is kept identical in
all the cases.Comment: 6 pages, 4 figures, Accepted in Phys. Rev.
Restoring site percolation on a damaged square lattice
We study how to restore site percolation on a damaged square lattice with
nearest neighbor (N) interactions. Two strategies are suggested for a
density of destroyed sites by a random attack at . In the first one, a
density of new sites are created with longer range interactions, either
next nearest neighbor (N) or next next nearest neighbor (N). In the
second one, new longer range interactions N or N are created for a
fraction of the remaining sites in addition to their N
interactions. In both cases, the values of and are tuned in order to
restore site percolation which then occurs at new percolation thresholds,
respectively , , and . Using Monte Carlo
simulations the values of the pairs , and , are calculated for the whole range . Our schemes are applicable to all regular lattices.Comment: 5 pages, revtex
Effect of discontinuity in threshold distribution on the critical behaviour of a random fiber bundle
The critical behaviour of a Random Fiber Bundle Model with mixed uniform
distribution of threshold strengths and global load sharing rule is studied
with a special emphasis on the nature of distribution of avalanches for
different parameters of the distribution. The discontinuity in the threshold
strength distribution of fibers non-trivially modifies the critical stress as
well as puts a restriction on the allowed values of parameters for which the
recursive dynamics approach holds good. The discontinuity leads to a
non-universal behaviour in the avalanche size distribution for smaller values
of avalanche size. We observe that apart from the mean field behaviour for
larger avalanches, a new behaviour for smaller avalanche size is observed as a
critical threshold distribution is approached. The phenomenological
understanding of the above result is provided using the exact analytical result
for the avalanche size distribution. Most interestingly,the prominence of
non-universal behaviour in avalanche size distribution depends on the system
parameters.Comment: 6 pages, 4 figures, text and figures modifie
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
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
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
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
Mapping functions and critical behavior of percolation on rectangular domains
The existence probability and the percolation probability of the
bond percolation on rectangular domains with different aspect ratios are
studied via the mapping functions between systems with different aspect ratios.
The superscaling behavior of and for such systems with exponents
and , respectively, found by Watanabe, Yukawa, Ito, and Hu in [Phys. Rev.
Lett. \textbf{93}, 190601 (2004)] can be understood from the lower order
approximation of the mapping functions and for and ,
respectively; the exponents and can be obtained from numerically
determined mapping functions and , respectively.Comment: 17 pages with 6 figure
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