554 research outputs found
Conductance of Finite-Scale Systems with Multiple Percolation Channels
We investigate properties of two-dimensional finite-scale percolation systems
whose size along the current flow is smaller than the perpendicular size.
Successive thresholds of appearing multiple percolation channels in such
systems have been determined and dependencies of the conductance on their size
and percolation parameter have been calculated. Various experimental
examples show that the finite-scale percolation system is the natural
mathematical model suitable for the qualitative and quantitative description of
different physical systems.Comment: 11 pages, 7 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
Phase diagram and universality of the Lennard-Jones gas-liquid system
The gas-liquid phase transition of the three-dimensional Lennard-Jones
particles system is studied by molecular dynamics simulations. The gas and
liquid densities in the coexisting state are determined with high accuracy. The
critical point is determined by the block density analysis of the Binder
parameter with the aid of the law of rectilinear diameter. From the critical
behavior of the gas-liquid coexsisting density, the critical exponent of the
order parameter is estimated to be . Surface tension is
estimated from interface broadening behavior due to capillary waves. From the
critical behavior of the surface tension, the critical exponent of the
correlation length is estimated to be . The obtained values of
and are consistent with those of the Ising universality class.Comment: 8 pages, 8 figures, new results are adde
Stochastic Renormalization Group in Percolation: I. Fluctuations and Crossover
A generalization of the Renormalization Group, which describes
order-parameter fluctuations in finite systems, is developed in the specific
context of percolation. This ``Stochastic Renormalization Group'' (SRG)
expresses statistical self-similarity through a non-stationary branching
process. The SRG provides a theoretical basis for analytical or numerical
approximations, both at and away from criticality, whenever the correlation
length is much larger than the lattice spacing (regardless of the system size).
For example, the SRG predicts order-parameter distributions and finite-size
scaling functions for the complete crossover between phases. For percolation,
the simplest SRG describes structural quantities conditional on spanning, such
as the total cluster mass or the minimum chemical distance between two
boundaries. In these cases, the Central Limit Theorem (for independent random
variables) holds at the stable, off-critical fixed points, while a ``Fractal
Central Limit Theorem'' (describing long-range correlations) holds at the
unstable, critical fixed point. This first part of a series of articles
explains these basic concepts and a general theory of crossover. Subsequent
parts will focus on limit theorems and comparisons of small-cell SRG
approximations with simulation results.Comment: 33 pages, 6 figures, to appear in Physica A; v2: some typos corrected
and Eqs. (26)-(27) cast in a simpler (but equivalent) for
Use of Ambr15 as a high throughput model to speed up perfusion bioprocess development
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Excess number of percolation clusters on the surface of a sphere
Monte Carlo simulations were performed in order to determine the excess
number of clusters b and the average density of clusters n_c for the
two-dimensional "Swiss cheese" continuum percolation model on a planar L x L
system and on the surface of a sphere. The excess number of clusters for the L
x L system was confirmed to be a universal quantity with a value b = 0.8841 as
previously predicted and verified only for lattice percolation. The excess
number of clusters on the surface of a sphere was found to have the value b =
1.215(1) for discs with the same coverage as the flat critical system. Finally,
the average critical density of clusters was calculated for continuum systems
n_c = 0.0408(1).Comment: 13 pages, 2 figure
Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters
Percolation models with multiple percolating clusters have attracted much
attention in recent years. Here we use Monte Carlo simulations to study bond
percolation on planar random lattices, duals of random
lattices, and square lattices with free and periodic boundary conditions, in
vertical and horizontal directions, respectively, and with various aspect ratio
. We calculate the probability for the appearance of
percolating clusters, the percolating probabilities, , the average
fraction of lattice bonds (sites) in the percolating clusters,
(), and the probability distribution function for the fraction
of lattice bonds (sites), in percolating clusters of subgraphs with
percolating clusters, (). Using a small number of
nonuniversal metric factors, we find that , ,
(), and () for random lattices, duals
of random lattices, and square lattices have the same universal finite-size
scaling functions. We also find that nonuniversal metric factors are
independent of boundary conditions and aspect ratios.Comment: 15 pages, 11 figure
Pseudorandom Number Generators and the Square Site Percolation Threshold
A select collection of pseudorandom number generators is applied to a Monte
Carlo study of the two dimensional square site percolation model. A generator
suitable for high precision calculations is identified from an application
specific test of randomness. After extended computation and analysis, an
ostensibly reliable value of pc = 0.59274598(4) is obtained for the percolation
threshold.Comment: 11 pages, 6 figure
Temporal Series Analysis Approach to Spectra of Complex Networks
The spacing of nearest levels of the spectrum of a complex network can be
regarded as a time series. Joint use of Multi-fractal Detrended Fluctuation
Approach (MF-DFA) and Diffusion Entropy (DE) is employed to extract
characteristics from this time series. For the WS (Watts and Strogatz)
small-world model, there exist a critical point at rewiring probability . For a
network generated in the range, the correlation exponent is in the range of .
Above this critical point, all the networks behave similar with that at . For
the ER model, the time series behaves like FBM (fractional Brownian motion)
noise at . For the GRN (growing random network) model, the values of the
long-range correlation exponent are in the range of . For most of the GRN
networks the PDF of a constructed time series obeys a Gaussian form. In the
joint use of MF-DFA and DE, the shuffling procedure in DE is essential to
obtain a reliable result. PACS number(s): 89.75.-k, 05.45.-a, 02.60.-xComment: 10 pages, 9 figures, to appear in PR
Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensional XY model
We study a percolation problem on a substrate formed by two-dimensional XY
spin configurations, using Monte Carlo methods. For a given spin configuration
we construct percolation clusters by randomly choosing a direction in the
spin vector space, and then placing a percolation bond between nearest-neighbor
sites and with probability ,
where governs the percolation process. A line of percolation thresholds
is found in the low-temperature range , where
is the XY coupling strength. Analysis of the correlation function , defined as the probability that two sites separated by a distance
belong to the same percolation cluster, yields algebraic decay for , and the associated critical exponent depends on and .
Along the threshold line , the scaling dimension for is,
within numerical uncertainties, equal to . On this basis, we conjecture
that the percolation transition along the line is of the
Berezinskii-Kosterlitz-Thouless type.Comment: 23 pages, 14 figure
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