19,557 research outputs found
Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses
In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136},
054106 (2012)], analytical results concerning the continuum percolation of
overlapping hyperparticles in -dimensional Euclidean space
were obtained, including lower bounds on the percolation threshold. In the
present investigation, we provide additional analytical results for certain
cluster statistics, such as the concentration of -mers and related
quantities, and obtain an upper bound on the percolation threshold . We
utilize the tightest lower bound obtained in the first paper to formulate an
efficient simulation method, called the {\it rescaled-particle} algorithm, to
estimate continuum percolation properties across many space dimensions with
heretofore unattained accuracy. This simulation procedure is applied to compute
the threshold and associated mean number of overlaps per particle
for both overlapping hyperspheres and oriented hypercubes for . These simulations results are compared to corresponding upper
and lower bounds on these percolation properties. We find that the bounds
converge to one another as the space dimension increases, but the lower bound
provides an excellent estimate of and , even for
relatively low dimensions. We confirm a prediction of the first paper in this
series that low-dimensional percolation properties encode high-dimensional
information. We also show that the concentration of monomers dominate over
concentration values for higher-order clusters (dimers, trimers, etc.) as the
space dimension becomes large. Finally, we provide accurate analytical
estimates of the pair connectedness function and blocking function at their
contact values for any as a function of density.Comment: 24 pages, 10 figure
Effect of Dimensionality on the Percolation Thresholds of Various -Dimensional Lattices
We show analytically that the , and Pad{\'e}
approximants of the mean cluster number for site and bond percolation on
general -dimensional lattices are upper bounds on this quantity in any
Euclidean dimension , where is the occupation probability. These results
lead to certain lower bounds on the percolation threshold that become
progressively tighter as increases and asymptotically exact as becomes
large. These lower-bound estimates depend on the structure of the
-dimensional lattice and whether site or bond percolation is being
considered. We obtain explicit bounds on for both site and bond
percolation on five different lattices: -dimensional generalizations of the
simple-cubic, body-centered-cubic and face-centered-cubic Bravais lattices as
well as the -dimensional generalizations of the diamond and kagom{\'e} (or
pyrochlore) non-Bravais lattices. These analytical estimates are used to assess
available simulation results across dimensions (up through in some
cases). It is noteworthy that the tightest lower bound provides reasonable
estimates of in relatively low dimensions and becomes increasingly
accurate as grows. We also derive high-dimensional asymptotic expansions
for for the ten percolation problems and compare them to the
Bethe-lattice approximation. Finally, we remark on the radius of convergence of
the series expansion of in powers of as the dimension grows.Comment: 37 pages, 5 figure
Diversity of Dynamics and Morphologies of Invasive Solid Tumors
Complex tumor-host interactions can significantly affect the growth dynamics
and morphologies of progressing neoplasms. The growth of a confined solid tumor
induces mechanical pressure and deformation of the surrounding
microenvironment, which in turn influences tumor growth. In this paper, we
generalize a recently developed cellular automaton model for invasive tumor
growth in heterogeneous microenvironments [Y. Jiao and S. Torquato, PLoS
Comput. Biol. 7, e1002314 (2011)] by incorporating the effects of pressure.
Specifically, we explicitly model the pressure exerted on the growing tumor due
to the deformation of the microenvironment and its effect on the local
tumor-host interface instability. Both noninvasive-proliferative growth and
invasive growth with individual cells that detach themselves from the primary
tumor and migrate into the surrounding microenvironment are investigated. We
find that while noninvasive tumors growing in "soft" homogeneous
microenvironments develop almost isotropic shapes, both high pressure and host
heterogeneity can strongly enhance malignant behavior, leading to finger-like
protrusions of the tumor surface. Moreover, we show that individual invasive
cells of an invasive tumor degrade the local extracellular matrix at the
tumor-host interface, which diminishes the fingering growth of the primary
tumor. The implications of our results for cancer diagnosis, prognosis and
therapy are discussed.Comment: 21 pages, 5 figures, invited article for the special issue "Physics
of Cancer" in AIP Advances, in pres
Constant curvature solutions of Grassmannian sigma models: (1) Holomorphic solutions
We present a general formula for the Gaussian curvature of curved holomorphic
2-spheres in Grassmannian manifolds G(m, n). We then show how to construct such
solutions with constant curvature. We also make some relevant conjectures for
the admissible constant curvatures in G(m, n) and give some explicit
expressions, in particular, for G(2, 4) and G(2, 5).Comment: 14 page
Accretion onto a Kiselev black hole
We consider accretion onto a Kiselev black hole. We obtain the fundamental
equations for accretion without the back-reaction. We determine the general
analytic expressions for the critical points and the mass accretion rate and
find the physical conditions the critical points should fulfill. The case of
polytropic gas are discussed in detail. It turns out that the quintessence
parameter plays an important role in the accretion process.Comment: 7 page
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