435 research outputs found
Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results
We achieve a detailed understanding of the -sided planar Poisson-Voronoi
cell in the limit of large . Let be the probability for a cell to
have sides. We construct the asymptotic expansion of up to
terms that vanish as . We obtain the statistics of the lengths of
the perimeter segments and of the angles between adjoining segments: to leading
order as , and after appropriate scaling, these become independent
random variables whose laws we determine; and to next order in they have
nontrivial long range correlations whose expressions we provide. The -sided
cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where
is the cell density; hence Lewis' law for the average area of
the -sided cell behaves as with . For
the cell perimeter, expressed as a function of the polar
angle , satisfies , where is known Gaussian
noise; we deduce from it the probability law for the perimeter's long
wavelength deviations from circularity. Many other quantities related to the
asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure
Note on a q-modified central limit theorem
A q-modified version of the central limit theorem due to Umarov et al.
affirms that q-Gaussians are attractors under addition and rescaling of certain
classes of strongly correlated random variables. The proof of this theorem
rests on a nonlinear q-modified Fourier transform. By exhibiting an invariance
property we show that this Fourier transform does not have an inverse. As a
consequence, the theorem falls short of achieving its stated goal.Comment: 10 pages, no figure
Some Open Points in Nonextensive Statistical Mechanics
We present and discuss a list of some interesting points that are currently
open in nonextensive statistical mechanics. Their analytical, numerical,
experimental or observational advancement would naturally be very welcome.Comment: 30 pages including 6 figures. Invited paper to appear in the
International Journal of Bifurcation and Chao
Heuristic theory for many-faced d-dimensional Poisson-Voronoi cells
We consider the d-dimensional Poisson-Voronoi tessellation and investigate
the applicability of heuristic methods developed recently for two dimensions.
Let p_n(d) be the probability that a cell have n neighbors (be `n-faced') and
m_n(d) the average facedness of a cell adjacent to an n-faced cell. We obtain
the leading order terms of the asymptotic large-n expansions for p_n(d) and
m_n(3). It appears that, just as in dimension two, the Poisson-Voronoi
tessellation violates Aboav's `linear law' also in dimension three. A
confrontation of this statement with existing Monte Carlo work remains
inconclusive. However, simulations upgraded to the level of present-day
computer capacity will in principle be able to confirm (or invalidate) our
theory.Comment: 17 pages, 6 figure
Frozen shuffle update for an asymmetric exclusion process on a ring
We introduce a new rule of motion for a totally asymmetric exclusion process
(TASEP) representing pedestrian traffic on a lattice. Its characteristic
feature is that the positions of the pedestrians, modeled as hard-core
particles, are updated in a fixed predefined order, determined by a phase
attached to each of them. We investigate this model analytically and by Monte
Carlo simulation on a one-dimensional lattice with periodic boundary
conditions. At a critical value of the particle density a transition occurs
from a phase with `free flow' to one with `jammed flow'. We are able to
analytically predict the current-density diagram for the infinite system and to
find the scaling function that describes the finite size rounding at the
transition point.Comment: 16 page
The perimeter of large planar Voronoi cells: a double-stranded random walk
Let be the probability for a planar Poisson-Voronoi cell to have
exactly sides. We construct the asymptotic expansion of up to
terms that vanish as . We show that {\it two independent biased
random walks} executed by the polar angle determine the trajectory of the cell
perimeter. We find the limit distribution of (i) the angle between two
successive vertex vectors, and (ii) the one between two successive perimeter
segments. We obtain the probability law for the perimeter's long wavelength
deviations from circularity. We prove Lewis' law and show that it has
coefficient 1/4.Comment: Slightly extended version; journal reference adde
Single-site approximation for reaction-diffusion processes
We consider the branching and annihilating random walk and with reaction rates and , respectively, and hopping rate
, and study the phase diagram in the plane. According
to standard mean-field theory, this system is in an active state for all
, and perturbative renormalization suggests that this mean-field
result is valid for ; however, nonperturbative renormalization predicts
that for all there is a phase transition line to an absorbing state in the
plane. We show here that a simple single-site
approximation reproduces with minimal effort the nonperturbative phase diagram
both qualitatively and quantitatively for all dimensions . We expect the
approach to be useful for other reaction-diffusion processes involving
absorbing state transitions.Comment: 15 pages, 2 figures, published versio
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