518 research outputs found
Asymptotic statistics of the n-sided planar Voronoi cell: II. Heuristics
We develop a set of heuristic arguments to explain several results on planar
Poisson-Voronoi tessellations that were derived earlier at the cost of
considerable mathematical effort. The results concern Voronoi cells having a
large number n of sides. The arguments start from an entropy balance applied to
the arrangement of n neighbors around a central cell. It is followed by a
simplified evaluation of the phase space integral for the probability p_n that
an arbitrary cell be n-sided. The limitations of the arguments are indicated.
As a new application we calculate the expected number of Gabriel (or full)
neighbors of an n-sided cell in the large-n limit.Comment: 22 pages, 10 figure
Sylvester's question and the Random Acceleration Process
Let n points be chosen randomly and independently in the unit disk.
"Sylvester's question" concerns the probability p_n that they are the vertices
of a convex n-sided polygon. Here we establish the link with another problem.
We show that for large n this polygon, when suitably parametrized by a function
r(phi) of the polar angle phi, satisfies the equation of the random
acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise.
On the basis of this relation we derive the asymptotic expansion log p_n = -2n
log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms
agree with a rigorous result due to Barany. The nonanalyticity in n of the
third term is a new result. The value 1/5 of the exponent follows from recent
work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show
that the n-sided polygon is effectively contained in an annulus of width \sim
n^{-4/5} along the edge of the disk. The distance delta_n of closest approach
to the edge is exponentially distributed with average 1/(2n).Comment: 29 pages, 4 figures; references added and minor change
New Monte Carlo method for planar Poisson-Voronoi cells
By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a
planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is
developed on the basis of earlier theoretical work; it exploits, in particular,
the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have
between four and six significant digits. Accurate n dependent averages, second
moments, and variances are obtained for the cell area and the cell perimeter.
The numerical large n behavior of these quantities is analyzed in terms of
asymptotic power series in 1/n. Snapshots are shown of typical occurrences of
extremely rare events implicating cells of up to n=1600 sides embedded in an
ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic
features of such many-sided cells and their immediate environment. Their
relevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table
Two interacting Ising chains in relative motion
We consider two parallel cyclic Ising chains counter-rotating at a relative
velocity v, the motion actually being a succession of discrete steps. There is
an in-chain interaction between nearest-neighbor spins and a cross-chain
interaction between instantaneously opposite spins. For velocities v>0 the
system, subject to a suitable markovian dynamics at a temperature T, can reach
only a nonequilibrium steady state (NESS). This system was introduced by Hucht
et al., who showed that for v=\infty it undergoes a para- to ferromagnetic
transition, essentially due to the fact that each chain exerts an effective
field on the other one. The present study of the v=\infty case determines the
consequences of the fluctuations of this effective field when the system size N
is finite. We show that whereas to leading order the system obeys detailed
balancing with respect to an effective time-independent Hamiltonian, the higher
order finite-size corrections violate detailed balancing. Expressions are given
to various orders in 1/N for the interaction free energy between the chains,
the spontaneous magnetization, the in-chain and cross-chain spin-spin
correlations, and the spontaneus magnetization. It is shown how finite-size
scaling functions may be derived explicitly. This study was motivated by recent
work on a two-lane traffic problem in which a similar phase transition was
found.Comment: 30 pages, 1 figur
Large-n conditional facedness m_n of 3D Poisson-Voronoi cells
We consider the three-dimensional Poisson-Voronoi tessellation and study the
average facedness m_n of a cell known to neighbor an n-faced cell. Whereas
Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic
expansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazar
et al., based on a very large data set, now clearly rule out Aboav's law. In
this work we determine the numerical value of k_1 and compare the expansion to
the Monte Carlo data. The calculation of k_1 involves an auxiliary planar
cellular structure composed of circular arcs, that we will call the
Poisson-Moebius diagram. It is a special case of more general Moebius diagrams
(or multiplicatively weighted power diagrams) and is of interest for its own
sake. We obtain exact results for the total edge length per unit area, which is
a prerequisite for the coefficient k_1, and a few other quantities in this
diagram.Comment: 18 pages, 5 figure
A parity breaking Ising chain Hamiltonian as a Brownian motor
We consider the translationally invariant but parity (left-right symmetry)
breaking Ising chain Hamiltonian \begin{equation} {\cal H} = -U_2\sum_{k}
s_{k}s_{k+1} - U_3\sum_{k} s_{k}s_{k+1}s_{k+3} \nonumber \end{equation} and let
this system evolve by Kawasaki spin exchange dynamics. Monte Carlo simulations
show that perturbations forcing this system off equilibrium make it act as a
Brownian molecular motor which, in the lattice gas interpretation, transports
particles along the chain. We determine the particle current under various
different circumstances, in particular as a function of the ratio and
of the conserved magnetization . The symmetry of the term
in the Hamiltonian is discussedComment: 11 pages, 4 figure
Theory of the critical state of low-dimensional spin glass
We analyse the critical region of finite-()-dimensional Ising spin glass,
in particular the limit of closely above the lower critical dimension
. At criticality the thermally active degrees of freedom are surfaces
(of width zero) enclosing clusters of spins that may reverse with respect to
their environment. The surfaces are organised in finite interacting structures.
These may be called {\em protodroplets}\/, since in the off-critical limit they
reduce to the Fisher and Huse droplets. This picture provides an explanation
for the phenomenon of critical chaos discovered earlier. It also implies that
the spin-spin and energy-energy correlation functions are multifractal and we
present scaling laws that describe them. Several of our results should be
verifiable in Monte Carlo studies at finite temperature in .Comment: RevTeX, 33 pages + 1 PostScript figure (uuencoded). Uses german.sty
and an input file def.tex, joined. Three additional figures may be requested
from the author
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