3,218,698 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
Modular Groups of Quantum Fields in Thermal States
For a quantum field in a thermal equilibrium state we discuss the group
generated by time translations and the modular action associated with an
algebra invariant under half-sided translations. The modular flows associated
with the algebras of the forward light cone and a space-like wedge admit a
simple geometric description in two dimensional models that factorize in
light-cone coordinates. At large distances from the domain boundary compared to
the inverse temperature the flow pattern is essentially the same as time
translations, whereas the zero temperature results are approximately reproduced
close to the edge of the wedge and the apex of the cone. Associated with each
domain there is also a one parameter group with a positive generator, for which
the thermal state is a ground state. Formally, this may be regarded as a
certain converse of the Unruh-effect.Comment: 28 pages, 4 figure
The Dependence of Dynamo -Effect on Reynolds Numbers, Magnetic Prandtl Number, and the Statistics of MHD Turbulence
We generalize the derivation of dynamo coefficient of Field et al
(1999) to include the following two aspects: first, the de-correlation times of
velocity field and magnetic field are different; second, the magnetic Prandtl
number can be arbitrary. We find that the contributions of velocity field and
magnetic field to the effect are not equal, but affected by their
different statistical properties. In the limit of large kinetic Reynolds number
and large magnetic Reynolds number, -coefficient may not be small if
the de-correlation times of velocity field and magnetic field are shorter than
the eddy turn-over time of the MHD turbulence. We also show that under certain
circumstances, for example if the kinetic helicity and current helicity are
comparable, depends insensitively on magnetic Prandtl number, while if
either the kinetic helicity or the current helicity is dominated by the other
one, a different magnetic Prandtl number will significantly change the dynamo
effect.Comment: 44 pages, 4 figures, to appear in ApJ (vol. 552
Stochastic Biasing and Galaxy-Mass Density Relation in the Weakly Non-linear Regime
It is believed that the biasing of the galaxies plays an important role for
understanding the large-scale structure of the universe. In general, the
biasing of galaxy formation could be stochastic. Furthermore, the future galaxy
survey might allow us to explore the time evolution of the galaxy distribution.
In this paper, the analytic study of the galaxy-mass density relation and its
time evolution is presented within the framework of the stochastic biasing. In
the weakly non-linear regime, we derive a general formula for the galaxy-mass
density relation as a conditional mean using the Edgeworth expansion. The
resulting expression contains the joint moments of the total mass and galaxy
distributions. Using the perturbation theory, we investigate the time evolution
of the joint moments and examine the influence of the initial stochasticity on
the galaxy-mass density relation. The analysis shows that the galaxy-mass
density relation could be well-approximated by the linear relation. Compared
with the skewness of the galaxy distribution, we find that the estimation of
the higher order moments using the conditional mean could be affected by the
stochasticity. Therefore, the galaxy-mass density relation as a conditional
mean should be used with a caution as a tool for estimating the skewness and
the kurtosis.Comment: 22 pages, 7 Encapusulated Postscript Figures, aastex, The title and
the structure of the paper has been changed, Results and conclusions
unchanged, Accepted for publication in Ap
RG flow of the Polyakov-loop potential: First status report
We study SU(2) Yang-Mills theory at finite temperature in the framework of
the functional renormalization group. We concentrate on the effective potential
for the Polyakov loop which serves as an order parameter for confinement. In
this first status report, we focus on the behaviour of the effective
Polyakov-loop potential at high temperatures. In addition to the standard
perturbative result, our findings provide information about the ``RG improved''
backreactions of Polyakov-loop fluctuations on the potential. We demonstrate
that these fluctuations establish the convexity of the effective potential.Comment: 10 pages, 2 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
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