1,081 research outputs found
Vectorized multigrid Poisson solver for the CDC CYBER 205
The full multigrid (FMG) method is applied to the two dimensional Poisson equation with Dirichlet boundary conditions. This has been chosen as a relatively simple test case for examining the efficiency of fully vectorizing of the multigrid method. Data structure and programming considerations and techniques are discussed, accompanied by performance details
From the area under the Bessel excursion to anomalous diffusion of cold atoms
Levy flights are random walks in which the probability distribution of the
step sizes is fat-tailed. Levy spatial diffusion has been observed for a
collection of ultra-cold Rb atoms and single Mg+ ions in an optical lattice.
Using the semiclassical theory of Sisyphus cooling, we treat the problem as a
coupled Levy walk, with correlations between the length and duration of the
excursions. The problem is related to the area under Bessel excursions,
overdamped Langevin motions that start and end at the origin, constrained to
remain positive, in the presence of an external logarithmic potential. In the
limit of a weak potential, the Airy distribution describing the areal
distribution of the Brownian excursion is found. Three distinct phases of the
dynamics are studied: normal diffusion, Levy diffusion and, below a certain
critical depth of the optical potential, x~ t^{3/2} scaling. The focus of the
paper is the analytical calculation of the joint probability density function
from a newly developed theory of the area under the Bessel excursion. The
latter describes the spatiotemporal correlations in the problem and is the
microscopic input needed to characterize the spatial diffusion of the atomic
cloud. A modified Montroll-Weiss (MW) equation for the density is obtained,
which depends on the statistics of velocity excursions and meanders. The
meander, a random walk in velocity space which starts at the origin and does
not cross it, describes the last jump event in the sequence. In the anomalous
phases, the statistics of meanders and excursions are essential for the
calculation of the mean square displacement, showing that our correction to the
MW equation is crucial, and points to the sensitivity of the transport on a
single jump event. Our work provides relations between the statistics of
velocity excursions and meanders and that of the diffusivity.Comment: Supersedes arXiv: 1305.008
The Distribution of the Area under a Bessel Excursion and its Moments
A Bessel excursion is a Bessel process that begins at the origin and first
returns there at some given time . We study the distribution of the area
under such an excursion, which recently found application in the context of
laser cooling. The area scales with the time as ,
independent of the dimension, , but the functional form of the distribution
does depend on . We demonstrate that for , the distribution reduces as
expected to the distribution for the area under a Brownian excursion, known as
the Airy distribution, deriving a new expression for the Airy distribution in
the process. We show that the distribution is symmetric in , with
nonanalytic behavior at . We calculate the first and second moments of the
distribution, as well as a particular fractional moment. We also analyze the
analytic continuation from . In the limit where from
below, this analytically continued distribution is described by a one-sided
L\'evy -stable distribution with index and a scale factor
proportional to
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