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 $T$. We study the distribution of the area under such an excursion, which recently found application in the context of laser cooling. The area $A$ scales with the time as $A \sim T^{3/2}$, independent of the dimension, $d$, but the functional form of the distribution does depend on $d$. We demonstrate that for $d=1$, 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 $d-2$, with nonanalytic behavior at $d=2$. We calculate the first and second moments of the distribution, as well as a particular fractional moment. We also analyze the analytic continuation from $d2$. In the limit where $d\to 4$ from below, this analytically continued distribution is described by a one-sided L\'evy $\alpha$-stable distribution with index $2/3$ and a scale factor proportional to $[(4-d)T]^{3/2}$