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

A look at profiler performance

Since about 1974, Doppler radars operating in UHF and VHF ranges have been used increasingly to study atmospheric winds. Historically, large systems capable of obtaining data from high altitudes have focused attention on the mesosphere and stratosphere, rather than on the troposphere wherein abides most of the weather considered by most meteorologists. Research address some questions the meteorologist must logically ask first, viz., what is the actual performance capability of these systems, how accurate is the wind data of interest to meteorologists, and from what altitudes in the troposphere are the data reliably obtained

Near-Infrared Spectroscopy of the Bright Kuiper Belt Object 2000 EB173

We have obtained a near-infrared spectrum of the bright Kuiper Belt object 2000 EB173; the spectrum appears featureless. The spectrum has a sufficient signal-to-noise ratio to rule out the 1.5 and 2.0 μm absorption from water ice even at the low level seen in the Centaur Chariklo. In addition, we can rule out a 2.3 μm absorption at the level seen in the Centaur Pholus

Scaling Green-Kubo relation and application to three aging systems

The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula valid for systems with long-range or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function $\langle v(t+\tau) v(t) \rangle$ asymptotically exhibits a certain scaling behavior and the diffusion is anomalous $\langle x^2(t) \rangle \simeq 2 D_\nu t^{\nu}$. We show how both the anomalous diffusion coefficient $D_\nu$ and exponent $\nu$ can be extracted from this scaling form. Our scaling Green-Kubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale invariant dynamics. This includes stationary systems with slowly decaying power law correlations as well as aging systems, whose properties depend on the the age of the system. Even for systems that are stationary in the long time limit, we find that the long time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity $D_{\nu}$ is not unique and we determine its values for a stationary respectively nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: Free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells and the L\'evy walk with numerous applications, in particular blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.Comment: 16 pages, 6 figures, 1 tabl

Nonlinear lattice model of viscoelastic Mode III fracture

We study the effect of general nonlinear force laws in viscoelastic lattice models of fracture, focusing on the existence and stability of steady-state Mode III cracks. We show that the hysteretic behavior at small driving is very sensitive to the smoothness of the force law. At large driving, we find a Hopf bifurcation to a straight crack whose velocity is periodic in time. The frequency of the unstable bifurcating mode depends on the smoothness of the potential, but is very close to an exact period-doubling instability. Slightly above the onset of the instability, the system settles into a exactly period-doubled state, presumably connected to the aforementioned bifurcation structure. We explicitly solve for this new state and map out its velocity-driving relation

Does the continuum theory of dynamic fracture work?

We investigate the validity of the Linear Elastic Fracture Mechanics approach to dynamic fracture. We first test the predictions in a lattice simulation, using a formula of Eshelby for the time-dependent Stress Intensity Factor. Excellent agreement with the theory is found. We then use the same method to analyze the experiment of Sharon and Fineberg. The data here is not consistent with the theoretical expectation.Comment: 4 page

The Universal Gaussian in Soliton Tails

We show that in a large class of equations, solitons formed from generic initial conditions do not have infinitely long exponential tails, but are truncated by a region of Gaussian decay. This phenomenon makes it possible to treat solitons as localized, individual objects. For the case of the KdV equation, we show how the Gaussian decay emerges in the inverse scattering formalism.Comment: 4 pages, 2 figures, revtex with eps

The Birth-Death-Mutation process: a new paradigm for fat tailed distributions

Fat tailed statistics and power-laws are ubiquitous in many complex systems. Usually the appearance of of a few anomalously successful individuals (bio-species, investors, websites) is interpreted as reflecting some inherent "quality" (fitness, talent, giftedness) as in Darwin's theory of natural selection. Here we adopt the opposite, "neutral", outlook, suggesting that the main factor explaining success is merely luck. The statistics emerging from the neutral birth-death-mutation (BDM) process is shown to fit marvelously many empirical distributions. While previous neutral theories have focused on the power-law tail, our theory economically and accurately explains the entire distribution. We thus suggest the BDM distribution as a standard neutral model: effects of fitness and selection are to be identified by substantial deviations from it

Comment on ``Solidification of a Supercooled Liquid in a Narrow Channel''

Comment on PRL v. 86, p. 5084 (2001) [cond-mat/0101016]. We point out that the authors' simulations are consistent with the known theory of steady-state solutions in this system