Fractional dynamics in the L\'evy quantum kicked rotor

We investigate the quantum kicked rotor in resonance subjected to momentum measurements with a L\'evy waiting time distribution. We find that the system has a sub-ballistic behavior. We obtain an analytical expression for the exponent of the power law of the variance as a function of the characteristic parameter of the L\'evy distribution and connect this anomalous diffusion with a fractional dynamics

Fractal time random walk and subrecoil laser cooling considered as renewal processes with infinite mean waiting times

There exist important stochastic physical processes involving infinite mean waiting times. The mean divergence has dramatic consequences on the process dynamics. Fractal time random walks, a diffusion process, and subrecoil laser cooling, a concentration process, are two such processes that look qualitatively dissimilar. Yet, a unifying treatment of these two processes, which is the topic of this pedagogic paper, can be developed by combining renewal theory with the generalized central limit theorem. This approach enables to derive without technical difficulties the key physical properties and it emphasizes the role of the behaviour of sums with infinite means.Comment: 9 pages, 7 figures, to appear in the Proceedings of Cargese Summer School on "Chaotic dynamics and transport in classical and quantum systems

Motional Broadening in Ensembles With Heavy-Tail Frequency Distribution

We show that the spectrum of an ensemble of two-level systems can be broadened through `resetting' discrete fluctuations, in contrast to the well-known motional-narrowing effect. We establish that the condition for the onset of motional broadening is that the ensemble frequency distribution has heavy tails with a diverging first moment. We find that the asymptotic motional-broadened lineshape is a Lorentzian, and derive an expression for its width. We explain why motional broadening persists up to some fluctuation rate, even when there is a physical upper cutoff to the frequency distribution.Comment: 6 pages, 4 figure

Phase transitions driven by L\'evy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations

Phase transitions and effects of external noise on many body systems are one of the main topics in physics. In mean field coupled nonlinear dynamical stochastic systems driven by Brownian noise, various types of phase transitions including nonequilibrium ones may appear. A Brownian motion is a special case of L\'evy motion and the stochastic process based on the latter is an alternative choice for studying cooperative phenomena in various fields. Recently, fractional Fokker-Planck equations associated with L\'evy noise have attracted much attention and behaviors of systems with double-well potential subjected to L\'evy noise have been studied intensively. However, most of such studies have resorted to numerical computation. We construct an {\it analytically solvable model} to study the occurrence of phase transitions driven by L\'evy stable noise.Comment: submitted to EP

The Stellar-Disk Electric (Short) Circuit: Observational Predictions for a YSO Jet Flow

We discuss the star-disk electric circuit for a young stellar object (YSO) and calculate the expected torques on the star and the disk. We obtain the same disk magnetic field and star-disk torques as given by standard magnetohydrodynamic (MHD) analysis. We show how a short circuit in the star-disk electric circuit may produce a magnetically-driven jet flow from the inner edge of a disk surrounding a young star. An unsteady bipolar jet flow is produced that flows perpendicular to the disk plane. Jet speeds of order hundreds of kilometres per second are possible, while the outflow mass loss rate is proportional to the mass accretion rate and is a function of the disk inner radius relative to the disk co-rotation radius.Comment: 6 pages, 8 figures, Accepted for publication in Astrophysics & Space Scienc

Levy distribution in many-particle quantum systems

Levy distribution, previously used to describe complex behavior of classical systems, is shown to characterize that of quantum many-body systems. Using two complimentary approaches, the canonical and grand-canonical formalisms, we discovered that the momentum profile of a Tonks-Girardeau gas, -- a one-dimensional gas of $N$ impenetrable (hard-core) bosons, harmonically confined on a lattice at finite temperatures, obeys Levy distribution. Finally, we extend our analysis to different confinement setups and demonstrate that the tunable Levy distribution properly reproduces momentum profiles in experimentally accessible regions. Our finding allows for calibration of complex many-body quantum states by using a unique scaling exponent.Comment: 7 pages, 6 figures, results are generalized, new examples are adde

Power-law tail distributions and nonergodicity

We establish an explicit correspondence between ergodicity breaking in a system described by power-law tail distributions and the divergence of the moments of these distributions.Comment: 4 pages, 1 figure, corrected typo

Optimal quantization for the pricing of swing options

In this paper, we investigate a numerical algorithm for the pricing of swing options, relying on the so-called optimal quantization method. The numerical procedure is described in details and numerous simulations are provided to assert its efficiency. In particular, we carry out a comparison with the Longstaff-Schwartz algorithm.Comment: 27

Bardeen-Petterson effect and the disk structure of the Seyfert galaxy NGC 1068

VLBA high spatial resolution observations of the disk structure of the active galactic nucleus NGC 1068 has recently revealed that the kinematics and geometry of this AGN is well characterized by an outer disk of H2O maser emission having a compact milliarcsecond (parsec) scale structure, which is encircling a thin rotating inner disk surrounding a ~10^7 M_\sun compact mass, likely a black hole. A curious feature in this source is the occurrence of a misalignment between the inner and outer parts of the disk, with the galaxy's radio jet being orthogonal to the inner disk. We interpret this peculiar configuration as due to the Bardeen-Petterson effect, a general relativistic effect that warps an initially inclined (to the black hole equator) viscous disk, and drives the angular momentum vector of its inner part into alignment with the rotating black hole spin. We estimate the time-scale for both angular momenta to get aligned as a function the spin parameter of the Kerr black hole. We also reproduce the shape of the parsec and kiloparsec scale jets, assuming a model in which the jet is precessing with a period and aperture angle that decrease exponentially with time, as expected from the Bardeen-Petterson effect.Comment: 12 pages, 3 figures, accepted for publication in The Astrophysical Journa

Stationary states for underdamped anharmonic oscillators driven by Cauchy noise

Using methods of stochastic dynamics, we have studied stationary states in the underdamped anharmonic stochastic oscillators driven by Cauchy noise. Shape of stationary states depend both on the potential type and the damping. If the damping is strong enough, for potential wells which in the overdamped regime produce multimodal stationary states, stationary states in the underdamped regime can be multimodal with the same number of modes like in the overdamped regime. For the parabolic potential, the stationary density is always unimodal and it is given by the two dimensional $\alpha$-stable density. For the mixture of quartic and parabolic single-well potentials the stationary density can be bimodal. Nevertheless, the parabolic addition, which is strong enough, can destroy bimodlity of the stationary state.Comment: 9 page