From the area under the Bessel excursion to anomalous diffusion of cold atoms

Abstract

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.

arXiv commentSupersedes arXiv: 1305.0081

References in corpus (37)

• Fractional Laplacian in Bounded Domains
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Dynamics of Sleep-Wake Transitions During Sleep
• Observation of anomalous diffusion and fractional self-similarity in one dimension
• On Fractional Kinetic Equations
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Average Time Spent by Levy Flights and Walks on an Interval with Absorbing Boundaries
• Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
• Power-law tail distributions and nonergodicity
• Theory of fractional-Lévy kinetics for cold atoms diffusing in optical lattices
• Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
• On the area under a continuous time Brownian motion till its first-passage time
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Asymptotic behavior of self-affine processes in semi-infinite domains
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• A fractional Feynman-Kac equation for weak ergodicity breaking
• Nonintersecting Brownian excursions
• Solution of the Fokker-Planck equation with a logarithmic potential
• Vibrational mechanics in an optical lattice: controlling transport via potential renormalization
• First-passage and first-exit times of a Bessel-like stochastic process
• Approach to equilibrium of diffusion in a logarithmic potential
• Fluctuations of time averages for Langevin dynamics in a binding force field
• Modeling long-range memory with stationary Markovian processes
• Scaling Green-Kubo relation and application to three aging systems
• Distributional Ergodicity in Stored-Energy-Driven Lévy Flights
• Anomalous spatial diffusion and multifractality in optical lattices
• Space-time velocity correlation function for random walks
• Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials
• Logarithmic oscillators: ideal Hamiltonian thermostats
• Anomalous Infiltration
• Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium
• Investigation of a generalized Obukhov Model for Turbulence
• Persistence distributions for non gaussian markovian processes
• Statistics of the occupation time of renewal processes
• The Spectrum of the Fractional Laplacian and First Passage Time Statistics
• One-Dimensional Directed Sandpile Models and the Area under a Brownian Curve
• Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution

Cited by in corpus (70)

• Lévy walks
• The inspection paradox in stochastic resetting
• The Single Big Jump Principle in Physical Modelling
• Anomalous diffusion and Lévy walks distinguish active from inertial turbulence
• Interlacing Relaxation and First-Passage Phenomena in Reversible Discrete and Continuous Space Markovian Dynamics
• Asymptotic densities of ballistic Lévy walks
• Duality Between Relaxation and First Passage in Reversible Markov Dynamics: Rugged Energy Landscapes Disentangled
• Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit
• The Airy distribution: experiment, large deviations and additional statistics
• Inference in non-equilibrium systems from incomplete information: the case of linear systems and its pitfalls
• Renewal theory with fat tailed distributed sojourn times: typical versus rare
• Phase Diagram in Stored-Energy-Driven Lévy Flight
• Lévy-walk-like Langevin dynamics
• Anomalous statistics of laser-cooled atoms in dissipative optical lattices
• Statistics of the longest interval in renewal processes
• Ergodic properties of heterogeneous diffusion processes in a potential well
• Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices
• Large-deviations for spatial diffusion of cold atoms
• Infinite invariant density in a semi-Markov process with continuous state variables
• Transitions in the ergodicity of subrecoil-laser-cooled gases
• Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion
• Thermodynamic speed limits for mechanical work
• A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk
• Feynman-Kac equation revisited
• Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs and equipartition: Statistics of confined cold atoms
• The Heat Distribution in a Logarithm Potential
• Generating discrete-time constrained random walks and Lévy flights
• Aging two-state process with Lévy walk and Brownian motion
• Anomalous Diffusion: Fractional Brownian Motion vs. Fractional Ito Motion
• Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential
• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Multimodal stationary states in symmetric single-well potentials driven by Cauchy noise
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• Extinctions of coupled populations, and rare-event dynamics under non-Gaussian noise
• Lévy flights and Lévy walks under stochastic resetting
• Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime
• Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory
• Universal Excursion and Bridge shapes in ABBM/CIR/Bessel processes
• Self-reinforcing directionality generates Lévy walks without the power-law assumption
• The Distribution of the Area under a Bessel Excursion and its Moments
• Measurements and analysis of response function of cold atoms in optical molasses
• Quantitative fluid approximation in transport theory: a unified approach
• Quantum Langevin dynamics of a charged particle in a magnetic field : Response function, position-velocity and velocity autocorrelation functions
• Area fluctuations on a subinterval of Brownian excursion
• Multimodal stationary states under Cauchy noise
• Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking
• Peculiarities of escape kinetics in the presence of athermal noises
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Long Time Tails in Quantum Brownian Motion of a charged particle in a magnetic field
• Stationary states for underdamped anharmonic oscillators driven by Cauchy noise
• Quantizing Lévy flights
• Deviations from generalized Equipartition in Confined, Laser Cooled Atoms
• Underdamped, anomalous kinetics in double-well potentials
• Length and area generating functions for height-restricted Motzkin meanders
• Dichotomous acceleration process in one dimension: Position fluctuations
• Effects of the kinetic energy in heat for overdamped systems
• Nonlinear friction in underdamped anharmonic stochastic oscillators
• Zero-Crossing Statistics for Non-Markovian Time Series
• Heat fluctuations in the logarithm-harmonic potential
• Short-time large deviation of constrained random acceleration process
• Dynamics of a randomly kicked particle
• The area under a spectrally positive stable excursion and other related processes
• Anatomy of an extreme event: What can we infer about the history of a heavy-tailed random walk?
• Role of long jumps in Lévy noise induced multimodality
• The p-Airy distribution
• Deterministic force-free resonant activation
• Sampling rare trajectories using stochastic bridges
• Short-time large deviations of first-passage functionals for high-order stochastic processes
• Drifted escape from the finite interval
• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion