Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling

Abstract

We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models show an accumulation of the momentum at zero in the long-time limit, and a formal steady state cannot be normalized, i.e., there exists an infinite invariant density. We obtain the exact form of the infinite invariant density and the scaling function for the exponential and deterministic models and devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically non-identical, it is natural to wonder whether their ergodic properties share common traits, given that they are all described by an infinite invariant density. We show that the answer to this question depends on the type of observable under study. If the observable is integrable, the ergodic properties such as the statistical behavior of the time averages are universal as they are described by the Darling-Kac theorem. In contrast, for non-integrable observables, the models in general exhibit non-identical statistical laws. This implies that focusing on non-integrable observables, we discover non-universal features of the cooling process, that hopefully can lead to a better understanding of the particular model most suitable for a statistical description of the process. This result is expected to hold true for many other systems, beyond laser cooling.

arXiv comment14 pages, 8 figures

References in corpus (25)

• Anomalous transport in the crowded world of biological cells
• In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Statistical Aging and Non Ergodicity in the Fluorescence of Single Nanocrystals
• Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity
• Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
• Infinite Ergodic Theory for Heterogeneous Diffusion Processes
• From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory
• Distributional Response to Biases in Deterministic Superdiffusion
• Ergodic properties of continuous-time random walks: finite-size effects and ensemble dependences
• Distributional Ergodicity in Stored-Energy-Driven Lévy Flights
• Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System
• Moses, Noah and Joseph Effects in Coupled Lévy Processes
• Infinite ergodic theory meets Boltzmann statistics
• Distributional Behavior of Diffusion Coefficients Obtained by Single Trajectories in Annealed Transit Time Model
• Transitions in the ergodicity of subrecoil-laser-cooled gases
• Infinite invariant density in a semi-Markov process with continuous state variables
• From laser cooling to aging: a unified Levy flight description
• Distributional behavior of time averages of non-$L^1$ observables in one-dimensional intermittent maps with infinite invariant measures
• Infinite density for cold atoms in shallow optical lattices
• Statistics of occupation time and connection to local properties of non-homogeneous random walks
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory
• Characteristic Sign Renewals of Kardar-Parisi-Zhang Fluctuations
• A work fluctuation theorem for a Brownian particle in a non confining potentia

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