Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem

Abstract

Strong anomalous diffusion, where $⟨|x(t){|}^q⟩\sim t^{q\nu (q)}$ with a nonlinear spectrum $ν(q) \neq \mbox{const}$, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomena is related to infinite covariant densities, i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multi-fractal anomalous diffusion, as it is complementary to the central limit theorem.

PRL, in press

References in corpus (3)

• Pesin-Type Identity for Weak Chaos
• Photon Counting Statistics For Blinking CdSe-ZnS Quantum Dots: A Levy Walk Process
• Space-time velocity correlation function for random walks

Cited by in corpus (58)

• Lévy walks
• The Continuous Time Random Walk, still trendy: Fifty-year history, state of art, and outlook
• The Single Big Jump Principle in Physical Modelling
• Single atom energy-conversion device with a quantum load
• Adaptive Finite Element Method for fractional differential equations using Hierarchical Matrices
• Infinite densities for Lévy walks
• Collectively enhanced thermalization via multiqubit collisions
• Asymptotic densities of ballistic Lévy walks
• Rare events in generalized Lévy Walks and the Big Jump principle
• Strange kinetics of bulk-mediated diffusion on lipid bilayers
• Emergence of Lévy Walks in Systems of Interacting Individuals
• Ultra-cold Single-Atom Quantum Heat Engines
• Lévy-walk-like Langevin dynamics
• Moses, Noah and Joseph Effects in Coupled Lévy Processes
• Ergodic properties of heterogeneous diffusion processes in a potential well
• Infinite ergodic theory meets Boltzmann statistics
• Single integro-differential wave equation for Lévy walk
• Large-deviations for spatial diffusion of cold atoms
• Infinite invariant density in a semi-Markov process with continuous state variables
• Large deviations in the presence of cooperativity and slow dynamics
• Lévy walk dynamics in an external harmonic potential
• Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling
• Infinite density for cold atoms in shallow optical lattices
• Rare events in stochastic processes with sub-exponential distributions and the Big Jump principle
• Ergodic property of Langevin systems with superstatistical, uncorrelated or correlated diffusivity
• A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics
• Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity
• Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes
• Dispersion of particles in an infinite-horizon Lorentz gas
• Extreme value theory for constrained physical systems
• Langevin picture of Lévy walk in a constant force field
• Strong anomalous diffusion in two-state process with Lévy walk and Brownian motion
• Langevin dynamics for Lévy walk with memory
• Anomalous transmission and drifts in one-dimensional Levy structures
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• Strong anomalous diffusion of the phase of a chaotic pendulum
• Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential
• Lévy-walk-like Langevin dynamics affected by a time-dependent force
• Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling
• Fluctuations around equilibrium laws in ergodic continuous-time random walks
• Generalized Lyapunov exponent as a unified characterization of dynamical instabilities
• Lévy walk dynamics in mixed potentials from the perspective of random walk theory
• Universality in the Onset of Super-Diffusion in Lévy Walks
• Large deviations of the ballistic Lévy walk model
• Anomalous temperature-dependent heat transport in one-dimensional momentum-conserving systems with soft-type interparticle interaction
• Limit theorems for Lévy walks in $d$ dimensions: rare and bulk fluctuations
• Heat perturbations spreading in the Fermi-Pasta-Ulam-$\beta$ system with next-nearest-neighbor coupling: Competition between phonon dispersion and nonlinearity
• Non-normalizable quasi-equilibrium solution of the Fokker-Planck equation for nonconfining fields
• On the detection of superdiffusive behaviour in time series
• Memory-dependent noise-induced resonance and diffusion in non-markovian systems
• IntLevPy: A Python library to classify and model intermittent and Lévy processes
• Breakdown of self-similarity in light transport
• Is Chaotic Advection Inherent to Heterogeneous Darcy Flow?
• Origin of universality in the onset of superdiffusion in Lévy walks
• In the folds of the Central Limit Theorem: Lévy walks, large deviations and higher-order anomalous diffusion
• Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain
• Asymptotic densities of planar Lévy walks: a non-isotropic case
• Strong overlap of deterministic and stochastic dynamics in a super-diffusive regime