Singularity of Levy walks in the lifted Pomeau-Manneville map
arXiv:2410.18997 · doi:10.1063/5.0243549
Abstract
Since groundbreaking works in the 1980s it is well-known that simple deterministic dynamical systems can display intermittent dynamics and weak chaos leading to anomalous diffusion. A paradigmatic example is the Pomeau-Manneville (PM) map which, suitably lifted onto the whole real line, was shown to generate superdiffusion that can be reproduced by stochastic Levy walks (LWs). Here we report that this matching only holds for parameter values of the PM map that are of Lebesgue measure zero in its two-dimensional parameter space. This is due to a bifurcation scenario that the map exhibits under variation of one parameter. Constraining this parameter to specific singular values at which the map generates superdiffusion by varying the second one, as has been done in previous literature, we find quantitative deviations between deterministic diffusion and diffusion generated by stochastic LWs in a particular range of parameter values, which cannot be cured by simple LW modifcations. We also explore the effect of aging on superdiffusion in the PM map and show that this yields a profound change of the diffusive properties under variation of the aging time, which should be important for experiments. Our findings demonstrate that even in this simplest well-studied setting, a matching of deterministic and stochastic diffusive properties is non-trivial.
arXiv comment19 pages, 10 figures. arXiv admin note: text overlap with arXiv:2208.09458
References in corpus (28)
- Anomalous transport in the crowded world of biological cells
- Lévy walks
- Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
- Simple Maps with Fractal Diffusion Coefficients
- Time averaged Einstein relation and fluctuating diffusivities for the Lévy walk
- Distributional Response to Biases in Deterministic Superdiffusion
- Simple deterministic dynamical systems with fractal diffusion coefficients
- Random time averaged diffusivities for Lévy walks
- Periodic Orbit Theory of Anomalous Diffusion
- Density-dependent diffusion in the periodic Lorentz gas
- Aging generates regular motions in weakly chaotic systems
- Superdiffusive dispersals impart the geometry of underlying random walks
- Understanding deterministic diffusion by correlated random walks
- Fractal structures of normal and anomalous diffusion in nonlinear nonhyperbolic dynamical systems
- Fractal Properties of Anomalous Diffusion in Intermittent Maps
- Normal and Anomalous Diffusion in Soft Lorentz Gases
- Negative and Nonlinear Response in an Exactly Solved Dynamical Model of Particle Transport
- Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent
- Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals
- Cycle expansions for intermittent diffusion
- Chaotic Diffusion in Delay Systems: Giant Enhancement by Time Lag Modulation
- Mean squared displacement in a generalized Lévy walk model
- Escape from intermittent repellers- Periodic orbit theory for crossover from exponential to algebraic decay
- Nonergodicity of -dimensional generalized Lévy walks and their relation to other space-time coupled models
- Computing the diffusion coefficient for intermittent maps: Resummation of stability ordered cycle expansions
- Diffusion coefficients for multi-step persistent random walks on lattices
- Capturing correlations in chaotic diffusion by approximation methods
- Fractal dimension of transport coefficients in a deterministic dynamical system