Exact Results for the Nonergodicity of 
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-Dimensional Generalized Lévy Walks

Albers, Tony; Radons, Günter

doi:10.1103/physrevlett.120.104501

Cited by 42 publications

(55 citation statements)

References 66 publications

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“…There the mean squared displacement is equal to the second derivative of the spatial distribution of the process. Results are known for processes of with PDFs of the form of equation (10) for Levy flights [25] and Levy walks [26]. The results in both cases are the same and therefore independent of the exact path the system takes during each waiting time.…”

Section: Model and Its Diffusive Behaviormentioning

confidence: 99%

Anomalous diffusion and the Moses effect in an aging deterministic model

et al. 2018

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We decompose the anomalous diffusive behavior found in an aging system into its fundamental constitutive causes. The model process is a sum of increments that are iterates of a chaotic dynamical system, the Pomeau-Manneville map. The increments can have long-time correlations, fat-tailed distributions and be non-stationary. Each of these properties can cause anomalous diffusion through what is known as the Joseph, Noah and Moses effects, respectively. The model can have either sub-or super-diffusive behavior, which we find is generally due to a combination of the three effects. Scaling exponents quantifying each of the three constitutive effects are calculated using analytic methods and confirmed with numerical simulations. They are then related to the scaling of the distribution of the process through a scaling relation. Finally, the importance of the Moses effect in the anomalous diffusion of experimental systems is discussed.

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Section: Model and Its Diffusive Behaviormentioning

confidence: 99%

Anomalous diffusion and the Moses effect in an aging deterministic model

et al. 2018

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“…For long times, in the former case, the particles spread like Richardson-Obukhov diffusion in turbulence, where the velocity follows a simple Brownian motion [34,35]. The Richardson-Obukhov diffusion is also discussed in recent papers on the diffusion of cold atoms in optical lattices [19,20] and the d-dimensional generalized Lévy walk [36]. In the latter case, the particle motion is like Lévy walk [37][38][39][40] of the sub-ballistic superdiffusion regime.…”

Section: Introductionmentioning

confidence: 99%

Lévy-walk-like Langevin dynamics

2019

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Continuous-time random walks and Langevin equations are two classes of stochastic models used to describe the dynamics of particles in the natural world. While some of the processes can be conveniently characterized by both of them, more often one model has significant advantages (or has to be used) compared with the other one. In this paper, we consider the weakly damped Langevin system coupled with a new subordinator-α-dependent subordinator with 1<α<2. We pay attention to the diffusive behavior of the stochastic process described by this coupled Langevin system, and find the super-ballistic diffusion phenomenon for the system with an unconfined potential on velocity but sub-ballistic superdiffusion phenomenon with a confined potential, which is like Lévy walk for long times. One can further note that the two-point distribution of inverse subordinator affects mean square displacement of this coupled weakly damped Langevin system in essential. 0 , model (1) yields subdiffusion. Nowadays, subordinator is a very effective tool to characterize various complex dynamical systems, especially some real-life data in biology [8]

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“…Jump lengths and waiting times may be coupled linearly, such that the resulting LW moves in a given direction with a constant speed until velocity reversal after a given waiting time, the velocity model [43], or the space-time coupling may have a power-law type [40]. The velocity may also be considered to change from one step to another [44,45]. Interestingly, for certain parameters even Lévy walks have an infinite variance, namely, when there is a distribution of velocities associated with different path lengths [45].…”

Section: Introductionmentioning

confidence: 99%

“…The velocity may also be considered to change from one step to another [44,45]. Interestingly, for certain parameters even Lévy walks have an infinite variance, namely, when there is a distribution of velocities associated with different path lengths [45]. LFs and LWs are non-ergodic in the sense that long time and ensemble averages of physical observables are different [19,20,[45][46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

“…Interestingly, for certain parameters even Lévy walks have an infinite variance, namely, when there is a distribution of velocities associated with different path lengths [45]. LFs and LWs are non-ergodic in the sense that long time and ensemble averages of physical observables are different [19,20,[45][46][47][48][49]. The linear response behaviour and time-averaged Einstein relation of LWs have been studied, as well [50,51].…”

Section: Introductionmentioning

confidence: 99%

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First passage and first hitting times of Lévy flights and Lévy walks

et al. 2019

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For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.

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Exact Results for the Nonergodicity of d -Dimensional Generalized Lévy Walks

Cited by 42 publications

References 66 publications

Anomalous diffusion and the Moses effect in an aging deterministic model

Anomalous diffusion and the Moses effect in an aging deterministic model

Lévy-walk-like Langevin dynamics

First passage and first hitting times of Lévy flights and Lévy walks

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