No-go theorem for ergodicity and an Einstein relation

Froemberg, D.; Barkai, Eli

doi:10.1103/physreve.88.024101

Cited by 28 publications

(15 citation statements)

References 33 publications

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“…A similar property was also found for subdiffusive continuous-time random walk models [12] and others anomalous diffusion processes [47,48].…”

Section: Generalized Einstein Relationsupporting

confidence: 79%

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Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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We introduce a class of discrete random walk model driven by global memory effects. At any time the right-left transitions depend on the whole previous history of the walker, being defined by an urn-like memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long time regime they remain random objects. While their average over realizations recover the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.

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“…A similar property was also found for subdiffusive continuous-time random walk models [12] and others anomalous diffusion processes [47,48].…”

Section: Generalized Einstein Relationsupporting

confidence: 79%

“…[46]. In addition, here a gener-alized Einstein fluctuation-dissipation relation is established [12,47,48] for the time-averaged moments.…”

Section: Introductionmentioning

confidence: 99%

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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“…which is consistent with the result in Ref. [47]. Here, α = 2 when 0 < α < 1 andα = 3 − α when 1 < α < 2.…”

Section: First Momentsupporting

confidence: 93%

Langevin picture of Lévy walk in a constant force field

2019

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Lévy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the Lévy walk with the exponent of the power-law distributed flight time α ∈ (0, 2). We add the term F η(s) (η(s) is the Lévy noise) on a subordinated Langevin system to characterize such a constant force, being effective on the velocity process for all physical time after the subordination. We clearly show the effect of the constant force F on this Langevin system and find this system is like the continuous limit of the collision model. The first moments of velocity processes for these two models are consistent. In particular, based on the velocity correlation function derived from our subordinated Langevin equation, we investigate more interesting statistical quantities, such as the ensemble-and time-averaged mean squared displacements. Under the influence of constant force, the diffusion of particles becomes faster. Finally, the super-ballistic diffusion and the non-ergodic behavior are verified by the simulations with different α.

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“…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]. We note that LFs and LWs have also been formulated in heterogeneous environments [52,53].…”

Section: Introductionmentioning

confidence: 99%

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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No-go theorem for ergodicity and an Einstein relation

Cited by 28 publications

References 33 publications

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Langevin picture of Lévy walk in a constant force field

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

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