Stationary states of non-linear oscillators driven by Lévy noise

Chechkin, Aleksei V.; Gonchar, V.Yu.; Klafter, J.; Metzler, Ralf; Tanatarov, L. V.

doi:10.1016/s0301-0104(02)00551-7

Cited by 140 publications

(212 citation statements)

References 21 publications

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“…This latter property is in contrast to an alternative LF model suggested in reference [33]. An a priori unexpected behaviour of LFs was obtained recently as solution of the fractional Fokker-Planck equation (4), namely the occurrence of bimodal solutions (PDFs with two maxima) and the finiteness of the second moment in the presence of superharmonic potentials of the form [34,35] …”

Section: Introductionmentioning

confidence: 91%

Lévy Flights in a Steep Potential Well

Chechkin

Gonchar

Klafter

et al. 2004

Journal of Statistical Physics

140

183

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Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial mono-modal to a terminal bimodal distribution occurs (Chechkin et al., Phys. Rev. E 67, 010102(R) (2003)). In this paper, we present a detailed study of Lévy flights in potentials of the type U (x) ∝ |x| c with c > 2. Apart from the bifurcation into bimodality, we find the interesting result that for c > 4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial δ-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient tri-modal distribution of the Lévy flight. These properties of LFs in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multi-modality and the numerical procedures to establish the probability distribution of the process.

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Section: Introductionmentioning

confidence: 91%

Lévy Flights in a Steep Potential Well

Chechkin

Gonchar

Klafter

et al. 2004

Journal of Statistical Physics

140

183

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“…(25)] still defines a Markov process, and it is therefore fairly straightforward to show that the corresponding fluctuation-averaged (deterministic) description is given in terms of the space-fractional Fokker-Planck equation (22) [54,60]. In what follows, we solve it with d-initial condition…”

Section: B Fractional Fokker-planck Equationmentioning

confidence: 99%

Fundamentals of Lévy Flight Processes

Chechkin

Gonchar

Klafter

et al. 2006

Advances in Chemical Physics

116

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“…To explore this phenomenon, unlike the standard Brownian motion, one needs to apply the Markovian theory of the fractional Fokker-Planck equation and to state non-trivial boundary conditions. As a result, even the steady-state probability density function of the particle coordinate can be found only for some simple potential profiles [13,15,31].…”

Section: Introductionmentioning

confidence: 99%

Spectral characteristics of steady-state Lévy flights in confinement potential profiles

Kharcheva¹,

Dubkov²,

Dybiec³

et al. 2016

J. Stat. Mech.

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Abstract. The steady-state correlation characteristics of superdiusion in the form of Lévy flights in one-dimensional confinement potential profiles are investigated both theoretically and numerically. Specifically, for Cauchy stable noise we calculate the steady-state probability density function for an infinitely deep rectangular potential well and for a symmetric steep potential well of the type ( ) ∝ U x x m 2 . For these potential profiles and arbitrary Lévy index α, we obtain the asymptotic expression of the spectral power density.

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Stationary states of non-linear oscillators driven by Lévy noise

Cited by 140 publications

References 21 publications

Lévy Flights in a Steep Potential Well

Lévy Flights in a Steep Potential Well

Fundamentals of Lévy Flight Processes

Spectral characteristics of steady-state Lévy flights in confinement potential profiles

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