Stationary states of non-linear oscillators driven by Lévy noise

A. Chechkin; V. Gonchar; J. Klafter; R. Metzler; L. Tanatarov

doi:10.1016/S0301-0104(02)00551-7

Stationary states of non-linear oscillators driven by Lévy noise

A. Chechkin^*, V. Gonchar, J. Klafter, R. Metzler, L. Tanatarov

^*Corresponding author for this work

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Abstract

We study the probability density function in the stationary state of non-linear oscillators which are subject to Lévy stable noise and confined within symmetric potentials of the general form U(x) ∝ x^2m-2/(2m + 2), m = 0, 1,2,.... For m ≥ 1, the probability density functions display a distinct bimodal character and have power-law tails which decay faster than those of the noise probability density. This is in contrast to the Lévy harmonic oscillator m = 0. For the particular case of an anharmonic Lévy oscillator with U(x) = ax²/2+bx⁴/4, a > 0, we find a turnover from unimodality to bimodality at stationarity.

Original language	English
Pages (from-to)	233-251
Number of pages	19
Journal	Chemical Physics
Volume	284
Issue number	1-2
DOIs	https://doi.org/10.1016/S0301-0104(02)00551-7
State	Published - 1 Nov 2002

Funding

Funders	Funder number
INTAS	00-0847
Deutscher Akademischer Austauschdienst
Deutsche Forschungsgemeinschaft

Keywords

Fractional derivative
Fractional kinetic equation
Lévy flights
Lévy stable noise
Non-linear oscillator

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10.1016/S0301-0104(02)00551-7

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title = "Stationary states of non-linear oscillators driven by L{\'e}vy noise",

abstract = "We study the probability density function in the stationary state of non-linear oscillators which are subject to L{\'e}vy stable noise and confined within symmetric potentials of the general form U(x) ∝ x2m-2/(2m + 2), m = 0, 1,2,.... For m ≥ 1, the probability density functions display a distinct bimodal character and have power-law tails which decay faster than those of the noise probability density. This is in contrast to the L{\'e}vy harmonic oscillator m = 0. For the particular case of an anharmonic L{\'e}vy oscillator with U(x) = ax2/2+bx4/4, a > 0, we find a turnover from unimodality to bimodality at stationarity.",

keywords = "Fractional derivative, Fractional kinetic equation, L{\'e}vy flights, L{\'e}vy stable noise, Non-linear oscillator",

author = "A. Chechkin and V. Gonchar and J. Klafter and R. Metzler and L. Tanatarov",

note = "Funding Information: The authors thank G. Voitsenya for help in numerical simulation, and R. Gorenflo, F. Mainardi and I. M. Sokolov for fruitful discussions. This work is supported by the INTAS Project 00-0847. RM acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) within the Emmy Noether programme. AC acknowledges support from the DAAD.",

year = "2002",

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doi = "10.1016/S0301-0104(02)00551-7",

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AU - Chechkin, A.

AU - Gonchar, V.

AU - Klafter, J.

AU - Metzler, R.

AU - Tanatarov, L.

N1 - Funding Information: The authors thank G. Voitsenya for help in numerical simulation, and R. Gorenflo, F. Mainardi and I. M. Sokolov for fruitful discussions. This work is supported by the INTAS Project 00-0847. RM acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) within the Emmy Noether programme. AC acknowledges support from the DAAD.

PY - 2002/11/1

Y1 - 2002/11/1

N2 - We study the probability density function in the stationary state of non-linear oscillators which are subject to Lévy stable noise and confined within symmetric potentials of the general form U(x) ∝ x2m-2/(2m + 2), m = 0, 1,2,.... For m ≥ 1, the probability density functions display a distinct bimodal character and have power-law tails which decay faster than those of the noise probability density. This is in contrast to the Lévy harmonic oscillator m = 0. For the particular case of an anharmonic Lévy oscillator with U(x) = ax2/2+bx4/4, a > 0, we find a turnover from unimodality to bimodality at stationarity.

AB - We study the probability density function in the stationary state of non-linear oscillators which are subject to Lévy stable noise and confined within symmetric potentials of the general form U(x) ∝ x2m-2/(2m + 2), m = 0, 1,2,.... For m ≥ 1, the probability density functions display a distinct bimodal character and have power-law tails which decay faster than those of the noise probability density. This is in contrast to the Lévy harmonic oscillator m = 0. For the particular case of an anharmonic Lévy oscillator with U(x) = ax2/2+bx4/4, a > 0, we find a turnover from unimodality to bimodality at stationarity.

KW - Fractional derivative

KW - Fractional kinetic equation

KW - Lévy flights

KW - Lévy stable noise

KW - Non-linear oscillator

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JF - Chemical Physics

IS - 1-2

ER -

Stationary states of non-linear oscillators driven by Lévy noise

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