Properties of Lévy flights on an interval with absorbing boundaries

S. V. Buldyrev; M. Gitterman; S. Havlin; A. Ya Kazakov; M. G.E. Da Luz; E. P. Raposo; H. E. Stanley; G. M. Viswanathan

doi:10.1016/s0378-4371(01)00461-7

Properties of Lévy flights on an interval with absorbing boundaries

S. V. Buldyrev, M. Gitterman, S. Havlin, A. Ya Kazakov, M. G.E. Da Luz, E. P. Raposo, H. E. Stanley, G. M. Viswanathan

Department of Physics - at Bar-Ilan University

Research output: Contribution to journal › Conference article › peer-review

60 Scopus citations

Abstract

We consider a Lévy flyer of order α that starts from a point x₀ on an interval [0,L] with absorbing boundaries. We find a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the flyer, such as mean first passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for these quantities in the continuous limit. We find numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the Lévy flights with different flight length distributions converge to the continuous approximations. We show that if x₀ is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near α = 1, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.

Original language	English
Pages (from-to)	148-161
Number of pages	14
Journal	Physica A: Statistical Mechanics and its Applications
Volume	302
Issue number	1-4
DOIs	https://doi.org/10.1016/s0378-4371(01)00461-7
State	Published - 15 Dec 2001
Event	International Workshop on Frontiers in the Physics of Complex Systems - Ramat-Gan, Israel Duration: 25 Mar 2001 → 28 Mar 2001

Bibliographical note

Funding Information:
We are grateful to M.F. Shlesinger, J. Klafter, C. Tsallis, A. Marshak, A. Davis, and E. Barkai for helpful suggestions and critical comments. We thank NSF and CNPq for financial support.

Funding

We are grateful to M.F. Shlesinger, J. Klafter, C. Tsallis, A. Marshak, A. Davis, and E. Barkai for helpful suggestions and critical comments. We thank NSF and CNPq for financial support.

Funders	Funder number
National Science Foundation
Conselho Nacional de Desenvolvimento Científico e Tecnológico

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abstract = "We consider a L{\'e}vy flyer of order α that starts from a point x0 on an interval [0,L] with absorbing boundaries. We find a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the flyer, such as mean first passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for these quantities in the continuous limit. We find numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the L{\'e}vy flights with different flight length distributions converge to the continuous approximations. We show that if x0 is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near α = 1, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.",

author = "Buldyrev, {S. V.} and M. Gitterman and S. Havlin and Kazakov, {A. Ya} and {Da Luz}, {M. G.E.} and Raposo, {E. P.} and Stanley, {H. E.} and Viswanathan, {G. M.}",

note = "Funding Information: We are grateful to M.F. Shlesinger, J. Klafter, C. Tsallis, A. Marshak, A. Davis, and E. Barkai for helpful suggestions and critical comments. We thank NSF and CNPq for financial support.; International Workshop on Frontiers in the Physics of Complex Systems ; Conference date: 25-03-2001 Through 28-03-2001",

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AU - Buldyrev, S. V.

AU - Gitterman, M.

AU - Havlin, S.

AU - Kazakov, A. Ya

AU - Da Luz, M. G.E.

AU - Raposo, E. P.

AU - Stanley, H. E.

AU - Viswanathan, G. M.

N1 - Funding Information: We are grateful to M.F. Shlesinger, J. Klafter, C. Tsallis, A. Marshak, A. Davis, and E. Barkai for helpful suggestions and critical comments. We thank NSF and CNPq for financial support.

PY - 2001/12/15

Y1 - 2001/12/15

N2 - We consider a Lévy flyer of order α that starts from a point x0 on an interval [0,L] with absorbing boundaries. We find a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the flyer, such as mean first passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for these quantities in the continuous limit. We find numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the Lévy flights with different flight length distributions converge to the continuous approximations. We show that if x0 is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near α = 1, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.

AB - We consider a Lévy flyer of order α that starts from a point x0 on an interval [0,L] with absorbing boundaries. We find a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the flyer, such as mean first passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for these quantities in the continuous limit. We find numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the Lévy flights with different flight length distributions converge to the continuous approximations. We show that if x0 is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near α = 1, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.

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ER -

Properties of Lévy flights on an interval with absorbing boundaries

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Bibliographical note

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