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Properties of Lévy flights on an interval with absorbing boundaries

Author

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

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.

Suggested Citation

  • Buldyrev, S.V. & Gitterman, M. & Havlin, S. & Kazakov, A.Ya. & da Luz, M.G.E. & Raposo, E.P. & Stanley, H.E. & Viswanathan, G.M., 2001. "Properties of Lévy flights on an interval with absorbing boundaries," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 302(1), pages 148-161.
    • Handle: RePEc:eee:phsmap:v:302:y:2001:i:1:p:148-161
    DOI: 10.1016/S0378-4371(01)00461-7
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    Cited by:

    1. Ferreira, A.S. & Raposo, E.P. & Viswanathan, G.M. & da Luz, M.G.E., 2012. "The influence of the environment on Lévy random search efficiency: Fractality and memory effects," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3234-3246.
    2. E P Raposo & F Bartumeus & M G E da Luz & P J Ribeiro-Neto & T A Souza & G M Viswanathan, 2011. "How Landscape Heterogeneity Frames Optimal Diffusivity in Searching Processes," PLOS Computational Biology, Public Library of Science, vol. 7(11), pages 1-8, November.

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