Dynamical Generators of Lévy Statistics in Biology

  • B. J. West
  • P. Allegrini
  • P. Grigolini
Part of the Mathematics and Biosciences in Interaction book series (MBI)


It is remarkable that it has only been two decades since Mandelbrot coined the term fractal, and only a decade since that term began to penetrate into the biomedical community in a significant way. This term captured the imagination of a generation of scientists in such a way that they were able to see the interconnections among a large class of physical, biological and physiological phenomena that traditional statistical physics was not equipped to describe, much less to explain. The common feature of these phenomena is that they are complex, nonlinear and appear to fluctuate randomly in space and/or time. The spectra of such systems, rather than being dominated by a narrow band of frequencies, spread with an inverse power law, so that correlations persist over very long time scales, see for example Bassighthwaighte et al. [1]. By the same token, the statistics of the fluctuations are found to deviate strongly from that normally expected using the Central Limit Theorem (CLT), for example, the second moments of many processes diverge. A generalized version of the CLT yields Lévy stable distributions to describe the statistical fluctuations in these systems; see, for example, Montroll and West [2]. Subsequently, it has been found that both the inverse power law spectra and the Lévy statistical distribution are a consequence of scaling and fractals, see [3].


Random Walk Dynamical Generator Detrended Fluctuation Analysis Anomalous Diffusion Velocity Autocorrelation Function 
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  1. [1]
    J.B. Bassingthwaighte, L.S. Liebovitch and B.J. West, Fractal Physiology, Oxford University Press, Oxford (1994).Google Scholar
  2. [2a]
    E.W. Montroll and B.J. West, in Fluctuation Phenomena, edited by E.W. Montroll and J.L. Lebowitz, North-Holland Personal Library, 1st Edition (1979)Google Scholar
  3. [2b]
    E.W. Montroll and B.J. West, in Fluctuation Phenomena, edited by E.W. Montroll and J.L. Lebowitz, North-Holland Personal Library, 2nd Edition (1987).Google Scholar
  4. [3]
    B.B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman and Co., San Francisco (1977).Google Scholar
  5. [4]
    P. Allegrini, M. Barbi, P. Grigolini and B.J. West, Dynamical model for DNA sequences, Phys. Rev. E 52, 5281 (1995).CrossRefGoogle Scholar
  6. [5]
    B.J. West and W. Deering, Fractal Physiology for Physicists: Lévy Statistics, Phys. Repts. 246, 1 (1994).CrossRefGoogle Scholar
  7. [6]
    C.K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F. Sciortino, M. Simons and H.G. Stanley, Long-range correaltions in nucleotide sequences, Nature 356, 168 (1992).PubMedCrossRefGoogle Scholar
  8. [7]
    C.K. Peng, S.V. Buldgrev, S. Havlin, M. Simons, H.E. Stanley and A.L. Goldberger, Mosaic of DNA nucleotides, Phys. Rev. E 49, 1685 (1994).CrossRefGoogle Scholar
  9. [8]
    R. Voss, Evolution of long-range fractal correlations and 1/f noise in DNA base sequences, Phys. Rev. Lett. 68, 3805–809 (1992).CrossRefGoogle Scholar
  10. [9]
    H.E. Stanley, S.V. Buldryev, A.L. Goldberger, Z.D. Goldberg, S. Havlin, R.N. Mantegna, S.M. Ossadnik, C.K. Peng, and M. Simons, Implications of thermodynamics of protein folding for evolution of primary sequences, Physica A 205, 214 (1994).PubMedCrossRefGoogle Scholar
  11. [10a]
    B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, Fractional diffusion and Lévy stable processes, Phys. Rev. E 55, 99 (1997)CrossRefGoogle Scholar
  12. [10b]
    P. Allegrini, P. Grigolini and B.J. West, Dynamical approach to Lévy Processes, Phys. Rev. E 54, 4760 (1996).CrossRefGoogle Scholar
  13. [11]
    P. Allegrini, P. Grigolini and B.J. West, A dynamical approach to DNA sequences. Phys. Lett. A 211, 217–22 (1996).CrossRefGoogle Scholar
  14. [12]
    A. Grosberg, Y. Rabin, S. Havlin and A. Neer, Europhys. Lett. 23, 373 (1993).CrossRefGoogle Scholar
  15. [13]
    Genetic Constraints on Adaptive Evolution, edited by B. Loeschcke, Springer-Verlag, Berlin (1987).Google Scholar
  16. [14]
    A. Lima-de-Faria, Evolution without Selection: Form and Function by Auto-evolution, Elsevier, Amsterdam (1987).Google Scholar
  17. [15]
    E.I. Shakhnovich and A.M. Gutin, Statistical mechanics in biology: How ubiquitous are long-range correlations? Nature 346, 773 (1990).PubMedCrossRefGoogle Scholar
  18. [16]
    M. Kimura, The Neutral Theory of Molecular Evolution, Cambridge University Press, Cambridge (1983).CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • B. J. West
    • 1
  • P. Allegrini
    • 1
  • P. Grigolini
    • 2
  1. 1.Center for Nonlinear ScienceUniversity of North TexasDentonUSA
  2. 2.Istituto di Biofisica del Consiglio Nazionale della RicherchePisaItaly

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