Dynamical Generators of Lévy Statistics in Biology
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. . 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 . 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 .
KeywordsRandom Walk Dynamical Generator Detrended Fluctuation Analysis Anomalous Diffusion Velocity Autocorrelation Function
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