Abstract
We consider the density fluctuations of an ideal Brownian gas of particles performing Lévy flìghts characterized by the indexf. We find that the fluctuations scale as δN(t)∼ tH, where the Hurst exponentH locks onto the universal value 1/4 for Lévy flights with a finite root-mean-square range (f>2). For Lévy flights with a finite mean range but infinite root-mean-square range (1<f<2) the Hurst exponent H=1/(2f). For infinite-range Lévy flights (f<1) the Hurst exponent locks onto the value 1/2. The corresponding power spectrum scales with an exponent 1 + 2H, independent of dimension.
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Fogedby, H.C., Bohr, T. & Jensen, H.J. Fluctuations in a lévy flight gas. J Stat Phys 66, 583–593 (1992). https://doi.org/10.1007/BF01060082
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DOI: https://doi.org/10.1007/BF01060082