Full-text resources of PSJD and other databases are now available in the new Library of Science.
Visit https://bibliotekanauki.pl


Preferences help
enabled [disable] Abstract
Number of results


2009 | 7 | 3 | 387-394

Article title

q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations


Title variants

Languages of publication



We study a strictly scale-invariant probabilistic N-body model with symmetric, uniform, identically distributed random variables. Correlations are induced through a transformation of a multivariate Gaussian distribution with covariance matrix decaying out from the unit diagonal, as ρ/r
α for r =1, 2, ..., N-1, where r indicates displacement from the diagonal and where 0 ⩽ ρ ⩽ 1 and α ⩾ 0. We show numerically that the sum of the N dependent random variables is well modeled by a compact support q-Gaussian distribution. In the particular case of α = 0 we obtain q = (1-5/3 ρ) / (1- ρ), a result validated analytically in a recent paper by Hilhorst and Schehr. Our present results with these q-Gaussian approximants precisely mimic the behavior expected in the frame of non-extensive statistical mechanics. The fact that the N → ∞ limiting distributions are not exactly, but only approximately, q-Gaussians suggests that the present system is not exactly, but only approximately, q-independent in the sense of the q-generalized central limit theorem of Umarov, Steinberg and Tsallis. Short range interaction (α > 1) and long range interactions (α < 1) are discussed. Fitted parameters are obtained via a Method of Moments approach. Simple mechanisms which lead to the production of q-Gaussians, such as mixing, are discussed.


  • Department of Mathematics, SUNY Institute of Technology, Utica, NY, 13504, USA
  • Department of Computer and Information Sciences, SUNY Institute of Technology, Utica, NY, 13504, USA
  • Raytheon Integrated Defense Systems, Principal Systems Engineer, San Diego, CA, USA


  • [1] G. Grimmett, D. Stirzaker, Probability and Random Processes, 3rd edition (Oxford University Press, Oxford, England, 2001)
  • [2] C. Tsallis, J. Stat. Phys. 52, 479 (1988) http://dx.doi.org/10.1007/BF01016429[Crossref]
  • [3] M. Gell-Mann, C. Tsallis, Nonextensive Entropy: Interdisciplinary Applications (Oxford University Press, New York, 2004)
  • [4] S. Umarov, C. Tsallis, S. Steinberg, Milan Journal of Mathematics 76, 307 (2008) http://dx.doi.org/10.1007/s00032-008-0087-y[Crossref]
  • [5] K. Pearson, Philos. T. R. Soc. A 186, 343 (1895) http://dx.doi.org/10.1098/rsta.1895.0010[Crossref]
  • [6] P. Bickel, K. Doksum, Mathematical Statistics (Prentice Hall, Upper Saddle River, NJ, 2001)
  • [7] C. Shalizi, Phys. Rev. E, arXiv:math/0701854
  • [8] H. J. Hilhorst, G. Schehr, J. Stat. Mech.-Theory E P06003 (2007)
  • [9] A. Pluchino, A. Rapisarda, C. Tsallis, Europhys. Lett. 80, 26002 (2007) http://dx.doi.org/10.1209/0295-5075/80/26002[Crossref]
  • [10] A. Pluchino, A. Rapisarda, C. Tsallis, Physica A, 387, 3121 (2008) http://dx.doi.org/10.1016/j.physa.2008.01.112[Crossref]
  • [11] A. Rodríguez, V. Schwämmle, C. Tsallis, J. Stat. Mech.-Theory E. P09006 (2008)
  • [12] V. E. Bening, V. Y. Korolev, Theor. Probab. Appl.+ 49, 377 (2004) http://dx.doi.org/10.1137/S0040585X97981159[Crossref]
  • [13] C. Vignat, A. Plastino, Phys. Lett. A 360, 415 (2007) http://dx.doi.org/10.1016/j.physleta.2006.07.005[Crossref]

Document Type

Publication order reference


YADDA identifier

JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.