Full-text resources of PSJD and other databases are now available in the new Library of Science.
Visit https://bibliotekanauki.pl
Visit https://bibliotekanauki.pl
Journal
Article title
Content
Full texts:
![http://www.degruyter.com/view/j/phys.2009.7.issue-3/s11534-009-0054-4/s11534-009-0054-4.pdf [remote]](images/mime-icons/pdf.gif "pdf")
Title variants
Languages of publication
Abstracts
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.
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.
Discipline
- 02.50.-r: Probability theory, stochastic processes, and statistics(see also section 05 Statistical physics, thermodynamics, and nonlinear dynamical systems)
- 02.60.-x: Numerical approximation and analysis
- 02.60.Cb: Numerical simulation; solution of equations
- 02.70.-c: Computational techniques; simulations(for quantum computation, see 03.67.Lx; for computational techniques extensively used in subdivisions of physics, see the appropriate section; for example, see 47.11.-j Computational methods in fluid dynamics)
- 05.10.-a: Computational methods in statistical physics and nonlinear dynamics(see also 02.70.-c in mathematical methods in physics)
Publisher
Journal
Year
Volume
Issue
Pages
387-394
Physical description
Dates
published
1 - 9 - 2009
online
25 - 6 - 2009
Contributors
author
- Department of Mathematics, SUNY Institute of Technology, Utica, NY, 13504, USA
author
- Department of Computer and Information Sciences, SUNY Institute of Technology, Utica, NY, 13504, USA
author
- Raytheon Integrated Defense Systems, Principal Systems Engineer, San Diego, CA, USA
author
References
- [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
Identifiers
YADDA identifier
bwmeta1.element.-psjd-doi-10_2478_s11534-009-0054-4
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.