On the Nature of the Tsallis–Fourier Transform
Abstract
:1. Introduction
2. Reviewing an Alternative qFT Definition
3. q-FT in the Limit
4. Illustration
5. An Open Problem
6. Conclusions
Author Contributions
Conflicts of Interest
References
- Gell-Mann, M.; Tsallis, C. (Eds.) Nonextensive Entropy: Interdisciplinary Applications; Oxford University Press: New York, NY, USA, 2004.
- Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
- Plastino, A.R.; Plastino, A. Tsallis Entropy, Erhenfest Theorem and Information Theory. Phys. Lett. A 1993, 177, 177–179. [Google Scholar] [CrossRef]
- Tsallis, C.; Gell-Mann, M.; Sato, Y. Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive. Proc. Natl. Acad. Sci. USA 2005, 102, 15377–15382. [Google Scholar] [CrossRef] [PubMed]
- Douglas, P.; Bergamini, S.; Renzoni, F. Tunable Tsallis Distributions in Dissipative Optical Lattices. Phys. Rev. Lett. 2006, 96, 110601. [Google Scholar] [CrossRef]
- Liu, B.; Goree, J. Superdiffusion and Non-Gaussian Statistics in a Driven-Dissipative 2D Dusty Plasma. Phys. Rev. Lett. 2008, 100. [Google Scholar] [CrossRef]
- DeVoe, R.G. Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas. Phys. Rev. Lett. 2009, 102, 063001. [Google Scholar] [CrossRef]
- Pickup, R.M.; Cywinski, R.; Pappas, C.; Farago, B.; Fouquet, P. Generalized Spin-Glass Relaxation. Phys. Rev. Lett. 2009, 102, 097202. [Google Scholar] [CrossRef]
- Burlaga, L.F.; Ness, N.F. Compressible turbulence observed in the helioheath by Voyager 2. Astrophys. J. 2009, 703, 311. [Google Scholar] [CrossRef]
- Caruso, F.; Pluchino, A.; Latora, V.; Vinciguerra, S.; Rapisarda, A. Analysis of self-organized criticality in the Olami-Feder-Christensen model and in real earthquakes. Phys. Rev. E 2007, 75, 055101(R). [Google Scholar] [CrossRef]
- The CMS Collaboration. Transverse-Momentum and Pseudorapidity Distributions of Charged Hadrons in pp Collisions at s=7 TeV. Phys. Rev. Lett. 2010, 105, 022002. [Google Scholar] [CrossRef]
- Adare, A.; Afanasiev, S.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Al-Bataineh, H.; Alexander, J.; Aoki, K.; Aphecetche, L.; Armendariz, R.; et al. Measurement of neutral mesons in p+p collisions at s=200 GeV and scaling properties of hadron. Phys. Rev. D 2011, 83, 052004. [Google Scholar] [CrossRef]
- Lyra, M.L.; Tsallis, C. Nonextensivity and Multifractality in Low- Dimensional Dissipative Systems. Phys. Rev. Lett. 1998, 80, 53. [Google Scholar] [CrossRef]
- Borland, L. Option Pricing Formulas Based on a Non-Gaussian Stock Price Model. Phys. Rev. Lett. 2002, 89, 098701. [Google Scholar] [CrossRef]
- Plastino, A.R.; Plastino, A. Tsallis Entropy and Stellar Polytropes. Phys. Lett A 1993, 174, 384–386. [Google Scholar] [CrossRef]
- Plastino, A.R.; Plastino, A. Non-extensive statistical mechanics and gen- eralized Fokker-Planck equation. Phys. A 1995, 222, 347–354. [Google Scholar] [CrossRef]
- Capurro, A.; Diambra, L.; Lorenzo, D.; Macadar, O.; Martin, M.T.; Mostaccio, C.; Plastino, A. Tsallis entropy and cortical dynamics: The analysis of EEG signals. Phys. A 1998, 257, 149–155. [Google Scholar] [CrossRef]
- Martin, M.T.; Plastino, A.R.; Plastino, A. Tsallis-like information mea- sures and the analysis of complex signals. Phys. A 2000, 275, 262–271. [Google Scholar] [CrossRef]
- Plastino, A.R.; Plastino, A. Non-extensive solutions to the Vlasov equation. Braz. J. Phys. 1999, 29, 79. [Google Scholar]
- Umarov, S.; Tsallis, C.; Steinberg, S. On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics. Milan J. Math. 2008, 76, 307. [Google Scholar] [CrossRef]
- Plastino, A.; Rocca, M.C. Inversion of Tsallis’ q-Fourier Transform and the complex-plane generalization. Phys. A 2012, 391, 4740–4747. [Google Scholar] [CrossRef]
- Sebastiao e Silva, J. Les fonctions analytiques comme ultra-distributions dans le calcul operationnel. Math. Ann. 1958, 136, 58–96. [Google Scholar] [CrossRef]
- Plastino, A.; Rocca, M.C. q-Fourier Transform and its inversion- problem. Milan J. Math. 2012, 80, 243–249. [Google Scholar] [CrossRef]
- Hasumi, M. Note on the n-dimensional tempered ultra-distributions. Tôhoku Math. J. 1961, 13, 94–104. [Google Scholar] [CrossRef]
- Schwartz, L. Théorie des Distributions; Hermann: Paris, France, 1966. [Google Scholar]
- Bollini, C.G.; Escobar, T.; Rocca, M.C. Convolution of ultradistri-butions and Âŕeld theory. Int. J. Theor. Phys. 2003, 43. [Google Scholar] [CrossRef]
© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Plastino, A.; Rocca, M.C. On the Nature of the Tsallis–Fourier Transform. Mathematics 2015, 3, 644-652. https://doi.org/10.3390/math3030644
Plastino A, Rocca MC. On the Nature of the Tsallis–Fourier Transform. Mathematics. 2015; 3(3):644-652. https://doi.org/10.3390/math3030644
Chicago/Turabian StylePlastino, A., and Mario C. Rocca. 2015. "On the Nature of the Tsallis–Fourier Transform" Mathematics 3, no. 3: 644-652. https://doi.org/10.3390/math3030644