We devise mappings between Gaussian distributions and power-law distributions, nowadays also called Tsallis distributions. To a given Tsallis distributed vector X, one can associate a Gaussian distributed vector N in the fashion where a is a random variable independent of X whose properties we are going to characterize here. We not only show that this mapping is invertible but also construct the adequate inversion operation. As an application of this stochastic mapping, we revisit the problem posed to Tsallis practitioners by the zeroth law of thermodynamics, that has bedeviled them for 15 years.
Nous définissons des relations bijectives entre les distributions Gaussiennes et les distributions de Tsallis ; à un vecteur aléatoire X suivant une distribution de Tsallis, il est possible d'associer un vecteur aléatoire Gaussien N de la façon suivante : où a est une variable aléatoire indépendante de X dont nous caractérisons les propriétés. Nous montrons que cette association est bijective et construisons explicitement l'association inverse. Nous appliquons ce résultat au problème du principe zéro de la thermodynamique tel qu'il se pose dans le cadre des statistiques de Tsallis.
Mots-clés : Suprastatistiques, Application stochastique, Entropie de Tsallis
Christophe Vignat 1; A. Plastino 2, 3
@article{CRPHYS_2006__7_3-4_442_0, author = {Christophe Vignat and A. Plastino}, title = {Stochastic invertible mappings for {Tsallis} distributions}, journal = {Comptes Rendus. Physique}, pages = {442--448}, publisher = {Elsevier}, volume = {7}, number = {3-4}, year = {2006}, doi = {10.1016/j.crhy.2006.01.012}, language = {en}, }
Christophe Vignat; A. Plastino. Stochastic invertible mappings for Tsallis distributions. Comptes Rendus. Physique, Statistical mechanics of non-extensive systems, Volume 7 (2006) no. 3-4, pp. 442-448. doi : 10.1016/j.crhy.2006.01.012. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2006.01.012/
[1] Physica A, 322 (2003), p. 267
[2] Elementary Principles in Statistical Mechanics, Collected Works, Yale Univ. Press, New Haven, 1948
[3] Statistical and Thermal Physics, Statistical Mechanics, McGraw-Hill, New York, 1965
[4] Foundations of Physics, Dover, New York, 1957
[5] Phys. Lett. A, Nonextensive Entropy: Interdisciplinary Applications, 36 (2005), p. 140 (Special issue and references therein)
[6] Physica A, 305 (2002) no. Special (and references therein)
[7] Phys. Lett. A, 193 (1994), p. 251
[8] Lectures on Phase Transitions and the Renormalization Group, Addison–Wesley, New York, 1992
[9] Braz. J. Phys., 29 (1999), p. 1
[10] Phys. Lett. A, 287 (2001), p. 240
[11] Phys. Rev. E, 63 (2001), p. 035503
[12] Physica A, 286 (2000), p. 156
[13] Phys. Lett. A, 310 (2003), p. 372
[14] J. Math. Phys., 37 (1996), p. 1776
[15] Nonextensive statistical mechanics and thermodynamics. Historical background and present status (S. Abe; Y. Okamoto, eds.), Nonextensive Statistical Mechanics and Its Applications, Lecture Notes in Physics, Springer-Verlag, Berlin, 2000
[16] Physica A, 295 (2001), p. 224
[17] J. Phys. A, 35 (2002), p. 7003
[18] Phys. Lett. A, 281 (2001), p. 126
[19] Physica A, 269 (1999), p. 403
[20] Geometric origin of probabilistic distributions in statistical mechanics | arXiv
[21] A. Jeffrey (Ed.), Gradshteyn and Ryzhik's Table of Integrals, Series, and Products, fifth ed., January 1994
[22] F. Barthe, M. Csornyei, A. Naor, A note on simultaneous polar and Cartesian decomposition, in: Geometric Aspects of Functional Analysis, in: Springer Lecture Notes in Math., vol. 1807, pp. 1–19
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