Stochastic invertible mappings for Tsallis distributions

Christophe Vignat; A. Plastino

doi:10.1016/j.crhy.2006.01.012

Stochastic invertible mappings for Tsallis distributions
[Mappings stochastiques inverses pour les distributions de Tsallis]

Christophe Vignat ¹ ; A. Plastino ^2,³

¹ Laboratoire de Production Microtechnique, E.P.F.L., Lausanne, Switzerland
² La Plata Physics Institute, Exact Sciences Faculty, National University La Plata, La Plata, Argentina
³ Argentine National Research Council—CONICET, C. C. 727, 1900 La Plata, Argentina

Comptes Rendus. Physique, Volume 7 (2006) no. 3-4, pp. 442-448.

Résumé
Abstract

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 : $N = a X$ 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.

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 $N = a X$ 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.

Publié le : 2006-04-01

DOI : 10.1016/j.crhy.2006.01.012

Keywords: Superstatistics, Stochastic mappings, Tsallis entropies
Mot clés : Suprastatistiques, Application stochastique, Entropie de Tsallis

Affiliations des auteurs :

Christophe Vignat ¹ ; A. Plastino ^{2, 3}

¹ Laboratoire de Production Microtechnique, E.P.F.L., Lausanne, Switzerland
² La Plata Physics Institute, Exact Sciences Faculty, National University La Plata, La Plata, Argentina
³ Argentine National Research Council—CONICET, C. C. 727, 1900 La Plata, Argentina

@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},
}

TY  - JOUR
AU  - Christophe Vignat
AU  - A. Plastino
TI  - Stochastic invertible mappings for Tsallis distributions
JO  - Comptes Rendus. Physique
PY  - 2006
SP  - 442
EP  - 448
VL  - 7
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crhy.2006.01.012
LA  - en
ID  - CRPHYS_2006__7_3-4_442_0
ER  -

%0 Journal Article
%A Christophe Vignat
%A A. Plastino
%T Stochastic invertible mappings for Tsallis distributions
%J Comptes Rendus. Physique
%D 2006
%P 442-448
%V 7
%N 3-4
%I Elsevier
%R 10.1016/j.crhy.2006.01.012
%G en
%F CRPHYS_2006__7_3-4_442_0

Christophe Vignat; A. Plastino. Stochastic invertible mappings for Tsallis distributions. Comptes Rendus. Physique, 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/

Bibliographie
Cité par

[1] C. Beck; E.G.D. Cohen Physica A, 322 (2003), p. 267

[2] J.W. Gibbs Elementary Principles in Statistical Mechanics, Collected Works, Yale Univ. Press, New Haven, 1948

[3] F. Reif; R.K. Pathria Statistical and Thermal Physics, Statistical Mechanics, McGraw-Hill, New York, 1965

[4] R.B. Lindsay; H. Margenau Foundations of Physics, Dover, New York, 1957

[5] A.R. Plastino; A. Plastino 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] A.R. Plastino; A. Plastino Phys. Lett. A, 193 (1994), p. 251

[8] N. Goldenfeld Lectures on Phase Transitions and the Renormalization Group, Addison–Wesley, New York, 1992

[9] C. Tsallis Braz. J. Phys., 29 (1999), p. 1

[10] C. Beck Phys. Lett. A, 287 (2001), p. 240

[11] C. Beck; G.S. Lewis; H.L. Swinney Phys. Rev. E, 63 (2001), p. 035503

[12] I. Bediaga; E.M.F. Curado; J.M. de Miranda Physica A, 286 (2000), p. 156

[13] C. Tsallis; J.C. Anjos; E.P. Borges Phys. Lett. A, 310 (2003), p. 372

[14] G.R. Guerberoff; G.A. Raggio J. Math. Phys., 37 (1996), p. 1776

[15] C. Tsallis 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] S. Martinez; F. Pennini; A. Plastino Physica A, 295 (2001), p. 224

[17] Q.A. Wang; L. Nivenen; A. Le Méhauté; M. Perezil J. Phys. A, 35 (2002), p. 7003

[18] S. Abe; S. Martinez; F. Pennini; A. Plastino Phys. Lett. A, 281 (2001), p. 126

[19] S. Abe Physica A, 269 (1999), p. 403

[20] C. Vignat; A. Plastino 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

Cité par Sources :

Commentaires - Politique