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Introduction to Nonextensive Statistical Mechanics

Introduction to Nonextensive Statistical Mechanics. Grupo Interdisciplinar de Sistemas Complejos. A. Rodríguez. Dpto . Matemática Aplicada y Estadística. UPM. August 26th , 2009. EPSRC Symposium Workshop on Quantum Simulations. Outline. General concepts q - entropy

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Introduction to Nonextensive Statistical Mechanics

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  1. IntroductiontoNonextensiveStatisticalMechanics Grupo Interdisciplinar de Sistemas Complejos A. Rodríguez Dpto. Matemática Aplicada y Estadística. UPM August 26th, 2009 EPSRC Symposium Workshop on Quantum Simulations.

  2. Outline • General concepts • q-entropy • ConexionwithThermodynamics • Someapplications • XY Model • Scaleinvariantprobabilisticmodel • Summary

  3. q-Entropy Boltzmann-Gibbsentropy Tsallisentropy q-logarithm: q-exponential:

  4. ConexionwithThermodynamics Boltzmann’sprinciple

  5. ConexionwithThermodynamics Boltzmanndistribution

  6. ConexionwithThermodynamics Gaussian: q-Gaussian:

  7. q-Gaussians q = 1

  8. q-Gaussians q = 0

  9. q-Gaussians q = -1

  10. q-Gaussians

  11. q-Gaussians q = 2

  12. q-Gaussians q = 2.9

  13. q-Gaussians

  14. Central LimitTheorem Nrandom variables: a) Independence: CLT (L-G)-CLT b) Global correlations: q-CLT

  15. Outline • General concepts • q-entropy • ConexionwithThermodynamics • Someapplications • XY Model • Scaleinvariantprobabilisticmodel • Summary

  16. XY Model asymmetry intensity of H g=0: XX model. |g|=1: Isingmodel. 0<|g|<1: XY model. F. Carusso and C. Tsallis, Phys. Rev. E 78, 021102 (2008)

  17. XY Model von NeumannEntropy: TsallisEntropy: F. Carusso and C. Tsallis, Phys. Rev. E 78, 021102 (2008)

  18. XY Model Ising (g=l=1) F. Carusso and C. Tsallis, Phys. Rev. E 78, 021102 (2008)

  19. XY Model central charge =1 • Conformalfieldtheory: (P. Calabrese et al, J. Stat. Mech.: TheoryExp. (2004), P06002 • Ising and XY : • XX : F. Carusso and C. Tsallis, Phys. Rev. E 78, 021102 (2008)

  20. Outline • General concepts • q-entropy • ConexionwithThermodynamics • Someapplications • XY Model • Scaleinvariantprobabilisticmodel • Summary

  21. x1 0 1 p 1-p Scaleinvariance Ndistinguisablebinaryindependent variables N=1 1 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  22. x2 0 1 p2 1 p(1-p) 0 p(1-p) (1-p)2 Scaleinvariance Ndistinguisablebinaryindependent variables N=2 x1 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  23. Scaleinvariance Ndistinguisablebinaryindependent variables N=2 x1 x2 0 1 p2 p 1 p(1-p) 0 p(1-p) (1-p)2 1-p p 1-p A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  24. p2(1-p) p(1-p)2 p(1-p)2 (1-p)3 p2 p(1-p) p(1-p) (1-p)2 p3 p2(1-p) p2(1-p) p(1-p)2 Scaleinvariance N=3 x3=0 x3=1 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  25. p2(1-p) p(1-p)2 p(1-p)2 p2(1-p) p2 p(1-p) p(1-p) Scaleinvariance N=3 (1-p)3 1 p 1-p N=0 N=1 N=2 p3 p2(1-p) p(1-p)2 (1-p)2 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  26. + + + p2 p(1-p) Scaleinvariance Leibniz rule 1 p 1-p N=0 N=1 N=2 (1-p)2 p2(1-p) p(1-p)2 p3 (1-p)3 N=3 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  27. p2 p(1-p) Scaleinvariance CLT 1 1 p 1-p N=0 N=1 N=2 1 1 1 (1-p)2 2 1 p2(1-p) p(1-p)2 p3 (1-p)3 1 3 3 1 N=3 Pascal triangle A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  28. p2 p(1-p) Invarianttriangles 1 p 1-p N=0 N=1 N=2 (1-p)2 p2(1-p) p(1-p)2 p3 (1-p)3 N=3 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  29. Invarianttriangles Leibniz triangle N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  30. Invarianttriangles Leibniz triangle N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  31. Invarianttriangles Leibniz triangle N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  32. Invarianttriangles N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  33. Invarianttriangles N=0 N=1 N=2 N=3 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  34. Invarianttriangles A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  35. Invarianttriangles A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

  36. Invarianttriangles

  37. Outline • General concepts • q-entropy • ConexionwithThermodynamics • Someapplications • XY Model • Scaleinvariantprobabilisticmodel • Summary

  38. Summary • NonextensiveStatisticalMechanicsallowstoaddressnon-equilibriumstationarystates (in Physics, Biology, Economics…) withconcepts and methods similar tothose of the BG StatisticalMechanics. • Theentropy (bridge betweenmechanicalmicroscopiclaws and classicalthermodynamics) mayadoptdifferentexpressionsdependingonthesystem: orothers. • Thevalue of q isdeterminedbythemicroscopicdynamics of thesystem, whichisfrequentlyunknown, so itgenerallycannotbepredictedbyfirstprinciples. • Itis a currentlydevelopingtheory, withmany open questions as thereslationshipbetweenscaleinvariance, extensivity and q-Gaussianity.

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