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

2. Outline General concepts General concepts q-entropy q-entropy Conexion with Thermodynamics Conexion with Thermodynamics Some applications Some applications XY Model XY Model Scale invariant probabilistic model Scale invariant probabilistic model Summary Summary

3. q-Entropy q-logarithm: q-exponential: Boltzmann-Gibbs entropyTsallis entropy

4. Conexion with Thermodynamics Boltzmann’s principle

5. Conexion with Thermodynamics Boltzmann distribution

6. Conexion with Thermodynamics Gaussian: q-Gaussian:

7. q-Gaussians q = 1

8. q-Gaussians q = 0

9. q-Gaussians q = -1

10. q-Gaussians

11. q = 2

12. q-Gaussians q = 2.9

13. q-Gaussians

14. Central Limit Theorem N random variables: a) Independence: CLT b) Global correlations: q-CLT (L-G)-CLT

15. Outline General concepts General concepts q-entropy q-entropy Conexion with Thermodynamics Conexion with Thermodynamics Some applications Some applications XY Model XY Model Scale invariant probabilistic model Scale invariant probabilistic model Summary Summary

16. XY Model XY Model  =0: XX model.  =0: XX model.  =1: Ising model.  =1: Ising model.  XY model.  XY model. intensity of H asymmetry F. Carusso and C. Tsallis, Phys. Rev. E 78, 021102 (2008)

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

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

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

20. Outline General concepts General concepts q-entropy q-entropy Conexion with Thermodynamics Conexion with Thermodynamics Some applications Some applications XY Model XY Model Scale invariant probabilistic model Scale invariant probabilistic model Summary Summary

21. Scale invariance x1x1x1x1p 1-p 1 0 N=1 N distinguisable binary independent variables 1 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

22. Scale invariance x1x1x1x1 p2p2p2p2 p (1-p) (1-p) 2 1 0 1 0 x2x2x2x2 N=2 N distinguisable binary independent variables A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

23. Scale invariance x1x1x1x1 p2p2p2p2 p (1-p) (1-p) 2 1 0 1 0 x2x2x2x2 p 1-p 1-p p 1-p N=2 N distinguisable binary independent variables A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

24. p3p3p3p3 p 2 (1-p) p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 Scale invariance p2p2p2p2 p(1-p) (1-p) 2 p 2 (1-p) p(1-p) 2 (1-p) 3 x 3 =1 x 3 =0 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

25. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 p 2 (1-p) N=3 Scale invariance p2p2p2p2 p(1-p) p(1-p) (1-p) 2 p(1-p) (1-p) 3 p 2 (1-p) p(1-p) 2 1 p 1-p N=0 N=1 N=2 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

26. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 Scale invariance p2 p2 p2 p2 p(1-p) p(1-p) (1-p) 2 (1-p) 2 (1-p) 3 (1-p) 3 1 p 1-p N=0 N=1 N=2 + + + Leibniz rule A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

27. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 Scale invariance p2 p2 p2 p2 p(1-p) p(1-p) (1-p) 2 (1-p) 2 (1-p) 3 (1-p) 3 1 p 1-p N=0 N=1 N=2 1 1 1 2 1 1 3 3 1 1 Pascal triangle CLT A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

28. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 Invariant triangles p2 p2 p2 p2 p(1-p) p(1-p) (1-p) 2 (1-p) 2 (1-p) 3 (1-p) 3 1 p 1-p N=0 N=1 N=2 A. Rodríguez, C. Tsallis and V. Schammle. J. Stat. Mech: theor. exp. P09006 (2008)

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

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

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

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

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

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

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

36. Invariant triangles

37. Outline General concepts General concepts q-entropy q-entropy Conexion with Thermodynamics Conexion with Thermodynamics Some applications Some applications XY Model XY Model Scale invariant probabilistic model Scale invariant probabilistic model Summary Summary

38. Summary Nonextensive Statistical Mechanics allows to address non-equilibrium stationary states (in Physics, Biology, Economics…) with concepts and methods similar to those of the BG Statistical Mechanics. Nonextensive Statistical Mechanics allows to address non-equilibrium stationary states (in Physics, Biology, Economics…) with concepts and methods similar to those of the BG Statistical Mechanics. The entropy (bridge between mechanical microscopic laws and classical thermodynamics) may adopt different expressions depending on the system: or others. The entropy (bridge between mechanical microscopic laws and classical thermodynamics) may adopt different expressions depending on the system: or others. The value of q is determined by the microscopic dynamics of the system, which is frequently unknown, so it generally cannot be predicted by first principles. The value of q is determined by the microscopic dynamics of the system, which is frequently unknown, so it generally cannot be predicted by first principles. It is a currently developing theory, with many open questions as the reslationship between scale invariance, extensivity and q-Gaussianity. It is a currently developing theory, with many open questions as the reslationship between scale invariance, extensivity and q-Gaussianity.