1. Non-Extensive Black Hole Thermodynamics estimate for Power-Law Particle Spectra Power-law tailed spectra and their explanations Abstract thermodynamics Event horizon thermodynamics Estimate from a wish Talk by T.S.Biró at the 10. Zimányi School, Budapest, Hungary, November 30 – December 3, 2010 arXiV: 1011.3442

2. Power-law tailed spectra particles and heavy ions: (SPS) RHIC, LHC fluctuations in financial returns natural catastrophes (earthquakes, etc.) fractal phase space filling network behavior some noisy electronics near Bose condensates citation of scientific papers….

3. Heavy ion collision: theoretical picture URQMD ( Univ. Frankfurt: Sorge, Bass, Bleicher…. )

4. Experimental picture … RHIC

5. Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data

6. Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data

8. Abstract thermodynamics S(E) = max (Jaynes-) principle nontrivial composition of e.g. the energy E 0-th law requires: factorizing form T1(E1) = T2(E2) This is equivalent to the existence and use of an additive function of energy L(E)! Repeated compositions asymptotically lead to such a form! ( formal logarithm ) Enrtopy formulas and canonical distributions

9. Jaynes’ entropy maximum principle Differentials are NOT independent!

10. Zeroth Law:  (E1,…)=  (E2,…) Which composition laws are compatible with this? empirical temperature with Péter Ván

11. Zeroth Law compatible composition of energy with Péter Ván

12. Zeroth Law compatible composition of energy same function! with Péter Ván

13. Zeroth Law compatible composition of energy with Péter Ván

14. The solution with Péter Ván

15. An example all L( ) functions are the same!

16. How may Nature do this?

17. In small steps!

18. Composition Laws

19. Formal logarithm: Additive quantity: Asymptotic composition rule:

20. Composition Laws: summary Such asymptotic rules are: 1.commutative x  y = y  x 2. associative (x  y)  z = x  (y  z) 3. zeroth-law compatible

21. Lagrange method

22. Superstatistics

23. Canonical Power-Law Footnote: w(t) is an Euler-Gamma distribution in this case.

24. Composition Laws In this family of entanglement all statistical phyics methods and results apply ! Non-extensive Boltzmann equation Nonlinear Fokker-Planck equation Coupled Langevin equations Lagrange multiplier method Superstatistics: shaken Monte Carlo

25. (Non-)additivity and (non-)extensivity

26. Tsallis Rényi Boltzmann Entropy formulas

27. Tsallis Rényi Boltzmann

28. Function of Entropy Tsallis Rényi Rényi = additive version of Tsallis

29. Canonical distribution with Rényi entropy This cut power-law distribution is an excellent fit to particle spectra in high-energy experiments!

30. The cut power-law distribution is an excellent fit to particle spectra in high-energy experiments! How to caluclate (predict) T, q, etc… ?

31. What is universal in collisons? HorizonEvent Horizon due to stopping Schwinger formula + Newton + Unruh = Boltzmann Dima Kharzeev

32. Horizon thermodynamics Information loss ~ entropy ~ horizon area Additive energy, non-additive horizon Temperature: Unruh, Hawking Based on Clausius’ entropy formula Since the 1970 - s

33. Quantum and Gravity Units Scales: in c = 1 units

34. Unruh temperature entirely classical special relativity suffices An observer with constant acceleration Fourier analyses a monochromatic EM - wave from a far, static system in terms of its proper time: the intensity distribution is proportional to the Planck distribution ! Unruh

35. Unruh temperature entirely classical special relativity suffices An observer with constant acceleration Fourier analyses a monochromatic EM - wave from a far, static system in terms of its proper time: the intensity distribution is proportional to the Planck distribution ! Unruh Max Planck

36. Unruh temperature Galilei Rindler

37. Unruh temperature

39. Interpret this as a black body radiation: Planck distribution of the frequency

40. Unruh temperature Planck-interpretation: Temperature in Planck units: Temperature in familiar units:

41. Unruh temperature On Earth’ surface it is 10^(-19) eV

42. Unruh temperature Stopping from 0.88 c to 0 in L = ħ/mc Compton wavelength distance: kT ~ 170 MeV for mc² ~ 940 MeV (proton)

43. Clausius’ entropy

44. Bekenstein-Hawking entropy Use Unruh temperature at horizon Use Clausius’ concept with that temperature Hawking Bekenstein

45. Acceleration at static horizons Maupertuis action for test masspoints Euler-Lagrange eom: geodesic Arc length is defined by the metric Maupertuis

46. Acceleration at static horizons This acceleration is the red-shift corrected surface gravity.

47. BH entropy inside static horizons This is like a shell in a phase space!

48. BH entropy for static horizons This is like a shell in a phase space!

49. BH entropy: Schwarzschild This area law is true for all cases when f(r,M) = 1 – 2M / r + a( r ) !!! Hawking-Bekenstein result Schwarzschild

50. Schwarzschild BH: EoS Hawking-Bekenstein entropy  instable eos S E T > 0 c < 0 Planck units: k = 1, ħ = 1, G = 1, c = 1 B

51. Schwarzschild BH: deformed entropy Tsallis-deformed HB entropy  for large E stable eos ☻S E T > 0 c < 0 T > 0 c > 0 a = q - 1 arXiV: 1011.3442

52. Schwarzschild BH: quantum zero point EoS stability limit is at / below the quantum zero point motion energy ☻S E T > 0 c < 0 T > 0 c > 0 STAR, PHENIX, CMS: a ~ 0.20 - 0.22 inflection point E 0 arXiV: 1011.3442 Bekenstein bound

53. Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data

54. Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data

55. Blast wave fits and quark coalescence SQM 2008, Beijing with Károly Ürmössy

56. Summary Thermodynamics build on composition laws Deformed entropy formulas Hawking entropy: phase space of f ( r ) = 0: horizon ‘size’ Schwarzschild BH: Boltzmann entropy  unstable eos Rényi entropy: stable BH eos at high energy ( T > Tmin ) Estimate for q: from the instability being in the Trans- Planckian domain

58. All particle types follow power-law E L(E) WRONG! R I G H T ! with Károly Ürmössy