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:
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….
Heavy ion collision: theoretical picture URQMD ( Univ. Frankfurt: Sorge, Bass, Bleicher…. )
Experimental picture … RHIC
Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data
Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data
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
Jaynes’ entropy maximum principle Differentials are NOT independent!
Zeroth Law: (E1,…)= (E2,…) Which composition laws are compatible with this? empirical temperature with Péter Ván
Zeroth Law compatible composition of energy with Péter Ván
Zeroth Law compatible composition of energy same function! with Péter Ván
Zeroth Law compatible composition of energy with Péter Ván
The solution with Péter Ván
An example all L( ) functions are the same!
How may Nature do this?
In small steps!
Composition Laws
Formal logarithm: Additive quantity: Asymptotic composition rule:
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
Lagrange method
Superstatistics
Canonical Power-Law Footnote: w(t) is an Euler-Gamma distribution in this case.
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
(Non-)additivity and (non-)extensivity
Tsallis Rényi Boltzmann Entropy formulas
Tsallis Rényi Boltzmann
Function of Entropy Tsallis Rényi Rényi = additive version of Tsallis
Canonical distribution with Rényi entropy This cut power-law distribution is an excellent fit to particle spectra in high-energy experiments!
The cut power-law distribution is an excellent fit to particle spectra in high-energy experiments! How to caluclate (predict) T, q, etc… ?
What is universal in collisons? HorizonEvent Horizon due to stopping Schwinger formula + Newton + Unruh = Boltzmann Dima Kharzeev
Horizon thermodynamics Information loss ~ entropy ~ horizon area Additive energy, non-additive horizon Temperature: Unruh, Hawking Based on Clausius’ entropy formula Since the s
Quantum and Gravity Units Scales: in c = 1 units
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
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
Unruh temperature Galilei Rindler
Unruh temperature
Interpret this as a black body radiation: Planck distribution of the frequency
Unruh temperature Planck-interpretation: Temperature in Planck units: Temperature in familiar units:
Unruh temperature On Earth’ surface it is 10^(-19) eV
Unruh temperature Stopping from 0.88 c to 0 in L = ħ/mc Compton wavelength distance: kT ~ 170 MeV for mc² ~ 940 MeV (proton)
Clausius’ entropy
Bekenstein-Hawking entropy Use Unruh temperature at horizon Use Clausius’ concept with that temperature Hawking Bekenstein
Acceleration at static horizons Maupertuis action for test masspoints Euler-Lagrange eom: geodesic Arc length is defined by the metric Maupertuis
Acceleration at static horizons This acceleration is the red-shift corrected surface gravity.
BH entropy inside static horizons This is like a shell in a phase space!
BH entropy for static horizons This is like a shell in a phase space!
BH entropy: Schwarzschild This area law is true for all cases when f(r,M) = 1 – 2M / r + a( r ) !!! Hawking-Bekenstein result Schwarzschild
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
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:
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 ~ inflection point E 0 arXiV: Bekenstein bound
Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data
Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing with Károly Ürmössy RHIC data
Blast wave fits and quark coalescence SQM 2008, Beijing with Károly Ürmössy
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
All particle types follow power-law E L(E) WRONG! R I G H T ! with Károly Ürmössy