NonExtensive Black Hole Thermodynamics estimate for PowerLaw Particle

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 ar. Xi. V: 1011. 3442

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 RHIC data with Károly Ürmössy SQM 2008, Beijing

Abstract thermodynamics • S(E) = max (Jaynes-) principle • nontrivial composition of e. g. the energy E • 0 -th law requires: factorizing form T 1(E 1) = T 2(E 2) • 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

Zeroth Law: (E 1, …)= (E 2, …) empirical temperature with Péter Ván Which composition laws are compatible with this?

Zeroth Law compatible composition of energy with Péter Ván

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

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

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.

Entropy formulas Boltzmann Tsallis Rényi

Function of Entropy Tsallis 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? • Event 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 1970 - 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 Unruh Planck distribution !

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 Unruh Planck distribution ! Max Planck

Unruh temperature Galilei Rindler

Unruh temperature

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) e. V

Unruh temperature Stopping from 0. 88 c to 0 in L = ħ/mc Compton wavelength distance: k. T ~ 170 Me. V for mc² ~ 940 Me. V (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 Hawking-Bekenstein result This area law is true for all cases when f(r, M) = 1 – 2 M / r + a( r ) !!!

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

Schwarzschild BH: deformed entropy ☻S T>0 c>0 T>0 c<0 E ar. Xi. V: 1011. 3442 Tsallis-deformed HB a=q-1 entropy for large E stable eos

Schwarzschild BH: quantum zero point ☻S inflection point T>0 c>0 T>0 c<0 Bekenstein bound E 0 STAR, PHENIX, CMS: a ~ 0. 20 - 0. 22 E ar. Xi. V: 1011. 3442 Eo. S stability limit is at / below the quantum zero point motion energy

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

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

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