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Power-law tailed spectra from equilibrium

Power-law tailed spectra from equilibrium. Particle spectra Non-extensive Boltzmann eq. (NEBE) Particular distributions Parton cascade demo Lattice strategy. T.S.Bir ó and G.Purcsel ( KF KI RMKI , Giessen) A.Jakovác (BMGE), G.Györgyi (ELTE), Zs.Schram (DE).

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Power-law tailed spectra from equilibrium

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  1. Power-law tailed spectra from equilibrium • Particle spectra • Non-extensive Boltzmann eq. (NEBE) • Particular distributions • Parton cascade demo • Lattice strategy T.S.Biróand G.Purcsel (KFKI RMKI,Giessen) A.Jakovác (BMGE), G.Györgyi (ELTE), Zs.Schram (DE) Quark Matter, Budapest, 3-8August, 2005

  2. Hadron statistics hep-ph/0409157 JPG31:1, 2005 • Gibbs thermodynamics: exponential • Non-extensive thermodynamics: power-law(pQCD jets are angle correlated) • Collective flow effects: scaling breakdown • low pt and high pt: connected?

  3. Particle spectra and Eq. of State  3 3 (2h) d N  = d, k)f(/T) 3 V dk   Spectrum Spectral function thermodynamics  Gibbs Tsallis . . . Peak: particle bgd.: field Shifted peak: quasiparticle

  4. t Experimental spectra: pp t mesons, 30 GeV, p -tail v = 10.1 ± 0.3 pions, 30 GeV, m -tail v = 9.8 ± 0.1 pions, 540 GeV, m -tail v = 8.1 ± 0.1 quarkonia, 1.8 TeV, m -tail v = 7.7 ± 0.4 t tt tt tt t tt Gazdiczki + Gorenstein (hep-ph / 0103010)

  5. t Experimental spectra: AuAu t pi, K, p, 200 GeV, m -scaling (i.e. E = m ) v = 16.3 (E = 2.71 GeV, T = 177 MeV) t t t c t Schaffner-Bielich, McLerran, Kharezeev (NPA 705, 494, 2002)

  6. RHIC Au Au heavy ion collision 200 GeV 0  from AuAu at 200 GeV (PHENIX) pions 1E-0 min. bias 1E-1 1E-2 v = 9.527 ± 0.181 E = 1.008 ± 0.0973 GeV 2 1E-3 c d  2p dp dy 1E-4 q = 1.11727 T = 118 ± 9 MeV t t 1E-5 1E-6 1E-7 1E-8 T = 364 ± 18 MeV 1E-9 0 2 4 6 8 10 12 14 p (GeV) t

  7. t Experimental spectra: cosmic rays t before knee, m -scaling (i.e. E = m ) v = 5.65 (E = 0.50 GeV, T = 107 MeV) in ankle, v = 5.50 (E = 0.48 GeV, T = 107 MeV) t t t c c Ch. Beck cond-mat / 0301354

  8. t Experimental spectra: e-beam t integral over longitudinal momenta TASSO 14 GeV v = 51 (E = 6.6 GeV) TASSO 34 GeV v = 9.16 (E = 0.94 GeV) DELPHI 91 GeV v = 5.50 (E = 0.56 GeV) DELPHI 161 GeV v = 5.65 (E = 0.51 GeV) t c c c t c Bediaga et.al. hep-ph / 9905255

  9. General Langevin problem TSB+GGy+AJ+GP, JPG31, 2005 < z(t) > = 0 . ∂E p = z - G(E) < z(t)z(t') > = 2 D(E) (t-t') ∂p In the Fokker – Planck equation: K (p) = D(E) 2 ∂E = K (p) -G(E) 1 ∂p Stationary distribution: ( ) A dE ∫ f(p) = exp - G(E) D(E) D(E)

  10. General inverse slope Stationary distribution: ( ) dE ∫ f(p) = A exp - T(E) 1) Gibbs: T(E) = T  exp(-E/T) 2) Tsallis: T(E) = T/q + (1-1/q) E  -q /(q-1) ( 1 + (q-1) E / T) T( T ) = T : a fixed point of the sliding slope

  11. Test v = 1 + E / T c Gaussian fit to parton distribution: < p > = D / G = 1 ... 1.5 GeV Power-tail in e+e- experiment (ZEUS): v = 5.8 ± 0.5 -> G / C = 9.6 ± 1 Derived inclination point at p = √ D / C = 3 ... 4 GeV. 2 t ☺ c

  12. All central transverse slopes All central transverse slopes Flow

  13. Transverse flow correction Energy in flowing cell:  E = u p = (m cosh(y-) - v p cos(-) )  T T Most detected: forward flying (blue shifted) at = y, = . E = (m - v p ) T T Spectrum ~ ∫d d f(E)

  14. Transverse flow corrected slopes forward blast wave!

  15. Average transverse momentum

  16. T. S. Bíró and G. Purcsel (University of Giessen, KFKI RMKI Budapest) Non-Extensive Boltzmann Equation hep-ph/0503204 • Non-extensive thermodynamics • 2-body Boltzmann Equation + non-ext. rules • Unconventional distributions: quasi-energy • H-theorem and non-extensive entropy • Principles of the numerical simulation

  17. Generalized sum f = f f statistical independence 1 12 2 non-extensive addition rule E = h ( E , E ) 1 12 2 non-extensive addition rules for energy, entropy, etc. h ( x, y ) ≠x + y

  18. 2,3 1,2 Sober addition rules associativity: 3 1 h ( h ( x, y ) , z ) = h ( x, h ( y, z ) ) general math. solution: maps it to additivity X ( h ) = X ( x ) + X ( y ) X( t ) is a strict monotonic, continous real function, X(0) = 0

  19. Boltzmann equation  ∫ f = w ( f f - f f ) 3 4 1 2 1 1234 234 2 w = M ( p + p - p - p )  2 3 4 1234 1234 1 (h( E , E ) - h( E , E )) 2 3 4 1

  20. canonical equilibrium: f ~ exp ( -X(E ) / T ) • 2-body collisions: X(E ) +X(E ) = X(E ) +X(E ) • non-extensive entropy density: s = f X (-ln f ) • H-theorem for X( S ) = - f ln f Consequences 1 2 3 4 -1 s ∫ tot s

  21. NEBE control sums Non-extensive entropy:

  22. Entropies Boltzmann Kaniadakis (NLBE) Tsallis entropy Rényi entropy NEBE

  23. T h e r m o d y n a m i c s e s rule additive equilibrium entropy name h ( x, y ) X ( E ) f ( E ) s [ f ] general x + y E exp( - E / T) - f ln f Gibbs -1/aT q 1 x + y + a xy ln(1+aE) (1+aE) (f - f)/(q-1) Tsallis a q = 1 - aT b 1/b b incomplete gamma fct. b b ( x + y ) (aE) / a exp(-(aE) /aT) Lévy 1 - 1/ aT 1 q a x y ln aE (aE) ln ∫f Rényi 1- q a q = 1 - aT

  24. Test particle simulation y h(x,y) = const. E E 2 E 4 E x E 3 E 1 E 3 -1 ∫ uniform random: Y(E ) = (  h/  y) dx 3 h=const 0

  25. Movie: Tsallis a = 0 y=2

  26. Movie: Tsallis a = 1 y=2

  27. Movie: Tsallis a = -0.25, y=2

  28. Tsallis distribution

  29. Limiting temperature with Tsallis distribution ( with A. Peshier, Giessen ) Massless particles, d-dim. momenta, N-fold Hagedorn d <X(E)> TE  c ; = T = E / d c H j=1 E – j T c N For N  2: Tsallis partons  Hagedorn hadrons

  30. Lattice Monte Carlo with fluctuating (Gamma-distributed) temperature (Research with Zs. Schramm, Debrecen) G-distributed a /a ratio Average effective action Large system limit Metropolis strategy s t

  31. q = 1 + 1 / c Canonical distribution: POWER – LAW TAILED -(c+1) f  exp( - X / T ) = ( 1 + E / cT ) This equals to Gamma distributed Gibbs factors: -(c+1) c 1  -t -xt/c ( 1 + x / c ) = dt t e e (c+1) Interpretations: fluctuating temperature, energy imbalance, multiplicative + additive noise, . . .

  32. Lattice theory Expectation values of observables: -S(t,U) DU dt w (t) e t A(U) ∫ ∫ v c A = -S(t,U) DU dt w (t) e ∫ ∫ c Action: S(t,U) = a(U) t + b(U) / t t= a / a asymmetry parameter t s

  33. Lattice theory: effective action ∞ c c+v-1 -(a+c)t - b/t c ∫ S = dt t e - ln G(c) eff 0 Plaquette sums: space-space: a = ∑ (1 – Re tr P ss) space-time: b = ∑ (1 – Re tr P ts) Evaluation methods: • exact analytical • saddle point • numerical (Gauss-Laguerre)

  34. Lattice theory: effective action (c+v)/2 c c b ( ( ) ) S = 2K (2  b(a+c) ) - ln a+c c+v eff G(c) Asymptotics: • large a,b finite c: 2  ab • large a,b,c and a-b<< (a+b): a + b

  35. Order parameter: Re Polyakov

  36. Strong – weak coupling transition action: MC a+b, TS < at + b/t >

  37. Strong – weak coupling transition Polyakov line: t-independent both for MC and TS

  38. Summary • Power-law distributions are predicted by non-extensive thermodynamics • Multiplicative noise  sliding slope T(E) • NEBE also leads to power-law tails • Tsallis distribution can be modelled by Gamma distributed inv. temperature • Non-extensive statistical properties are worth to study

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