T. Osada （ Tokyo City Univ. ） and G. Wilk （ Andrzej So ł tan Inst. ）

1. Dissipative effects on relativistic hydrodynamics in the presence of long range interactionsbased on arXiv:0710.1905[nucl-th] T.O and G. Wilk PRC77 044903, (2008). T. Osada（Tokyo City Univ.） and G. Wilk（Andrzej Sołtan Inst.） mini-Workshop at Dept. of Phys. TCU. June 24, 2009

2. RHICExperiments The detector of PHENIXCollab.

3. 初期宇宙における高温・高密度状態の再現・解明に向けて初期宇宙における高温・高密度状態の再現・解明に向けて 高エネルギー原子核衝突実験⇔極限状態での物質の形態 日経サイエンス　２００６年０８月号

4. hydro scale; definable local equilibrium or stationary Non- extensivity for system with correlations 1 quarks, gluons or hadrons produced by high-energy nucleus-nucleus collisions correlation and/or fluctuation scale; of composing particles

5. extensivity: entropy, measure of information about the particle distribution in the states available to the system, is extensive in the sense that the total entropy of two independent subsystems is the sum of their entropies. Assumptions in Boltzmann-Gibbs statistics 2 some assumptions leading to the Boltzmann-Gibbs (BG) statistics may be too tight * absence of memory effects, * negligible local correlation * absence of long-range interaction + Boltzmann H-theorem (based on the extensive entropy)

6. Non-extensive entropy 3 Tsallis’s non-extensive entropy ・In 1988 Tsallis proposed a generalization of the entropy of the BG entropy C. Tsallis, J.Stat.Phys.52(1988) 749.

7. Why Nonextensive hydrodynamics? 4 *‘standard’ thermodynamics exponential particle spectra, experiments definitely power-law tail @ high pT. ↦usual hydro+ (other dynamical origins…) or non-hydrodynamic approach ↦Nonextensive hydro + (other dynamical origins…) ↑ including (momentum) correlation *Nonextensive (perfect) hydro ⇌(usual q=1) dissipative hydro link via nonextensive/dissipative correspondence

8. Relativistic non-extensive kinetic theory 5 non-extensive version of Boltzmann equation: A. Lavagno, Phys.Lett.A301(2002) 13. ★correlation functionsame space-time x but different p

9. q-generalized entropy current revised by Osada and Wilk therm. dyn. rel. OK Non-extensive H theorem 6 q-generalized Boltzmann Stosszahlansatz J. A. S. Lima, R. Silva and A.R. Plastino,Phys.Rev.Lett86(2001),2983

10. q- equilibrium 7 collision invariant: q- equilibrium distribution function: q- energy-momentum tensor:

11. q- hydrodynamical model 8

12. Tsallis hydrodynamics, locally conserve q-entropy current: standard hydrodynamics, locally conserve (BG) entropy current: Standard hydrodynamics vs. q- hydrodynamics 9 without correlations between cells including correlations

13. Single particle spectra by q-hydro model 10

14. Single particle spectra by q-hydro model 11

15. Perfect q-hydrodynamics ⇌ (q=1) dissipative hydrodynamics 12 Nonextensive/dissipative correspondence (perfect) nonextensive hydrodynamics dissipative hydrodynamics

16. 13 Entropy production q > 1↦ near equilibrium; stationary stateq = 1 ↦ true equilibrium

17. Bulk and shear viscosities 14 perfect q-hydrodynamics + nonextensive/dissipative correspondence

18. Summary 15 • Nonextensive (Tsallis’s ) hydrodynamical model is formulated by based on the relativistic nonextensive kinetic theory. • perfect q- hydrodynamics may be connected with the dissipative ‘standard’( i.e., q =1) hydrodynamics. What’s the next ? • 2+1 q-hydro model with QGP EoS. ↦ elliptic flow (pT-dependence) and the HBT puzzle • q-energy momentum tensor in Einstein equation ↦ effects on the cosmological constant L