1. Duke University Chiho NONAKA in Collaboration with Masayuki Asakawa (Kyoto University) Hydrodynamical Evolution near the QCD Critical End Point November, 2003@Collective Flow and QGP properties, BNL

2. C.NONAKA 11/19/2003 2 Critical End Point in QCD ? Critical End Point in QCD ? NJL model (Nf = 2) Lattice QCD K. Yazaki and M.Asakawa., NPA 1989 Suggestions 2SC CF L T  RHIC GSI Critical end point Imaginary chemical potential Forcrand and Philipsen hep-lat/0307020 Reweighting Z. Fodor and S. D. Katz (JHEP 0203 (2002) 014)

3. C.NONAKA 11/19/2003 3 Phenomenological Consequence ? Phenomenological Consequence ? Divergence of Fluctuation Correlation Length critical end point M. Stephanov, K. Rajagopal, and E.Shuryak, PRL81 (1998) 4816 Still we need to study EOS Focusing Dynamics (Time Evolution) Hadronic Observables : NOT directly reflect properties at E Fluctuation, Collective Flow If expansion is adiabatic.

4. C.NONAKA 11/19/2003 4 How to Construct EOS with CEP? Assumption Critical behavior dominates in a large region near end point Near QCD end point singular part of EOS Mapping Matching with known QGP and Hadronic entropy density Thermodynamical quantities EOS with CEP  r h T QGP Hadronic 3d Ising Model Same Universality Class QCD

5. C.NONAKA 11/19/2003 5 EOS of 3-d Ising Model Parametric Representation of EOS Guida and Zinn-Justin NPB486(97)626 h : external magnetic field QCD Mapping  T r h

6. C.NONAKA 11/19/2003 6 Thermodynamical Quantities Singular Part of EOS near Critical Point Gibbs free energy Entropy density Matching Entropy density Thermodynamical quantities Baryon number density, pressure, energy density  r h T QGP Hadronic

7. C.NONAKA 11/19/2003 7 Equation of State CEP Entropy Density Baryon number density

8. C.NONAKA 11/19/2003 8 Focusing and CEP

9. C.NONAKA 11/19/2003 9 Comparison with Bag + Excluded Volume EOS Comparison with Bag + Excluded Volume EOS With End Point Bag Model + Excluded Volume Approximation (No End Point) Focused Not Focused = Usual Hydro Calculation n /s trajectories in T-  plane B

10. C.NONAKA 11/19/2003 10 Sound Velocity Clear difference between n /s=0.01 and 0.03 B Effect on Time Evolution Collective flow EOS Sound velocity along n /s B /L TOTAL /L TOTAL

11. C.NONAKA 11/19/2003 11 Slowing out of Equilibrium Slowing out of Equilibrium B. Berdnikov and K. Rajagopal, Phys. Rev. D61 (2000) 105017 Berdnikov and Rajagopal’s Schematic Argument along r = const. line Correlation length longer than  eq h  faster (shorter) expansion r h slower (longer) expansion Effect of Focusing on  ? Focusing Time evolution : Bjorken’s solution along n B /s    fm, T 0 = 200 MeV   eq

12. C.NONAKA 11/19/2003 12 Correlation Length (I) Widom’s scaling low  eq depends on n /s.  Max. Trajectories pass through the region where  is large. (focusing) eq B r h

13. C.NONAKA 11/19/2003 13 Correlation Length (II)  time evolution (1-d) Model C (Halperin RMP49(77)435)  is larger than  at Tf. Differences among  s on n /s are small. In 3-d, the difference between  and  becomes large due to transverse expansion. eq B 

14. C.NONAKA 11/19/2003 14 Consequences in Experiment (I) CERES:Nucl.Phys.A727(2003)97 Fluctuations CERES 40,80,158 AGeV Pb+Au collisions No unusually large fluctuation CEP exists in near RHIC energy region ? Mean P T Fluctuation

15. C.NONAKA 11/19/2003 15 Consequences in Experiment (II) Xu and Kaneta, nucl-ex/0104021(QM2001) Kinetic Freeze-out Temperature J. Cleymans and K. Redlich, PRC, 1999 ? Low T comes from large flow. f ? Entropy density EOS with CEP EOS with CEP gives the natural explanation to the behavior of T. f

16. C.NONAKA 11/19/2003 16 CEP and Its Consequences Realistic hydro calculation with CEP Future task Slowing out of equilibrium Large fluctuation Freeze out temperature at RHIC Fluctuation Its Consequences Focusing