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Initial conditons, equations of state and final state in hydrodynamics

Initial conditons, equations of state and final state in hydrodynamics. Csanád Máté Eötvös University Department of Atomic Physics. Hydro models IS, EoS, FOC and FS Observables. Little vocabulary of hydrodynamics. Exact solution

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Initial conditons, equations of state and final state in hydrodynamics

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  1. Initial conditons, equations of state and final state in hydrodynamics Csanád Máté Eötvös University Department of Atomic Physics Hydro models IS, EoS, FOC and FS Observables

  2. Little vocabulary of hydrodynamics • Exact solution • Solution of hydro equations analytically, without approximation • Parametric solution • Exact solution, that has fit parameters • Hydro inspired parameterization • Distribution determined at freeze-out only, their time dependence is not considered • Numerical solution • Solution of hydro equations numerically M. Csanád, WPCF08 Krakow

  3. How analytic hydro works • Take hydro equations and EoS • Find a solution • Will contain parameters (like Friedmann, Schwarzschild etc.) • Will use a possible set of initial conditions • Use a freeze-out condition • Eg fixed proper time or fixed temperature • Generally a hyper-surface • Calculate the hadron source function • Calculate observables • E.g. spectra, flow, correlations • Straightforward calculation • Hydrodynamics: Initial conditions  dynamical equations  freeze-out conditions M. Csanád, WPCF08 Krakow

  4. Famous solutions • Landau’s solution (1D, developed for p+p): • Accelerating, implicit, complicated, 1D • L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51 • I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529 • L.D.Landau and S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309 • Hwa-Bjorken solution: • Non-accelerating, explicit, simple, 1D, boost-invariant • R.C. Hwa, Phys. Rev. D10, 2260 (1974) • J.D. Bjorken, Phys. Rev. D27, 40(1983) • Others • Chiu, Sudarshan and Wang • Baym, Friman, Blaizot, Soyeur and Czyz • Srivastava, Alam, Chakrabarty, Raha and Sinha M. Csanád, WPCF08 Krakow

  5. 3D solutions • Nonrelativistic, spherically symmetric solution • P. Csizmadia, T. Csörgő, B. Lukács, nucl-th/9805006 • Relativistic, spherically symmetric solution • T. Csörgő, L. Csernai, Y. Hama, T. Kodama, nucl-th/0306004 • Accelerationless • Hubble flow profile (flow proportional to distance) • Relativistic, spherically symmetric solution • T. Csörgő, M. Nagy, M. Csanád, nucl-th/0605070 • Accelerating • Realistic rapidity distributions (data described by it) • Advanced energy and lifetime estimate • All describe expanding fireballs • Sometimes: rings/shells of fire M. Csanád, WPCF08 Krakow

  6. Where we are • Other accelerationless solutions: • T. S. Biró, Phys. Lett. B 474, 21 (2000) • Yu. M. Sinyukov and I. A. Karpenko, nucl-th/0505041 • Solutions by coordinate transformations: • S. Pratt, nucl-th/0612010 • Revival of interest • Bialas, Janik, Peschanski: Phys.Rev.C76:054901,2007 • Borsch, Zhdanov: SIGMA 3:116,2007 • There are some exotic solutions as well • Need for solutions that are: • explicit • simple • accelerating • relativistic • realistic / compatible with the data • Buda-Lund type of solutions: each fulfilled • but not simultaneously M. Csanád, WPCF08 Krakow

  7. A Buda-Lund type of solution • For sake of simplicity, take the following nonrel. solution • Csörgő, Akkelin, Hama, Lukács, Sinyukov, Phys.Rev.C67:034904,2003 • Self similarly expanding ellipsoid, Gaussian IC • Flow profile: directional Hubble • Equation of motion for principal axes: • Freeze-out at constant temperature assumed M. Csanád, WPCF08 Krakow

  8. Dependence on IC+EoS (nonrel) • Evolution of principal axes of the ellipsoid M. Csanád, WPCF08 Krakow

  9. Dependence on IC+EoS (nonrel) • Evolution of expansion rates M. Csanád, WPCF08 Krakow

  10. Dependence on IC+EoS (nonrel) • Time evolution of temperature M. Csanád, WPCF08 Krakow

  11. Same in relativistic hydro Nagy, Csörgő, Csanád, Phys.Rev.C77:024908,2008, Csanád, Nagy, Csörgő, Eur.Phys.J.ST 155:19-26,2008 Same final state for different evolutions, even with viscosity (see T. Csörgő, WPCF’07) M. Csanád, WPCF08 Krakow

  12. Conjectured EoS dependence of e0 • Relativistic, accelerating solution → describe dn/dh • Energy density modified compared to Björken • With f/0 = 10, cs = 0.35 [nucl-ex/0608033], correction to 0 is about 2.9× • 0 = 14.5 GeV/fm3 in 200 GeV, 0-5 %Au+Au at RHIC M. Csanád, WPCF08 Krakow

  13. Predictions of the Buda-Lund models • Hydro predicts scaling (even viscous) • What does a scaling mean? • See Hubble’s law – or Newtonian gravity: • Data collapse • Collective, thermal behavior → Loss of information • Spectra slopes: • Elliptic flow: • HBT radii: M. Csanád, WPCF08 Krakow

  14. Elliptic flow • Prediction of 2003: scaling variable • If plotted against ‘w’, data collapse: • From 20 to 200 GeV • All centralities • Pion, kaon, proton • pt and h dependence • Prediction: Csanád, Csörgő, Lörstad, Ster et al. nucl-th/0512078 M. Csanád, WPCF08 Krakow

  15. Correlation radii Geometrical radii Thermal radii Prediction for HBT radii • Exact hydro result (nonrel shown) • Correlation radii = geometrical  thermal • Harmonic squared sum: 1/R2corr= 1/R2geom+ 1/R2therm • Geom.: Rgeom = X • Thermal: • Hubble-profile → Xth=Yth • Rout  Rside  Rlong M. Csanád, WPCF08 Krakow

  16. Azimuthal HBT and elliptic flow • AsHBT data described • Both governed by asymmetries • as: coordinate-space • r2: momentum-space • v2 depends only on r2 • Csanád, Tomasik, Csörgő Eur. Phys. J. A 37,111 (2008) M. Csanád, WPCF08 Krakow

  17. Azimuthal HBT and elliptic flow • Simultaneous description • Slopes as before (slide 12) • Elliptic flow as before (slide 13) • Correlation radii • Asymmetry parameters used: r2=0.17, as=0.997 • Csanád, Tomasik, Csörgő Eur. Phys. J. A 37,111 (2008) M. Csanád, WPCF08 Krakow

  18. Prediction for kaon HBT • Transverse mass scaling → same curve for pions and kaons if plotted versus mt • Other models? K M. Csanád, WPCF08 Krakow

  19. Beyond hydro: long source tails • HRC reproduces HBT (hydro as well) • But also long tails in two-pion source! • Anomalous diffusion (rescattering) • This goes beyond hydro • Hydro: regular mt scaling • Lévy-tails important here! • Tail depends on m.f.p., thus the cross-section • Kaons: lowest cross- section → heaviest tail T. Humanic, Int. J. Mod. Phys. E15 197 (2006) Csörgő, Braz.J.Phys.37:1002-1013,2007 M. Csanád, WPCF08 Krakow

  20. The HBT test • Models with acceptable results: • nucl-th/0204054 Multiphase Trasport model (AMPT)‏ Z. Lin, C. M. Ko, S. Pal • nucl-th/0205053 Hadron cascade model T. Humanic • hep-ph/9509213 Family of Buda-Lund hydro models T. Csörgő, B. Lörstad, A. Ster • hep-ph/0209054 Cracow (single freeze-out, thermal) W. Broniowski, W. Florkowski • nucl-ex/0307026 Blast wave model F. Retiére for STAR • 0801.4361 2 + 1 boost invariant rel. hydro, W. Broniowski, M. Chojnacki, W. Florkowski, A. Kisiel M. Csanád, WPCF08 Krakow

  21. Conclusions • Several types of hydro models • Success in spectra and flow • Few describe v2(h) or HBT • Hadronic final state: combination of IC, EoS and FC • Penetrating probes required • Similarities of successful models? • Gaussian IC, Hubble flow etc. • Compare Hubble-coefficients in models! • Search for decisive tests! M. Csanád, WPCF08 Krakow

  22. Thank you for your attention

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