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Tamás Csörgő 1 , Márton Nagy 2 and Máté Csanád 2

New exact solutions of perfect fluid hydrodynamics and an advanced estimate of the initial energy density and the life-time of the reaction. Tamás Csörgő 1 , Márton Nagy 2 and Máté Csanád 2. 1 MTA KFKI RMKI, Budapest, Hungary 2 Eötvös University, Budapest, Hungary.

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Tamás Csörgő 1 , Márton Nagy 2 and Máté Csanád 2

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  1. New exact solutions of perfectfluid hydrodynamics and an advanced estimate of theinitial energy density and thelife-time of the reaction Tamás Csörgő 1, Márton Nagy 2 and Máté Csanád 2 1 MTA KFKI RMKI, Budapest, Hungary 2 Eötvös University, Budapest, Hungary Accepted by Braz. J. Phys (Proceedings of ISMD 2007) NCRH2007, Frankfurt a. M. , Germany

  2. Contents • Introduction, motivation • The equations of relativistic hdrodynamics • Review of known solutions • Spherically symmetric flows • New simple solutions in 1+D dimensions • New simple solutions in 1+1 dimensions • Rapidity distribution • Advanced energy density and life-time estimation • Summary and outlook NCRH2007, Frankfurt a. M. , Germany

  3. Topic: Relativistic hydrodynamics, from high energy phenomenology point of view Interesting theoretical problem, with relevant applications History: • Fermi: basic idea of collective description of high-energy reactions, based on avg. free path arguments (1950) • Landau & co-workers: clarification of Fermi’s idea, formulation of relativistic hydrodynamics, exact solution, realistic rapidity profile calculation (1953-57) • Hwa & Bjorken: (very) simple relativistic solution, (unrealistic) rapidity distribution, initial energy density estimation (1974, 1983) Introduction NCRH2007, Frankfurt a. M. , Germany

  4. Nowadays: • CERN and RHIC: scaling laws observed in data • Hydrodynamics provides good frame of successful models • Buda-Lund model: successful description of data, based on exact (nonrelativistic and relativistic) analytic hydro solutions Important task: to find new relativistic analytic (accelerating) solutions Concepts of (relativistic) hydrodynamics: • Local thermal equilibrium • Local conservation of energy-momentum • No physical scale Introduction NCRH2007, Frankfurt a. M. , Germany

  5. Four-vector form NR notation comoving proper-time derivative comoving derivative Euler equation: Energy conservation: Charge conservation: For perfect fluid: The equations of relativistic hydrodynamics NCRH2007, Frankfurt a. M. , Germany

  6. Entropy conservation: as expected for perfect fluid. Alternative form of Euler equation: Thermodynamics: The equations of relativistic hydrodynamics EoS mean additional constraint NCRH2007, Frankfurt a. M. , Germany

  7. Landau-Khalatnikov solution L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51 I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529 L.D.Landau & S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309 Implicit1D solution, approx. Gaussian rapidity distribution Basic relations: Unknown variables: Auxiliary function: Expression of is very complicated NCRH2007, Frankfurt a. M. , Germany

  8. Landau-Khalatnikov solution Animation (courtesy of T. Kodama) NCRH2007, Frankfurt a. M. , Germany

  9. Rindler coordinates: Boost-invariance (for asymptotically high energies): depends on EoS. Hwa-Bjorken solution R.C. Hwa, Phys. Rev. D10, 2260 (1974) J.D. Bjorken, Phys. Rev. D27, 40(1983) Accelerationless, expanding 1D simple boost-invariant solution Euler equation satisfies NCRH2007, Frankfurt a. M. , Germany

  10. Hwa-Bjorken solution The Hwa-Bjorken solution / Rindler coordinates NCRH2007, Frankfurt a. M. , Germany

  11. Self-similar, ellipsoidal solutions T. Csörgő, L.P.Csernai, Y. Hama, T. Kodama, Heavy Ion Phys. A 21 (2004) 73 Spherically symmetric, accelerationless velocity profile: // inside the lightcone // Let Scaling function The A = const. surfaces: self-similarly expanding ellipsoids EoS: (massive) ideal gas Euler equation satisfies NCRH2007, Frankfurt a. M. , Germany

  12. Other accelerationless solutions: T. S. Biró, Phys. Lett. B 474, 21 (2000) Yu. M. Sinyukov & I. A. Karpenko, nucl-th/0505041 Coordinate transformations: S. Pratt, nucl-th/0612010 Self-similar, ellipsoidal solutions If , then is free: Buda-Lund type of solution Need for explicit simple accelerating solutions NCRH2007, Frankfurt a. M. , Germany

  13. Spherically symmetric flows Spherical symmetry: Scaling function: For the continuity equations: Solutions: NCRH2007, Frankfurt a. M. , Germany

  14. Full solution: Free functions: Constraints: Spherically symmetric flows For solving the Euler equation: pseudo-orthogonal complementer NCRH2007, Frankfurt a. M. , Germany

  15. Thermodynamics: After some calculation we obtain: Spherically symmetric flows We need EoS to specify a solution Simple choice: massless ideal gas NCRH2007, Frankfurt a. M. , Germany

  16. Spherically symmetric flows A final equation: NCRH2007, Frankfurt a. M. , Germany

  17. 1: for any , : well known 2: for : Wave equation: general solution exists 3: for , : NEW! Spherically symmetric flows / New solutions Some non-trivial solutions: NCRH2007, Frankfurt a. M. , Germany

  18. New simple solutions in 1+D dimensions The first new solution: If including temperature & conserved particle number: Trajectories are constantly accelerating (in rest frame): NCRH2007, Frankfurt a. M. , Germany

  19. New simple solutions in 1+D dimensions Fluid trajectories of the 1+D dimenisonal new solution NCRH2007, Frankfurt a. M. , Germany

  20. New simple solutions in 1+D dimensions The lines (black) and the lines (red) NCRH2007, Frankfurt a. M. , Germany

  21. If , general solution is possible A particular choice (similar to Hwa-Bjorken and the previous one): Hwa-Bjorken, Buda-Lund type New accelerating Important multidimensional (thanks to T. S. Biró) Special EoS, but general velocity New simple solutions in 1+1 dimensions Possible cases (one row of the tabular is one solution): NCRH2007, Frankfurt a. M. , Germany

  22. New simple solutions in 1+1 dimensions Animation of the temperature profile in the case NCRH2007, Frankfurt a. M. , Germany

  23. Rapidity distribution Specifying freeze-out conditions, numerical calculation is straightforward Analytic formula is available using saddle-point integration: Gaussian ,,width’’: Realistic, finite widths (or local minimum at mid–rapidity) Data analysis suggests peak at For the parameter it means NCRH2007, Frankfurt a. M. , Germany

  24. Rapidity distribution Rapidity distribution from the 1+1 dimensional solution, for . NCRH2007, Frankfurt a. M. , Germany

  25. Rapidity distribution Rapidity distribution from the 1+1 dimensional solution, for . NCRH2007, Frankfurt a. M. , Germany

  26. Rapidity distribution BRAHMS data fitted with the analytic formula of pseudorapidity distribution NCRH2007, Frankfurt a. M. , Germany

  27. Rapidity distribution BRAHMS data fitted with the analytic formula of rapidity distribution NCRH2007, Frankfurt a. M. , Germany

  28. Advanced energy density estimation Fit result: in collisions at RHIC Flow is accelerating Bjorken’s initial energy density estimation needs correction Correction terms: due to work and faster expansion For (accelerating) flows, both factor is bigger than 1 At RHIC energies the correction can be as high as a factor of 2! NCRH2007, Frankfurt a. M. , Germany

  29. Advanced energy density estimation NCRH2007, Frankfurt a. M. , Germany

  30. Advanced life-time estimation Life-time estimation: for Hwa-Bjorken type of flows A. Makhlin & Yu. Sinyukov, Z. Phys. C 39, 69 (1988) With this formula the life-time is underestimated, as pointed out by many people In our treatment: correction is because of acceleration: At RHIC energies: correction is about +20% NCRH2007, Frankfurt a. M. , Germany

  31. Summary and outlook Result: • Explicit simple accelerating solutions of relativistic hydrodynamics • Analytic (approximate) calculation of observables • Realistic rapidity distributions; BRAHMS data well described • Advanced estimate of initial energy density: up to a factor of 2 at RHIC energies • Advanced estimate of high-energy reaction lifetime: a correction by 20% at RHIC energies nucl-th/0605070, submitted to PRL Further aims • more general EoS • less symmetry, ellipsoidal solutions • asymptotically Hubble-like flows NCRH2007, Frankfurt a. M. , Germany

  32. Thank you for your attention! NCRH2007, Frankfurt a. M. , Germany

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