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Exact results in analytic hydrodynamics

Exact results in analytic hydrodynamics. UTILIZING THE FLUID NATURE OF QGP M. Csanád, T. Csörgő , M. I. Nagy ELTE MTA KFKI RMKI Budapest, Hungary Quark Matter 2008, Jaipur, Rajastan, India February 8, 200 8. High temperature superfluidity at RHIC!.

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Exact results in analytic hydrodynamics

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  1. Exact results in analytic hydrodynamics UTILIZING THE FLUID NATURE OF QGP M. Csanád, T. Csörgő, M. I. Nagy ELTE MTA KFKI RMKI Budapest, Hungary Quark Matter 2008, Jaipur, Rajastan, India February 8, 2008

  2. High temperature superfluidity at RHIC! All “realistic” hydrodynamic calculations for RHIC fluids to date have assumed zero viscosity = 0→perfect fluid But there is a conjectured quantum limit:“A Viscosity Bound Conjecture”, P. Kovtun, D.T. Son, A.O. Starinets, hep-th/0405231Where do “ordinary” fluids sit wrt this limit?(4 p) η/s > 10 ! RHIC’s perfectfluid (4 p) η/s~1 on thisscale: The hottest (T > 2 Terakelvin) and the most perfect fluidever made… (4

  3. Equations of relativistic hydro Four-momentum tensor: Relativistic Euler equation: Energy conservation: Charge conservation: Consequence is entropy conservation:

  4. Context Reknown exactsolutions Landau-Khalatnikov solution: dn/dy ~ Gaussian Hwasolution (PRD 10, 2260 (1974))-Bjorken e0 estimate (1983) Chiu, Sudarshan and Wang: plateaux Baym, Friman, Blaizot, Soyeur and Czyz: finite size parameter D Srivastava, Alam, Chakrabarty, Raha and Sinha: dn/dy ~ Gaussian Revival of interest: Buda-Lund model + exact solutions, Biró, Karpenko+Sinyukov, Pratt (2007), Bialas+Janik+Peschanski, Borsch+Zhdanov (2007) New simple solutions Evaluation of measurables Rapidity distribution Advanced initial energy density HBT radii Advanced life-time estimation

  5. Goal Need for solutions that are: explicit simple accelerating relativistic realistic / compatible with the data: lattice QCD EoS ellipsoidal symmetry (spectra, v2, v4, HBT) finite dn/dy Report on a new class that satisfies these criteria but not all simultaneously arXiv:0709.3677v1 [nucl-th]PRC(2008) in press

  6. Self similar, ellipsoidal solutions Publication (for example): T. Csörgő, L.P.Csernai, Y. Hama, T. Kodama, Heavy Ion Phys. A 21 (2004) 73 3D spherically symmetric velocity profile: No acceleration, i.e. Define a scaling variable (compatible to flow): Self-similarly expanding ellipsoids with principal axes of at, bt and ct Use EoS of a (massive) ideal gas Scaling function can be chosen freely

  7. New, simple, exact solutions Possible cases (one row of the table is one solution): New, accelerating, d dimension d dimensional with p=p(t,h) (thanks T. S. Biró) Hwa-Bjorken, Buda-Lund type Special EoS, but general velocity If k= d = 1 , general solution is obtained, for ARBITRARY initial conditions. It is STABLE !

  8. New simple solutions Different final states from similar initial states are reached by varyingl

  9. New simple solutions Similar final states from different initial states are reached by varyingl

  10. Rapidity distribution Rapidity distribution from the 1+1 dimensional solution, for .

  11. Pseudorapidity distribution BRAHMS data fitted with the analytic formula of Additionally: yη transformation

  12. Rapidity distribution BRAHMS data fitted with the analytic formula of

  13. Advanced energy density estimate Fit result: l > 1 Flows accelerate: do work initial energy density > Bjorken’s Corrections due towork & acceleration. Ref: For l > 1 (accelerating) flows, both factors > 1 At RHIC energies the correction can be as high as a factor of ~2!

  14. Advanced energy density estimate Correction depends on timescales, dependence is: With a tipical tf/t0 of ~8-10, one gets a correction factor of 2!

  15. Advanced life-time estimate Life-time estimation: for Hwa-Bjorken type of flows Makhlin & Sinyukov, Z. Phys. C 39, 69 (1988) Underestimates lifetime (Renk, CsT, Wiedemann, Pratt, …) New correction: dn/dy width related to acceleration and work At RHIC energies: correction is about +20%

  16. Conclusions Explicit simple accelerating relativistic hydrodynamics Analytic (approximate) calculation of observables Realistic rapidity distributions; BRAHMS data well described New estimate of initial energy density: ec/eBjup by factor of 2 @ RHIC dependence on cs estimated Estimate of work effects on lifetime: increase by 20% @ RHIC A lot to do … more general EoS less symmetry, ellipsoidal solutions asymptotically Hubble-like flows

  17. New simple solutions in 1+D dim Fluid trajectories of the 1+D dimenisonal new solution

  18. Back-up Slides

  19. Landau-Khalatnikov solution Publications: 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 Implicit 1D solution with approx. Gaussian rapidity distribution Basic relations: Unknown variables: Auxiliary function: Expression of is a true „tour de force”

  20. Landau-Khalatnikov solution Temperature distribution (animation courtesy of T. Kodama) „Tour de force” implicit solution: t=t(T,v), r=r(T,v)

  21. Hwa-Bjorken solution The Hwa-Bjorken solution / Rindler coordinates

  22. Hwa-Bjorken solution The Hwa-Bjorken solution / Temperature evolution

  23. Bialas-Janik-Peschanski solution Publications: A. Bialas, R. Janik, R. Peschanski, arXiv:0706.2108v1 Accelerating, expanding 1D solution interpolates between Landau and Bjorken Generalized Rindler coordinates:

  24. Hwa-Bjorken solution Publications: R.C. Hwa, Phys. Rev. D10, 2260 (1974) J.D. Bjorken, Phys. Rev. D27, 40(1983) Accelerationless, expanding 1D simple boost-invariant solution Rindler coordinates: Boost-invariance (valid for asymptotically high energies): depends on EoS, e.g.

  25. New simple solutions in 1+d dim The fluid lines (red) and the pseudo-orthogonal freeze-out surface (black)

  26. Rapidity distribution Rapidity distribution from the 1+1 dimensional solution, for .

  27. Suppression of high pt particle production in Au+Au collisions at RHIC 1st milestone: new phenomena

  28. 2nd milestone: new form of matter d+Au: no suppression Its not the nuclear effect on the structure functions Au+Au: new form of matter !

  29. 3rd milestone: Top Physics Story 2005 http://arxiv.org/abs/nucl-ex/0410003 PHENIX White Paper: second most cited in nucl-ex during 2006

  30. 4th Milestone: A fluid of quarks v2 for the D follows that of other mesons v2 for the φ follows that of other mesons Strange and even charm quarks participate in the flow

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