Perfect Fluid: Flow measurements are described by ideal hydro

1. with Abdel-Aziz in progress How Can We Measure Viscosity?Sean Gavin Wayne State University Perfect Fluid: Flow measurements are described by ideal hydro Problem: all fluids have viscosity Ask: is viscosity small or flow strong? I. Viscosity and its consequences II. Radial flow fluctuations: Dissipated by shear viscosity III. Contribution to transverse momentum fluctuations

2. Viscosity viscous contribution to stress tensor, flow velocity v << 1 shear viscosity  resists shear flow vx(z) bulk viscosity  resists expansion “hubble” flow important: typically  << 

3. How Ideal is the Perfect Fluid? How can we measure viscosity of parton and hadron fluids? • measure flow through a fixed geometry  elliptic and radial flow Teaney et al.; Muronga et al.; Heinz et al.; Baier et al. • fluid response to external probe  jet phenomena, Mach cone Stocker; Casalderry-Solana, Shuryak & Teaney • attenuation of sound waves  suggest:dissipation of fluctuations

4. z r Radial Flow Fluctuations small local variations in radial flow in each event viscous friction arises as neighboring fluid elements flow past each another shear viscosity drives velocity toward the average damping of radial flow fluctuations  viscosity

5. z Evolution of Fluctuations u momentum current for small fluctuations r u(z,t) ≈ vr  vr  shear stress momentum conservation diffusion equation for momentum current momentum diffusion length shear viscosity  energy density e, pressure P

6. Viscosity Broadens Rapidity Distribution analogous effect in charge diffusion Stephanov & Shuryak; Abdel-Aziz & Gavin; Koide; Sasaki et al.; Wolchin; Teaney & Moore; Bass, Pratt & Danielowicz viscous diffusion + Bjorken flow random walk in rapidityyvs. proper time V = 2s/0 V momentum diffusion lengths (e+P) formation at0 /0

7. fluctuations diffuse through volume global equilibrium – diffusion drivesr  rg Hydrodynamic Momentum Correlations momentum density correlation function equilibrium difference rg rg r g,eqsatisfies diffusion equation Van Kampen, Stochastic Processes in Physics and Chemistry, (Elsevier, 1997); Gardiner, Handbook of Stochastic Methods, (Springer, 2002) width in relative rapidity grows from initial value :

8. Dynamic Fluctuations variance minus thermal contribution multiplicityN meanpt Pruneau, Voloshin & S.G. correlation function:

9. ptFluctuations Energy Independent Au+Au, 5% most central collisions • sources of ptfluctuations: thermalization, flow, jets? • central collisions  thermalized • energy independent bulk quantity  jet contribution small

10. Hydrodynamic Density Correlations hydro: stress-energy tensor T and current j number density n = j0 and momentum density gi = T0i phase space density f(x,p)fluctuates: multiplicity fluctuations probedensity density correlation function Measured: NA49; PHENIX; PHOBOS

11. Hydrodynamic Momentum Correlations pt fluctuations probetransverse momentum density momentum density correlation function observable: measured pt1pt2 plus HIJING R rg and rn are comparable

12. How Much Viscosity? Hirano & Gyulassy • flow data doesn’t require small viscosity • Reynolds number must be large enough for ideal flow radial flow speed and length scales vr, R Abdel-Aziz, S.G - in progress momentum diffusion sticky V diffusion for extreme scenarios: perfect sticky liquidwQGP ~HRG~ 2 fm perfect liquidsQGP ~ (4 Tc)-1,HRG~ 2 fm  (fm)

13. initial perfect sticky  Rapidity Dependence of Momentum Fluctuations momentum correlation function near midrapidity • relative rapidity r 12 • weak dependence on a  12 fluctuationsin rapidity window Abdel-Aziz, S.G - in progress initial~ 0.25 balance function ~ 1 fm q~ 3 fm h~ 9 fm f~ 20 fm

14. Summary: small viscosity or strong flow?

15. Summary: small viscosity or strong flow?

16. Summary: small viscosity or strong flow? viscosity broadens momentum correlation function in rapidity pt fluctuations measure these correlations testing the perfect liquid  viscosity info • diffusion coefficient shear viscosity • compare rapidity width of momentum fluctuations for different projectile sizes and energies • cross-check: combine with other indirect viscosity measures schematic calculation -- lots to do: • Maxwell/Muronga type corrections • O(R-1) corrections • angular correlations

17. Thermalization + Flow M. Abdel-Aziz & S.G. blue-shift: • average increases • enhances equilibrium contribution flow added thermalization s1/2=200 GeV 130 GeV 20 GeV • blue-shift cancels in ratio participants

18. Hydrodynamic Momentum Correlations pt fluctuations probetransverse momentum density density correlations momentum density correlations observable: measured pt1pt2 plus HIJING R rg and rn are comparable

19. Hydrodynamic Density Correlations hydro: stress-energy tensor T and current j number density n = j0 and momentum density gi = T0i phase space density f(x,p)fluctuates: multiplicity fluctuations probedensity density correlation function Measured: NA49; PHENIX; PHOBOS

20. Hydrodynamic Momentum Correlations pt fluctuations probetransverse momentum density momentum density correlation function observable: measured pt1pt2 plus HIJING R rg and rn are comparable