Role of Viscosity in Relativistic Nuclear Collisions Joe Kapusta * University of Minnesota Montreal, 2007 * Collaborators: Laszlo Csernai, Larry McLerran...

What has RHIC told us about the equation of state?

How does RHIC connect to other fields like cosmology and condensed matter physics?

Big Experimental Motivation! PHENIX data + Huovinen et al. PHENIX: First Three Years of Operation of RHIC 2-body scattering insufficient to generate v 2 unless parton-parton cross section is 45 mb! (Molnar, Gyulassy)

Big Theoretical Motivation! Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics Kovtun, Son, Starinets PRL 94, (2005) Using the Kubo formula the low energy absorption cross section for gravitons on black holes, and the black hole entropy formula they found that and conjectured that this is a universal lower bound.

Is the RHIC data, in the form of elliptic and radial flow, telling us that the matter has very small viscosity, a perfect fluid ?

Atomic and Molecular Systems In classical transport theoryand so that as the density and/or cross section is reduced (dilute gas limit) the ratio gets larger. In a liquid the particles are strongly correlated. Momentum transport can be thought of as being carried by voids instead of by particles (Enskog) and the ratio gets larger.

Helium NIST data

Nitrogen NIST data

2D Yukawa Systems in the Liquid State Applications to dusty-plasmas and many other 2D condensed matter systems. Liu & Goree (2005)

QCD Chiral perturbation theory at low T (Prakash et al.): grows with decreasing T. Quark-gluon plasma at high T (Arnold, Moore, Yaffe): grows with increasing T.

QCD Low T (Prakash et al.) using experimental data for 2-body interactions. High T (Yaffe et al.) using perturbative QCD. η/s≈1/2 just above T c from lattice (Nakamura, Sakai) and classical quasiparticle model (Gelman, Shuryak, Zahed)

Large N c Limit at Low T Baryon masses are proportional to N c and can be neglected, meson masses are essentially independent of N c. Hagedorn temperature and critical temperature should not change by much. Meson-meson cross sections scale as 1/ N c 2, therefore η/s should scale as N c 2 in the hadronic phase. From Yaffe et al. η/s = A/[(g 2 N c ) 2 ln(Bg 2 N c )] with A and B known constants, therefore η/s has a finite limit as N c becomes large in the plasma phase. Implication: There is a jump in η/s of order N c 2 in going from the low to the high temperature phases.

Relativistic Dissipative Fluid Dynamics In the Eckart approach u is the velocity of baryon number flow.

Relativistic Dissipative Fluid Dynamics In the Landau-Lifshitz approach u is the velocity of energy transport.

How is this relevant for RHIC? For baryon-free matter: transverse waves sound waves Momentum diffusion constants: Bulk viscosity is generally small unless internal degrees of freedom (rotation, vibration) can easily be excited in collisions.

Viscous Heating of Expanding Fireballs JK, PRC 24, 2545 (1981) Viscosity smoothes out gradients in temperature, velocity, pressure, etc.

Shear vs. Bulk Viscosity Shear viscosity is relevant for change in shape at constant volume. Bulk viscosity is relevant for change in volume at constant shape. Bulk viscosity is zero for point particles and for a radiation equation of state. It is generally small unless internal degrees of freedom (rotation, vibration) can easily be excited in collisions. But this is exactly the case for a resonance gas – expect bulk viscosity to be large near the critical temperature!

Lennard-Jones potential Meier, Laesecke, Kabelac J. Chem. Phys. (2005) Pressure fluctuations give peak in bulk viscosity.

Why is the entropy per baryon of the universe as large as 10 9 ? Is it due to viscosity? Weinberg (1971) If the photon mean free time is much bigger than the mean free time for material particles then Shear viscosity and heat conductivity play no role in a Robertson-Walker model, only bulk viscosity.

Relativistic Thermal Nucleation Rate Probability per unit time per unit volume to nucleate a bubble of critical size in a fluid (or a droplet of critical size in a vapor) is proportional to a linear combination of dissipative coefficients because, for the fluctuation to grow, latent heat must be transported away from the interface. Csernai and Kapusta, extended to include heat conduction by Venugopalan and Vischer; reproduces famous Langer and Turski result in nonrelativistic limit and ignoring viscosities.

Suppose the bulk viscosity increases with decreasing temperature. Should be small compared to 1 Wins at large time

Suppose the bulk viscosity diverges at a critical temperature. Takes infinite time to reach critical temperature: Critical Slowing Down

Extracting η/s from RHIC data Elliptic flow (Teaney,…) HBT (Teaney,…) Momentum spectra (Teaney, Baier & Romatschke,…) Momentum fluctuations (Gavin & Abdel-Aziz,…) Photon & dilepton spectra Jet quenching

Conclusion Hadron/quark-gluon matter should have a minimum in shear viscosity and a maximum in bulk viscosity at or near the critical or crossover point in the phase diagram analogous to atomic and molecular systems. Sufficiently detailed calculations and experiments ought to allow us to infer the viscosity/entropy ratios. This are interesting dimensionless measures of dissipation relative to disorder.

Conclusion RHIC is a thermometer (hadron ratios, photon and lepton pair production) RHIC is a barometer (elliptic flow, transverse flow) RHIC may be a viscometer (deviations from ideal fluid flow) There is plenty of work for theorists (and experimentalists)!