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

.correlated are and But fT T

2-body scattering insufficient to generate v2 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, 111601 (2005)

Using the Kubo formula )0(),(1

lim20

1tracelesstraceless

4

0

ijijti TxTexd

the low energy absorption cross section for gravitons on blackholes, and the black hole entropy formula they found that

4/1/ s 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

vTls free~

nl

1~freeIn classical transport theory and

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. Momentumtransport can be thought of as being carried by voids insteadof by particles (Enskog) and the ratio gets larger.

Helium

NIST data

Nitrogen

NIST data

OH2

NIST data

2D Yukawa Systemsin the Liquid State

radius Seitz-Wigner1

17parameter coupling Coulomb

at located Minimum

2

2

na

aT

Q

Applications to dusty-plasmas andmany other 2D condensed mattersystems.

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.

4

4

16

15

T

f

s

)/42.2ln(

12.54 ggs

TT

TT

Tgln2ln

9

4ln

8

9

)(

1222

MeV 30T

QCDLow T (Prakash et al.)using experimentaldata for 2-bodyinteractions.

High T (Yaffe et al.)using perturbativeQCD.

η/s≈1/2 just above Tc

from lattice (Nakamura, Sakai)and classical quasiparticle model (Gelman, Shuryak, Zahed)

Large Nc Limit at Low T

• Baryon masses are proportional to Nc and can be neglected, meson masses are essentially independent of Nc. Hagedorn temperature and critical temperature should not change by much. Meson-meson cross sections scale as 1/ Nc

2, therefore η/s should scale as Nc

2 in the hadronic phase.

• From Yaffe et al. η/s = A/[(g2 Nc)2 ln(Bg2 Nc)] with A and B known constants, therefore η/s has a finite limit as Nc becomes large in the plasma phase.

• Implication: There is a jump in η/s of order Nc2 in going from the

low to the high temperature phases.

BBB JunJ

TuuwPgT

QuHuHuHuuT 3

2

uTuTQuuguuH ,,

TuT

susJB 1

,0

22

22

32

2 kkk

kk

kiji

jj

i uTTT

uT

uuuT

s

Relativistic Dissipative Fluid Dynamics

In the Eckart approach u is the velocity of baryon number flow.

BBB JunJ

TuuwPgT

uHuuT 32

uTuTQuuguuH ,,

BBBB

B JT

susTw

TnJ

,

2

22

22

32

2 kkk

kk

kiji

jj

i uTTT

uT

uuuT

s

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

02 kiDt

0222 kiDv ls

TsD

TsD lt

3/4Momentum diffusion constants:

Bulk viscosity is generally small unless internal degreesof freedom (rotation, vibration) can easily be excited incollisions.

m

TmT

1

03

2

Viscous Heating of Expanding Fireballs JK, PRC 24, 2545 (1981)

Viscosity smoothesout gradients intemperature, 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 radiationequation of state. It is generally small unless internal degreesof freedom (rotation, vibration) can easily be excited incollisions. 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, KabelacJ. Chem. Phys. (2005)

Pressure fluctuations give peak in bulk viscosity.

3

2224

4

3

4

903

14

timefreemean photon 15

4

aT

avaT

aT

s

Why is the entropy per baryon of the universe as large as 109? 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 rolein a Robertson-Walker model, only bulk viscosity.

TFw

RT

TI /exp

)(

)3/42/(

3

442

*

2/3

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 byVenugopalan and Vischer; reproduces famous Langer and Turskiresult in nonrelativistic limit and ignoring viscosities.

Suppose the bulk viscosity increases with decreasing temperature.

negligible )(,)(,)( 4 TT

TTBATTP

n

ii

i

iii

i

iii

i

n

i

n

i Tsn

n

Tsn

n

T

T

P

1

1

41

1

41

0d

d

onillustratifor expansion Bjorken

3/)4(4

2

Should be small compared to 1

Wins atlarge time

Suppose the bulk viscosity diverges at a critical temperature.

negligible )(,)(,)( 4 TTT

TTTBATTP

n

c

cii

as 1

0d

d

onillustratifor expansion Bjorken

/1/1

2

n

c

n

i

icic TsTTTT

P

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

RHIC.at 5.01.0/at suggest th studiesy Preliminar s

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)!