Transcript "Shear and Bulk Viscosities of Hot Dense Matter"

Shear and Bulk
Viscosities of Hot Dense
Matter
Joe Kapusta
University of Minnesota
New Results from LHC and RHIC, INT, 25 May 2010
Is the matter created at RHIC a
perfect fluid ?
Physics Today, May 2010
Atomic and Molecular Systems

1
In classical transport theory
and
s
n
so that as the density and/or cross section is reduced
(dilute gas limit) the ratio gets larger.
~ Tlfree v
lfree ~
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
L. Csernai, L. McLerran and J. K.
Nitrogen
NIST data
L. Csernai, L. McLerran and J. K.
H 2O
NIST data
L. Csernai, L. McLerran and J. K.
2D Yukawa Systems
in the Liquid State
Minimum located at
Q2

 Coulomb coupling parameter  17
aT
1
a2 
 Wigner - Seitz radius
n
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.

15 f4

s 16 T 4
• Quark-gluon plasma at high T (Arnold, Moore,
Yaffe): grows with increasing T.

s

5.12
g 4 ln( 2.42 / g )

 T 
 T
1
9
4
  2 ln  2 ln 
 2 ln 
2

g (T ) 8

9

 T
 T


 


T  30 MeV
QCD
Low T (Prakash et al.)
using experimental
data for 2-body
interactions.
High T (Yaffe et al.)
using perturbative
QCD.
L. Csernai, L. McLerran and J. K.
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.
QCD
• Chiral perturbation theory at low T
(Chen, Wang): grows with increasing T.
4
9   p 1   p 3  T
 2  ln
  ln
  4
s 8  T 4  T 8  f

• Quark-gluon plasma at high T (Arnold, Dogan,
Moore, ): decreases with increasing T.

g4

s 5000 ln( 6.34 / g )

 T 
 T
1
9
4
  2 ln  2 ln 
 2 ln 
2

g (T ) 8

9

 T
 T


 


T  30 MeV
QCD
Low T (Chen & Wang)
using chiral
perturbation theory.
High T (Arnold et al.)
using perturbative
QCD.
ς/s rises dramatically as Tc is approached from above (Karsch, Kharzeev, Tuchin)
Lattice w/o quarks (Meyer) → 0.008 at T/Tc=1.65 and 0.065 at T/Tc=1.24
QCD
Low T (Prakash et al.)
using experimental
data for 2-body
interactions.
High T (Arnold et al.)
using perturbative
QCD.
ς/s rises dramatically as Tc is approached from above (Karsch, Kharzeev, Tuchin)
Lattice w/o quarks (Meyer) → 0.008 at T/Tc=1.65 and 0.065 at T/Tc=1.24
Quasi-Particle Theory of Shear and
Bulk Viscosity of Hadronic Matter
•
•
•
•
•
Relativistic
Allows for an arbitrary number of hadron species
Allows for arbitrary elastic and inelastic collisions
Respects detailed balance
Allows for temperature-dependent mean fields
and quasi-particle masses
• The viscosities and equation of state are
consistent in the sense that the same
interactions are used to compute them.
P. Chakraborty & J. K.
Linear Sigma Model
Calculated in the self-consistent Phi-derivable approximation
= summation of daisy + superdaisy diagrams
= mean field plus fluctuations
m  600 MeV
P. Chakraborty & J. K.
m  900 MeV
Go beyond the mean field approximation by averaging
over the thermal fluctuations of the quasi-particles as
indicated by the angular brackets.
 v
mean field
1 2 2 1 2 2
(T )      U  m   m 
2
2
2
2

U

U
U
2
m2 
m

0

2
2


v
3
2


d
p
p
1
2

 2

2
m
(2 )3 E e E / T  1
2

d 3 p p2
1
2
 3
2
m
(2 )3 E e E / T  1
fluctuation
Linear Sigma Model
m  600 MeV
P. Chakraborty & J. K.
m  900 MeV
Linear Sigma Model
Solution to the integral equation:
P. Chakraborty & J. K.
     
     
  
Linear Sigma Model
Relaxation time approximation
1

15T
P. Chakraborty & J. K.
d 3 p p4
eq

(
E
)
f
a  (2 )3 E 2 a a a ( Ea / T )
a
Linear Sigma Model
Increasing the vacuum sigma mass causes the crossover
transition to look more like a second order transition.
P. Chakraborty & J. K.
Linear Sigma Model
2
 1 2 2 2 2

1
d p  a ( Ea ) eq
2 d ma 

  
f a ( Ea / T ) 3  vs p  vs  ma  T
2 
T a (2 )3 Ea2
d
T



3
Violation of conformality
P. Chakraborty & J. K.
2
Linear Sigma Model
2
 1 2 2 2 2

1
d p  a ( Ea ) eq
2 d ma 

  
f a ( Ea / T ) 3  vs p  vs  ma  T
2 
T a (2 )3 Ea2
d
T



3
Violation of conformality
P. Chakraborty & J. K.
2
Both /s and
Romatschkes 2007
Both η/s and ζ/s depend on T – they are not constant.
Beam energy scans at RHIC and LHC are necessary
to infer their temperature dependence.
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 and LHC are thermometers (hadron
ratios, photon and lepton pair production)
• RHIC and LHC are barometers (elliptic
flow, transverse flow)
• RHIC and LHC are viscometers
(deviations from ideal fluid flow)
• There is plenty of work for theorists and
experimentalists!