Viscosity at RHIC
Scott Pratt
& Kerstin Paech
Michigan State University
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Scott Pratt
Michigan State University
OUTLINE
• What is viscosity?
• Sources of viscosity
• Bulk viscosity near Tc
Scott Pratt
Michigan State University
Definition of Viscosity
After boosting and rotating,
T


 (  T )v
xx x


 (  Tyy )vy

 (  Tzz )vz
(  Txx )vx (  Tyy )vy (  Tzz )vz 

0
0
Txx

0
Tyy
0

0
0
Tzz


Txx  P  B  v  2 x vx  (2 / 3)  v
1
 xTxx
t vx  
  Txx
P, B and  are functions of 
Scott Pratt
Michigan State University
Definition of Viscosity
Txx  P  B  v  2 x vx  (2 / 3)  v
1
t vx  
 xTxx
  Txx
Viscosity = change in "pressure" to due expansion
Bulk viscosity = change due to isotropic expansion
Shear viscosity = change due to anisotropic expansion
Scott Pratt
Michigan State University
Sources of Viscosity
1. If ∂zvz >∂xvx ,
pz2  px2
   (4P / 3) collision
Vanishes if mean free path -> 0
Scott Pratt
Michigan State University
Sources of Viscosity
2. If Rinteraction > 0 ,
P
BP
2
Rint
 coll
2
Rint
 coll
Important at early times
Scott Pratt
Michigan State University
v2 from Boltzmann Calculation
Finite-range
effects
dampen v2
Scott Pratt
S. Cheng, S. P., P. Csizmadia, Y. Nara, D. Molnar, M. Gyulassy,
S.E. Vance & B. Zhang, PRC 65, 024901 (2002)
Michigan State University
Sources of Viscosity
3. Longitudinal Fields
Txx  Tyy  
Tzz   
Hyper-shear at very early times
Increases transverse acceleration
Scott Pratt
Michigan State University
Sources of Viscosity
4. Longitudinal Fields,
r
  E / 2, E  Ezˆ
Txx  Tyy  
2
Tzz  
Hyper-shear for < 0.5 fm/c
Increases transverse acceleration
Scott Pratt
Michigan State University
Sources of Viscosity
5. Chemical non-equilibrium
offset from
equilibrium
dN
 (N  N equil ) /  chem
dt
dN equil
 N   chem
dt
d(nequil / s) 2
dP
B
 chem
s
dn 
ds
Using some thermodynamics
Large when T falls or when m rises
Scott Pratt
Michigan State University
Sources of Viscosity
6. Mean fields
 2

2



m
(   equil )  R(t)
2
t
t
Langevin "force"
2
d x
dx
m 2 
 k(x  xequil )  R(t)
dt
dt
k x   x&equil

   2 &equil
m
 can blow up at phase transition!
Scott Pratt
Michigan State University
H



4
K.Paech & A.Dumitru, PLB 623, 200 (2005)
2
2
 f  m / 
2
2
 h  
2 2
q
quarks (T , m
 g )
Example:
Linear Sigma
Model
1st order
when g>3.55
Scott Pratt
Michigan State University
H

4
 4   2  h   quarks (T , m  g )
Example:
Linear Sigma
Model
1st order when
g > 3.5549
Scott Pratt
Michigan State University
Example: Linear
Sigma Model
For g=3.4, Txx -> 0
Scott Pratt
Michigan State University
How might this
affect
dynamics?
P
Txx
r
• "traffic jam"
• flash-like emission
Scott Pratt
Michigan State University
Rando
m
Lesson
s
• Numerous sources of viscosity
– Finite collision time (shear)
– Finite interaction range (shear & bulk)
– Longitudinal fields (shear)
– Chemical non-equilibrium (bulk)
– Non-equilibrium fields (bulk)
• Shear viscosity important at early times
– Affects elliptic flow
• Bulk viscosity important near Tc
– Affects dynamics ??
• Alternatively, effects can be included through
– Explicit chemical evolution
– Explicit evolution of fields
K.Paech & A.Dumitru, PLB 623, 200 (2005)
Scott Pratt
Michigan State University
Support your
local theorist!!
http://www.phy.duke.edu/~muller/RTI_Complete.pdf
Scott Pratt
Michigan State University