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Viscosity at RHIC
Scott Pratt& Kerstin Paech
Michigan State UniversityQuickTime™ and a
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OUTLINE
• What is viscosity?• Sources of viscosity
• Bulk viscosity near Tc
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Definition of Viscosity
Tαβ =
ε (ε +Txx)vx (ε +Tyy)vy (ε +Tzz)vz
(ε +Txx)vx Txx 0 0(ε +Tyy)vy 0 Tyy 0
(ε +Tzz)vz 0 0 Tzz
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Txx =P −B∇⋅v−2η∂xvx −(2 / 3)η∇⋅v
∂tvx =−1
ε +Txx∂xTxx
After boosting and rotating,
P, B and η are functions of ε
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Definition of Viscosity
Txx =P −B∇⋅v−2η∂xvx −(2 / 3)η∇⋅v
∂tvx =−1
ε +Txx∂xTxx
Viscosity = change in "pressure" to due expansionBulk viscosity = change due to isotropic expansionShear viscosity = change due to anisotropic expansion
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Sources of Viscosity
1. If ∂zvz >∂xvx ,
pz2 > px
2 → η =(4P / 3)τcollision
Vanishes if mean free path -> 0
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Sources of Viscosity
2. If Rinteraction > 0 ,
η ∝ PRint
2
τ coll
B ∝ PRint
2
τ coll
Important at early times
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
v2 from Boltzmann Calculation
Finite-range effects dampen v2
S. Cheng, S. P., P. Csizmadia, Y. Nara, D. Molnar, M. Gyulassy,S.E. Vance & B. Zhang, PRC 65, 024901 (2002)
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Sources of Viscosity
3. Longitudinal Fields
Txx =Tyy =ε
Tzz =−ε
Hyper-shear at very early timesIncreases transverse acceleration
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Sources of Viscosity
4. Longitudinal Fields,
ε =E2 / 2,rE = Ez
Txx = Tyy = ε
Tzz = −ε
Hyper-shear for τ< 0.5 fm/cIncreases transverse acceleration
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Sources of Viscosity
5. Chemical non-equilibrium
dN
dt=−(N −Nequil ) / τchem
δN =τchem
dNequil
dt
B=dPdn ε
τchem
d(nequil / s)ds
s2
Large when T falls or when m rises
offset fromequilibrium
Using some thermodynamics
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Sources of Viscosity
6. Mean fields
∂2σ∂t2
+ Γ∂σ
∂t+m2 (σ − σ equil ) = R(t)
Langevin "force"
md2x
dt2+γ dx
dt+ k(x−xequil ) =R(t)
kδx=−γ &xequil
δσ =−
Γm2
&σ equil
δσ can blow up at phase transition!
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Example: Linear Sigma Model
H =λ2
4σ 2 − fπ
2 + mπ2 / λ2( )
2−hqσ +εquarks(T,m=gσ )
1st order when g>3.55
K.Paech & A.Dumitru, PLB 623, 200 (2005)
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Example: Linear Sigma Model
H =λ4σ 4 −βσ 2 +hσ +εquarks(T,m=gσ )
1st order when g > 3.5549
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Example: Linear Sigma
Model
For g=3.4, Txx -> 0
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
How might this affect dynamics?
Txx
r
P
• "traffic jam"• flash-like emission
Scott Pratt Scott Pratt Michigan State UniversityMichigan State University
Random
Lessons
• 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 Scott Pratt Michigan State UniversityMichigan State University
Support your local theorist!!
http://www.phy.duke.edu/~muller/RTI_Complete.pdf
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