scott pratt michigan state university viscosity at rhic scott pratt & kerstin paech michigan...

Scott Pratt Scott Pratt Michigan State UniversityMichigan State University

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


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


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


Sources of Viscosity

1. If ∂zvz >∂xvx ,

pz2 > px

2 → η =(4P / 3)τcollision

Vanishes if mean free path -> 0



2. If Rinteraction > 0 ,

η ∝ PRint

2

τ coll

B ∝ PRint

2

τ coll

Important at early times


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)



3. Longitudinal Fields

Txx =Tyy =ε

Tzz =−ε

Hyper-shear at very early timesIncreases transverse acceleration



4. Longitudinal Fields,

ε =E2 / 2,rE = Ez

Txx = Tyy = ε

Tzz = −ε

Hyper-shear for τ< 0.5 fm/cIncreases transverse acceleration



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



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!


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)


Example: Linear Sigma Model

H =λ4σ 4 −βσ 2 +hσ +εquarks(T,m=gσ )

1st order when g > 3.5549


Example: Linear Sigma

Model

For g=3.4, Txx -> 0


How might this affect dynamics?

Txx

r

P

• "traffic jam"• flash-like emission


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)


Support your local theorist!!

http://www.phy.duke.edu/~muller/RTI_Complete.pdf

scott pratt michigan state university viscosity at rhic scott pratt & kerstin paech michigan...

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