Flow Results and Hydrodynamic Limit

Aihong Tang for the STAR Collaboration

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

1The story rewindedΓs =4
η(e + p)3Y. Lu et al. Journal of Phys. G 32 1121
(2006)
D. Teaney, PRC 68 034913 (2003)

Viscosity reduces v2
Viscosity needs to be small
in order to explain data
STAR, PRC 66 034904(2002)

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

v2/ε approaches the limit
of ideal hydrodynamics
Hadronic interaction
alone does not produce
enough v22

The story continues
H. Song and U. Heinz, PRC 77 064901 (2008)

T. Hirano, RHIC & AGS Users Mtg 06
M. Luzum and P. Romatschke arXiv0804.4015

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 20083

Are we saturated already ?

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

© Aihong Tang
4

Are we still far away from there ?

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

© Aihong Tang
5

How to view the hydro behavior better ? - Move away from it- Ideal fluid and low viscosity ⇔local equilibrium (small λ or largeσ)- To study the local equilibrium,
we have to move away from it,say, check what if we relax theconstraint of local equilibrium- How to get a complete view?
Study Boltzman equation for
diluted system. It recovers Hydro
when λ becomes small.

“To have a complete view of Lu Mountain,
one has to move away from it.”- Shi Su (1037~1101)
Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

6

Transport Theory and Hydrodynamics

Transport Theory

Hydrodynamics

Microscopic
Applicable out of equilibrium
Cannot describe phase transition
D<<1MacroscopicLocal equilibrium
Can treat phase transition
K<<1

D (Dilution parameter) =K (Knudsen number) =Typical distance between two particles
Mean free path

Mean free path
System size

Boltzmann Equation will be reduced toHydrodynamics when both D<<1 and K<<1
Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

7

Connecting Pieces

n −1/ 3D≡
= σ n2 / 3
λn: particle density
σ: parton cross section
R: system size
λ: mean free path

1 R≡
K λ
1
λ=
σn
1 1 dNn=ct S dyt  R / cs

Number of collisions.

11 dN cs=σK
S dy c1/k
Aihong Tang
2nd ATHIC, Tsukuba, Oct. 20088

v2 from Solving the Boltzmann Equation
Hydro limit is recovered
when D<<1 and K <<1C. Gombeaud and J-Y Ollitrault, nucl-th/0702075

v2 ∝ 1/K, and v2 saturates
eventually when the systemreaches local equilibrium⇒

v2 v2hydro1=ε
ε 1 + K / K0
K→0 Hydro limit
K>>1 Low density limit,
v2/ε ∝1/S dN/dy

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

9Choose the right {v2, ε} pairsv2 that are sensitive toanisotropy w.r.t. the Reaction
Plane v2:
v2{4}, v2{qDist},v2{qCumulant4}, v2{ZDCSMD}

v2 that are sensitive toanisotropy w.r.t. theParticipant Plane :v2{2},v2{EP},v2{uQ} etc.

ε that are sensitive toanisotropy w.r.t. the Reaction
Plane:
ε{std}, ε{4}ε That are sensitive toanisotropy w.r.t. theParticipant Plane:
ε{part} ε{2}R.Bhalerao and J-Y. Ollitrault, Phys. Lett. B 614 (2006) 260
S.Voloshin, A.Poskanzer,A.Tang and G.Wang, Phys. Lett. B 659 (2008) 537

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 200810

v2 methods

v2{4}=v2{qDist}
Data points from STAR, PRC 66
034904 (2002)

G. Wang (STAR) QM2005

v2{4}=v2{ZDC-SMD}

Au +Au 200 GeV
S TAlim
R preinary

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 200811

Flow increases

The pt where v2/ε peaks increases with pt - the applicable range
For hydrodynamics extends to large pt in central collisions.
v2/ε for the CGC case sees hints of saturation.
Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

12

How much deviation from ideal hydro ?
v2 v2hydro1=εε 1 + K / K0
K =λ/R1 σ dN=Cs
K S dyFitting function from Drescher, Dumitru, Gombeaud,J.Ollitrault, Phys. Rev. C76, 024905(2007)
CGC ε obtained from A.Adil, H-J Drescher,A.Dumitru,
A.Hayashigaki and Y.Nara, Phys. Rev. C 74 044905 (2006)

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 200813

Hydro limit for ID’d particles

π, k, p data points from Phenix (nucl-ex/0604011)

Fitted limit increases with mass
Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

14

How much deviation from ideal hydro ?

v2 v2hydro=εε1⎛ C 1 dN ⎞1 + ⎜σ s
⎝ c 40S dy ⎟⎠−1

1K0π, k, p data points from Phenix (nucl-ex/0604011)

With the assumption that parton cross section (σ) is same for all ID’d particles,
a universal trend of approaching hydro limit is established.
Anti-correlation found between σ and Cs.
Good constraint on EoS
Aihong Tang
2nd ATHIC, Tsukuba, Oct. 200815

How much deviation from ideal hydro ?

v2 v2hydro=εε1⎛ C 1 dN ⎞1 + ⎜σ s
⎝ c 40S dy ⎟⎠−11K0π, k, p data points from Phenix (nucl-ex/0604011)

With the assumption that parton cross section (σ) is same for all ID’d particles,
a universal trend of approaching hydro limit is established.
Anti-correlation found between σ and Cs.
Good constraint on EoS
Aihong Tang
2nd ATHIC, Tsukuba, Oct. 200816

Constraint on EoS⎛c ⎞σ ⎜ s ⎟ k0 = const.
⎝ c⎠

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 200817

Constraint on η/sηλTTTT
1
η/s ~4xquantumlimit.
= 0.316= 0.316= 0.316= 0.316
i1 dN1sc
cσ n
σ cs 1 dN
cσ needi to be understood.
Discrepancies between different approaches
S dydyR / (cs / c)Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

SR18

A slightly different approach on Knudsen number

For centrality 20-30%, Knudsen
number needs to be greater than
0.5 in order to explain data

Boltzmann calculation :Gombeaud and Ollitrault,
private communication

K=

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 2008

λR

K=0 : Ideal Hydro
K>>1 : Free Streaming

19Summary- v2/ε, if examined with transport motivated formulawith certain assumptions, approaches Hydro limit in asimilar way for different particle species.
- In central AuAu collisions, v2/ε is considerably awayfrom Hydro limit. This conclusion is independent ofPID, initial conditions, and choice of {v2, ε} pairs.
- Knudsen number extracted from fitting v2/ε vs. 1/S
dN/dy, as well as that extracted from v4/v22, stays finite
for central collisions - not as zero as required by idealhydrodynamics.
- Good constraint on EoS obtained.
- For the first time the centrality dependence of η/s ispresented.

Aihong Tang
2nd ATHIC, Tsukuba, Oct. 200820