2009-4-29 1

Elliptic flow and incomplete equilibration in AMPT

Jian-Li Liu

Harbin Institute of Technology

2009-4-29 2

Outline

Eccentricity scaled v2 and incomplete

equilibration

Variation of v2/εwith cross section

Variation of v2/εwith centrality

Summary

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Eccentricity scaled v2 and incomplete equilibration

Heiselberg & Levy, PRC59, 2716(1999)Collisionless limit:2 2

22 2

rel trv dNv y x

Sdy y x

σε

ε< > −< >

∝ =< > + < >

2 .v

constε

∝

Ideal hydrodynamic limit:

From collisionless limit to hydrodynamic limit:

. 12 2

1 10

hydrov v K

K Kε ε

−

− −=+

Bhalerao et al., PLB627, 49(2005)

work well in 2-D transport model. Gombeaud & Ollitrault, PRC77, 054904(2008)

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Knudsen coefficient

2 2

2 2

1

1 1 1

1:

14

: .ss

s

RK

R x y

dNS x y

Sdy

isotropi

Rc

c

coc

dNK

n

d

s

cS y

t

λ

λ σρσ

ρ πτ

τ

σ−

= = +

=

= = < >< >

=

⇒ =

%%

%Time scale of formation of elliptic flow:

Only longitudinal expansion: (dN/dy is constant)

Bhalerao et al., PLB627, 49(2005)

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AMPT model data

Differences:1. cs is not constant from parton to hadron, BUT cs is approximately constant in parton stage2. dN/dy is not constant.3. 3-D expansion4. Non-isotropic differential cross section.

1/ 3sc ≈

Molnar & Gyulassy, NPA697, 459(2002)

Lin et al., PRC72, 064901(2005)

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Relation between isotropic cross section and anisotropic cross section

For isotropic cross section:

0

3

2 trσ σ=

AMPT model:

μis turned to fixed total cross section.Transport cross section is related to μ and s.

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Variation of v2/εwith cross section for quark

Fitting parameters ：

Gombeaud et al. 2-D transport model (2008):

Ideal hydrodynamics ：

Initial dN/dy ：

final s ：

initial dN/dy ， final s ：

K0 is sensitive to parameters used

27%

19%

σ=3,6,10,14mb

Issah et al., arXiv:nucl-ex/0604011

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HIJINGhadroninitialpartonfinal

parton

hadron(c)

hadron(f)

Fitting parameters ：

15% 11%

ε ， S is calculated for hadron from HIJING

K0 （ c ） is much larger than

K0 for quark:

1. Multiplicity difference

2. Variation of v2 (c)/v2 (quark) from 1.27 to 1.1 for cross section from 3mb to 14mb.

27% 19%

Variation of v2/εwith cross section for hadron

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20%

14%

Fitting parameters ：HIJINGhadroninitialpartonfinal

parton

hadron(c)

hadron(f)

15%

11%

Keeping dN/dy unchanged and replace

V2(f) with V2 (c):

Deviation of elliptic flow from its

hydrodynamic limit is almost the same

as hadron(c).

Keeping elliptic flow unchanged and

Replace dN(f)/dy with dN(c)/dy:Deviation of elliptic flow from its hydrodynamic limit is almost unchanged.

ε ， S is calculated for hadron from HIJING

Variation of v2/εwith cross section for hadron

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Variation of v2/εwith centrality for quark

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Changing the calculation of Knudsen coefficient

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Changing the calculation of eccentricity

Calculate eccentricity for quarks in all rapidity range.

consistent with ideal hydrodynamic result

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Only changing the calculation of eccentricity

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1. Knudsen coefficient at initial stage ：

0τ τ=

Possible reasons for changed calculation of Knudsen coefficient

2. “Effective” calculation of Knudsen coefficient defined by Bhalerao et al.: The transverse expansion of system maybe important and is related to the size of system.

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Original calculation ofKnudsen coefficient

New calculation ofKnudsen coefficient

Dependence on cross section ：1. Relative distance between quark is related to cross section.2. Quarks are coalesced according to their relative cross section

Variation of v2/εwith centrality for hadron

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Original calculation ofKnudsen coefficient

New calculation ofKnudsen coefficient

Dependence on cross section.

Variation of v2/εwith centrality for hadron

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Summary

v2 /εvariation with cross section for fixed parameter in AMPT modelcould be described well by the formula suggested Bhalerao et al. Thedeviation of v2 /εof quark from its hydrodynamics limit is 19% ～ 27%for cross section from 6mb to 10mb.

v2 /εvariation with centrality for different cross section and collisionenergy in AMPT model could not be described by the formula suggested by Bhalerao et al, except the calculation of knudsencoefficient is changed.

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