Elliptic flow and incomplete equilibration
in AMPT
Jian-Li Liu
Harbin Institute of Technology
2009-4-29
1
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
Collisionless limit:
Heiselberg & Levy, PRC59, 2716(1999)
vrel tr dN


Sdy
 y 2    x2 

 y 2    x2 
v2
Ideal hydrodynamic limit:
v2

 const.
From collisionless limit to hydrodynamic limit:
Bhalerao et al., PLB627, 49(2005)
v2


hydro.
2
v

work well in 2-D transport model.
2009-4-29
1
K
K 1  K 01
Gombeaud & Ollitrault, PRC77, 054904(2008)
3
Knudsen coefficient
R
K

Only longitudinal expansion:
(dN/dy is constant)
Time scale of formation
of elliptic flow:

1
1 dN

 Sdy
R

cs
1
1
1
 2
2
x
y
 : isotropic

K
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
R
S  4  x 2  y 2 
cs : const.

 dN
S dy
cs
Bhalerao et al., PLB627, 49(2005)
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AMPT model data
Lin et al., PRC72, 064901(2005)
Differences:
1. cs is not constant from parton to hadron, BUT cs is
approximately constant in parton stage cs  1/ 3
2. dN/dy is not constant.
3. 3-D expansion
4. Non-isotropic differential cross section.
Molnar & Gyulassy, NPA697, 459(2002)
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Relation between isotropic cross section
and anisotropic cross section
For isotropic cross section:
3
 0   tr
2
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
σ=3,6,10,14mb
Fitting parameters：
19%
Gombeaud et al. 2-D transport
model (2008):
27%
Ideal hydrodynamics：
Issah et al., arXiv:nucl-ex/0604011
Initial dN/dy：
final s ：
initial dN/dy，final s ：
K0 is sensitive to parameters used
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Variation of v2/εwith cross section for hadron
ε，S is calculated for hadron from
HIJING
Fitting parameters：
K0（c）is much larger than
K0 for quark:
1. Multiplicity difference
HIJING
hadron
initial
parton
final
parton
hadron(c)
hadron(f)
19%
27%
11%
15%
2. Variation of v2 (c)/v2 (quark) from
1.27 to 1.1 for cross section from
3mb to 14mb.
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Variation of v2/εwith cross section for hadron
ε，S is calculated for hadron from
HIJING
HIJING
hadron
Fitting parameters：
initial
parton
final
Keeping dN/dy unchanged and replace parton
hadron(c)
V (f) with V (c):
2
2
Deviation of elliptic flow from its
hadron(f)
hydrodynamic limit is almost the same
as hadron(c).
11% 14%
15% 20%
Keeping elliptic flow unchanged and
Replace dN(f)/dy with dN(c)/dy:
Deviation of elliptic flow from its
hydrodynamic limit is almost unchanged.
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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
consistent with ideal
hydrodynamic result
Calculate eccentricity for quarks in all rapidity range.
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Only changing the calculation of eccentricity
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Possible reasons for changed calculation
of Knudsen coefficient
1. Knudsen coefficient at initial stage：
  0
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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Variation of v2/εwith centrality for hadron
Original calculation of
Knudsen coefficient
New calculation of
Knudsen 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
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Variation of v2/εwith centrality for hadron
Original calculation of
Knudsen coefficient
New calculation of
Knudsen coefficient
Dependence on cross section.
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Summary
v2 /εvariation with cross section for fixed parameter in AMPT model
could be described well by the formula suggested Bhalerao et al. The
deviation 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 collision
energy in AMPT model could not be described by the formula
suggested by Bhalerao et al, except the calculation of knudsen
coefficient is changed.
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