AN EXACT HYDRODYNAMIC SOLUTION FOR THE
ELLIPTIC FLOW
Emmanuel N. SaridakisNuclear and Particle Physics Section
Physics Department
University of Athens
E.N.Saridakis - Ischia, Sept 2009
In collaboration with Robi Peschanski
Institut de Physique Théorique, CEA,IPhT
2
Goal
� Understanding the dynamicalmechanisms of elliptic flow and extraction of exact solutions
� Tools:
The formulation of the transverse flow in terms of a hydrodynamic potential
E.N.Saridakis - Ischia, Sept 2009
3
Talk Plan
� 1) Introduction (why hydrodynamics?)
� 2) Assumptions and potential formulation of perfect fluid relativistic hydrodynamics
� 3) General and exact solution for spatial anisotropy and elliptic flow
� 4) Applications: Comparison with experiment and simulations
� 5) Assumptions Check
� 6) Conclusions-Prospects
E.N.Saridakis - Ischia, Sept 2009
4
Introduction
� Evidence that hydrodynamics of nearly perfect fluidmay be relevant for the description of mediumcreated in heavy-ion collisions
� Almost perfect fluid
(low viscosity)
E.N.Saridakis - Ischia, Sept 2009
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Assumptions
� 1) Transversally isentropic:
� 2) Quasi-stationary: Transverse flow smooth enough to be driven only by temperature change. Thus, Bernoulli equation:
( ) ( ) 021 21=∂+∂ susu xx
00 TTu =
E.N.Saridakis - Ischia, Sept 2009
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Hydrodynamic Potential
( ) ( ) 012 21=∂−∂ TuTu xx
),( ϕχ l
E.N.Saridakis - Ischia, Sept 2009
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Hydrodynamic Potential
� Hodograph Transform:(kinematical variables) (thermodyknamical variables)
),( ϕχ l
( ) ( ) 012 21=∂−∂ TuTu xx
),(),( 21 ϕlxx →
+=
−=
⊥
⊥2
22
0 1log
2
11log
2
1
u
u
T
Tl
1
2tanu
u=ϕ
2
2
2
1
2uuu +≡⊥
E.N.Saridakis - Ischia, Sept 2009
→
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Khalatnikov-Kamenshchik equation
( ) 01
111 2
22
22
2
2
2
2
=∂∂
−+
∂∂
−+∂∂
−
le
cle
c
e l
s
l
s
l χχϕχ
� Solvable!Tds
sdT
d
dpcs ==
ε2
E.N.Saridakis - Ischia, Sept 2009
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Potential equation
� Solvable!
( ) 01
111 2
22
22
2
2
2
2
=∂∂
−+
∂∂
−+∂∂
−
le
cle
c
e l
s
l
s
l χχϕχ
Tds
sdT
d
dpcs ==
ε2
∂∂
+
∂∂
=+− 22
2
0
22
2
2
1 ϕχχ
lT
exx
l
∂∂
∂∂
+=−1
1
2 arctanarctanlx
x χϕχ
ϕ
E.N.Saridakis - Ischia, Sept 2009
� (1+1)
[Khalatnikov, 1954]
� Transverse (cosmology)
[Khalatnikov-Kamenshchik, 2004]
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Elliptic Flow
� Cause Effect (Initial Spatial Anisotropy) (Entropy Anisotropy)
⇒
E.N.Saridakis - Ischia, Sept 2009
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Elliptic Flow
� Cause Effect (Initial Spatial Anisotropy) (Entropy Anisotropy)
Eccentricity Elliptic Flow coefficient
⇒
2
1
2
2
2
1
2
2
xx
xx
+
−=ε
∫
∫=
ϕϕ
ϕϕϕ
d
dSd
d
dSd
v
)2cos(
2
E.N.Saridakis - Ischia, Sept 2009
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Elliptic Flow
� Cause Effect (Initial Spatial Anisotropy) (Entropy Anisotropy)
Eccentricity Elliptic Flow coefficient
⇒
2
1
2
2
2
1
2
2
xx
xx
+
−=ε
∫
∫
∂∂
+
∂∂
∂∂
∂∂
+
∂∂
−
∂∂
=22
22
2sin22cos
ld
lld
χϕχ
ϕ
χϕχ
ϕχ
ϕχ
ϕϕ
ε
∫
∫=
ϕϕ
ϕϕϕ
d
dSd
d
dSd
v
)2cos(
2
( )
∂∂
−∂∂
−=
2
2
22
0 1 llceT
sT
d
dS
s
l
χχϕ
[E.N.S, R.Peschanski ‘09] E.N.Saridakis - Ischia, Sept 2009
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General and exact solution
∑∞
=+=
100 )2cos()()(),(p p pllcl ϕββϕχ
( ) 2
11
2
0
2
1)('sc
lel
−
−=β
( ) ( ) ( ) ( )+
++
−+−++
−−−+−= −
−−−
−−+ l
ss
sss
s
plp
pp epcpc
cpcpc
cpFecl222
22222
222
12
21)1(;21,
16
11
4
1,
16
11
4
1)1()(β
( ) ( )
−
+−
+−
+−
−−
+−
−−−
−−
pp
cpcc
cpcc
eGc sss
sss
l
p
22
22222
222
20,2
2,2
)2(
16
1
4
5,
16
1
4
5|
ρ≡0
)1(
1
c
cλ≡
)1(
1
)2(
1
c
c
E.N.Saridakis - Ischia, Sept 2009
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Centrality dependence (‘Callibration’)
� For fixed ⇒T
[E.N.S, R.Peschanski ‘09] [Hirano, et al ‘06]
maxmaxmax
11N
Nc
b
b−∝−∝∝
ρρ
E.N.Saridakis - Ischia, Sept 2009
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Initial conditions
� Source at a given temperature
� (fixes )
� (fixes )
IT
0)(2 =ITv
max)( =ITε
λ
ρ
E.N.Saridakis - Ischia, Sept 2009
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Temperature dependence
E.N.Saridakis - Ischia, Sept 2009
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Time dependence
� Temperature evolution Time evolution⇔
[Kolb, Heinz ‘03]
2
3
2
0 xx +≡τ2
0
0
sc
T
T
=
ττ
[E.N.S, R.Peschanski ‘09] E.N.Saridakis - Ischia, Sept 2009
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Time dependence22
00
−−
−
≡ss
c
I
c
T
T
T
Tθ
[Bhalerao, et al ‘05][E.N.S, R.Peschanski ‘09] E.N.Saridakis - Ischia, Sept 2009
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Centrality dependence
� For fixed ,T
maxmax
11N
Nc −∝−∝
ρρ
cv final −∝∝⇒ 12 ρ
[Hirano, et al ‘06] E.N.Saridakis - Ischia, Sept 2009
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Assumptions check
� 1) Transversally isentropic:( )( )
10
>>∂
∂= ⊥⊥
su
suR
x
τ
Transverse over time
typical
entropy gradient
E.N.Saridakis - Ischia, Sept 2009
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Assumptions check
� 2) Quasi-stationarity: 1)(
)(<<
∂
∂= ⊥
T
TxV
T
T
τ
Transverse over time, temperature-dependent
expansion rate
E.N.Saridakis - Ischia, Sept 2009
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Assumptions check
� 2) Quasi-stationarity: 1)(
)(<<
∂
∂= ⊥
T
TxV
T
T
τ
( )( )
( )( ) ( )
( )0
0
0
1
)(
)( su
su
V
T
Tx
su
su
su
suR
T
T
T
T
T
T
x
∂∂
=
∂
∂∂∂
=∂
∂= ⊥
⊥
⊥
⊥⊥
ττ
The two assumptions are connected
E.N.Saridakis - Ischia, Sept 2009
Transverse over time, temperature-dependent
expansion rate
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Conclusions
� i) Hydrodynamic potential allows to bypass the non-linearities of the initial equations.
� ii) Under transverse isentropicity and quasi-stationarity we obtain the general solution for spatial eccentricity and elliptic flow coefficient.
� iii) Time-evolution is acquired through temperature-evolution.
� iv) Assumptions are verified qualitatively, but not exactly
χ
E.N.Saridakis - Ischia, Sept 2009
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Outlook
� i) Describe the -dependence of elliptic flow.
� ii) Extend to more physical initial conditions(beyond fixed-temperature)
� iii) Extend to weak viscosity.
� iv) The rather simple mechanisms, may facilitate AdS/CFT approach of elliptic flow.
⊥p
E.N.Saridakis - Ischia, Sept 2009