an exact hydrodynamic solution for the ...an exact hydrodynamic solution for the elliptic flow...

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


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


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)


5

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 =


6

Hydrodynamic Potential

( ) ( ) 012 21=∂−∂ TuTu xx

),( ϕχ l


7

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 +≡⊥


→

8

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


9

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


� (1+1)

[Khalatnikov, 1954]

� Transverse (cosmology)

[Khalatnikov-Kamenshchik, 2004]

10

Elliptic Flow

� Cause Effect (Initial Spatial Anisotropy) (Entropy Anisotropy)

⇒


11

Elliptic Flow


Eccentricity Elliptic Flow coefficient

⇒

2

1

2

2

2

1

2

2

xx

xx

+

−=ε

∫

∫=

ϕϕ

ϕϕϕ

d

dSd

d

dSd

v

)2cos(

2


12

Elliptic Flow


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

13

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


14

Centrality dependence (‘Callibration’)

� For fixed ⇒T

[E.N.S, R.Peschanski ‘09] [Hirano, et al ‘06]

maxmaxmax

11N

Nc

b

b−∝−∝∝

ρρ


15

Initial conditions

� Source at a given temperature

� (fixes )

� (fixes )

IT

0)(2 =ITv

max)( =ITε

λ

ρ


16

Temperature dependence


17

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

18

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

19

Centrality dependence

� For fixed ,T

maxmax

11N

Nc −∝−∝

ρρ

cv final −∝∝⇒ 12 ρ

[Hirano, et al ‘06] E.N.Saridakis - Ischia, Sept 2009

20

Assumptions check

� 1) Transversally isentropic:( )( )

10

>>∂

∂= ⊥⊥

su

suR

x

τ

Transverse over time

typical

entropy gradient


21

Assumptions check

� 2) Quasi-stationarity: 1)(

)(<<

∂

∂= ⊥

T

TxV

T

T

τ

Transverse over time, temperature-dependent

expansion rate


22

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


Transverse over time, temperature-dependent

expansion rate

23

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


24

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


an exact hydrodynamic solution for the ...an exact hydrodynamic solution for the elliptic flow...

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