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

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

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 =

E.N.Saridakis - Ischia, Sept 2009

6

Hydrodynamic Potential

( ) ( ) 012 21=∂−∂ TuTu xx

),( ϕχ l

E.N.Saridakis - Ischia, Sept 2009

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 +≡⊥

E.N.Saridakis - Ischia, Sept 2009

→

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

E.N.Saridakis - Ischia, Sept 2009

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

E.N.Saridakis - Ischia, Sept 2009

� (1+1)

[Khalatnikov, 1954]

� Transverse (cosmology)

[Khalatnikov-Kamenshchik, 2004]

10

Elliptic Flow

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

⇒

E.N.Saridakis - Ischia, Sept 2009

11

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

12

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

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

E.N.Saridakis - Ischia, Sept 2009

14

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

15

Initial conditions

� Source at a given temperature

� (fixes )

� (fixes )

IT

0)(2 =ITv

max)( =ITε

λ

ρ

E.N.Saridakis - Ischia, Sept 2009

16

Temperature dependence

E.N.Saridakis - Ischia, Sept 2009

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

E.N.Saridakis - Ischia, Sept 2009

21

Assumptions check

� 2) Quasi-stationarity: 1)(

)(<<

∂

∂= ⊥

T

TxV

T

T

τ

Transverse over time, temperature-dependent

expansion rate

E.N.Saridakis - Ischia, Sept 2009

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

E.N.Saridakis - Ischia, Sept 2009

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

E.N.Saridakis - Ischia, Sept 2009

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

E.N.Saridakis - Ischia, Sept 2009