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Transport Theory for the
Quark-Gluon Plasma
V. Greco
UNIVERSITY of CATANIA
INFN-LNS
Quark-Gluon Plasma and Heavy-Ion Collisions – Turin (Italy), 7-12 March 2011
All the observables are in a way or the other
related with the evolution of the phase space
density :
z
y
dN
f ( x, p , t )  3 3
d xd p
x
Hydrodynamics
No microscopic descriptions
(mean free path  -> 0, h=0)
implying f=feq
What happens if we drop such assumptions?
There is a more “general” transport theory
valid also in non-equilibrium?


 T ( x)  0 + EoS P(e) Is there any motivation to look for it?



j

  B ( x)  0
Picking-up four main results at RHIC
 Nearly Perfect Fluid, Large Collective Flows:
 Hydrodynamics good describes dN/dpT + v2(pT) with mass ordering
 BUT VISCOSITY EFFECTS SIGNIFICANT (finite  and f ≠feq)
 High Opacity, Strong Jet-quenching:
 RAA(pT) <<1 flat in pT - Angular correlation triggered by jets pt<4 GeV
 STRONG BULK-JET TALK: Hydro+Jet model non applicable at pt<8-10 GeV
 Hadronization modified, Coalescence:
 B/M anomalous ratio + v2(pT) quark number scaling (QNS)
 MICROSCOPIC MECHANISM RELEVANT
 Heavy quarks strongly interacting:
 small RAA large v2 (hard to get both) pQCD fails: large scattering rates
 NO FULL THERMALIZATION ->Transport Regime
Initial Conditions
Quark-Gluon Plasma
BULK
(pT~T)
CGC (x<<1)
Gluon saturation
MINIJETS
(pT>>T,LQCD)
Hadronization
Microscopic
Mechanism
Matters!
Heavy Quarks
(mq>>T,LQCD)
 pT>> T , intermediate pT
 m >> T , heavy quarks
 h/s >>0 , high viscosity
 Initial time studies of thermalizations
 Microscopic mechanism for Hadronization can modify QGP observable
 Non-equilibrium + microscopic scale are relevant in all the subfields
Plan for the Lectures
 Classical and Quantum Transport Theory
- Relation to Hydrodynamics and dissipative effects
- density matrix and Wigner Function
 Relativistic Quantum Transport Theory
- Derivation for NJL dynamics
- Application to HIC at RHIC and LHC
 Transport Theory for Heavy Quarks
- Specific features of Heavy Quarks
- Fokker-Planck Equation
- Application to c,b dynamics
Classical Transport Theory
For a classical relativistic system of N particles
NN
 
f (rx, p,)t) 
 4 (3x(ix(i ()t )xx) ) 4 3((ppi (i(t)) pp))
i 1i 1
dP
dP
dd43xxdd43 pp
f(x,p) is a Lorentz scalar &
P0=(p2+m2)1/2
Gives the probability to find a particle in phase-space
If one is interested to the collective behavior or to the behavior of a typical particle
knowledge of f(x,p) is equivalent to the full solution … to study the correlations
among particles one should go to f(x1,x2,p1,p2) and so on…
Liouville Theorem:
if there are only conservative forces -> phase-space density is a constant o motion
df ( x, p )  dx  
dp   
0


f ( x, p )

 
d
d p 
 d x
 p 
 


 F ( x )   f ( x, p )

p 
 m x
Force
Relativistic Vlasov Equation
 p 
 
1



F
(
x
)
f
(
x
,
p
)

0

p



F
(
x
)




x
p  f ( x, p)


p 
m

 m x
The non-relativistic reduction
 
f  
 v  r f  F   p f  0
t
Liouville -> Vlasov -> No dissipation
+
Collision= Boltzmann-Vlasov
Dissipation
Entropy production
Allowing for scatterings
particles go in and out phase space
(d/dt) f(x,p)≠0
Collision term
1

p



F
(
x
)



x
p  f ( x, p )  C[ f ]( x, p )
m

The Collision Term
It can be derived formally from the reduction of the 2-body distribution
Function in the N-body BBGKY hierarchy.
The usual assumption in the most simple and used case:
1) Only two-body collisions
2) f(x1,x2,p1.p2)=f(x1,p1) f(x2,p2)
The collision term describe the change in f (x,p) because:
a) particle of momentum p scatter with p2 populating the phase space in (p’1,p’2)
Closs   d 3 p2 d 3 p1d 3 p2 w( p1 , p2 , p1, p2 ) f [ 2] ( x, p1 , p2 )
Sum over all the
Probability to make
momenta the kick-out
the transition
The particle in (x,p)
probability finding
2 particles in p e p2
and space x
w(12  1 2)d 3 p1d 3 p2  v12 d Collision Rate
C gain   d 3 p2 d 3 p1d 3 p2 w( p1, p2 , p1 , p2 ) f [ 2] ( x, p1, p2 )
In a more explicit form and covariant version:
gain
loss
At equilibrium in each phase-space region Cgain =Closs
C [ f 0 ( x, p)]  0
When one is close to equilibrium or when the mfp is very small
One can linearize the collision integral in f=f-f0 <<f
C[ f ]  
f  f0


What is the f0(x,p)=0?
1
 v


v
Relaxation time
time between 2 collisions
Local Equilibrium Solution
C[ f ]   d 3 p2 d 3 p1d 3 p2 w( p1 , p2 , p1, p2 )  f ( x, p1 ) f ( x, p2 )  f ( x, p1 ) f ( x, p2 )
The necessary and sufficient condition to have C[f]=0 is
f ( x, p1 ) f ( x, p2 )  f ( x, p1) f ( x, p2 )
Noticing that p1+p2=p’1+p’2 such a condition is satisfied by the relativistic
extension of the Boltzmann distribution:
f juttner ( p)  exp   ( p  u )   
=1/T temperature
u collective four velocity
 chemical potential
It is an equilibrium solution also with LOCAL VALUES of T(x), u(x), m(x)
The Vlasov part gives the constraint and the relation among T,u, locally
Main points:
• Boltzmann-Vlasov equation gives the right equilibrium distributions
• Close to equilibrium there can be many collisions with vanishing net effect
Relation to Hydrodynamics
General definitions
Ideal Hydro
j    d 4 p p  f ( x, p )


 T ( x)  0



  j ( x)  0
T    d 4 p p  p f ( x, p )
Inserting
Vlasov Eq.
Notice in Hydro appear only
p-integrated quantities
f  f0 
m 4


  j   d p p  f   d p  mF ( x) p f  m
   d p( f  f 0 )

 



4

4
Integral of a divergency
We can see that ideal Hydro can be satisfied only if f=feq , on the other hand
the underlying hypothesis of Hydro is that the mean free path is so small that
the f(x,p)is always at equilibrium during the evolution.
Similarly ∂T , for f≠feq and one can do the expansion in terms of transport
coefficients: shear and bulk viscosity , heat conductivity [P. Romatschke]
At the same time f≠feq is associated to the entropy production ->
Entropy Production <-> Thermal Equilibrium
S  ( x)   d 4 p p  f ( x, p)1  ln f ( x, p)
f
  S    d p p    f (1  ln f )    d p ( p   f )ln f  ...   d p( f  f 0 ) ln

f0
Boltzmann-Vlasov Eq.
( x  1) ln x  0 , x  0

4

x
4

x
m
4
Approach to thermal equlibrium is always associated to entropy production
All these results are always valid and do not rely
on the relaxation time approx. more generally:
s  
  S     s    d p C  ff 2  f1' f 2 ' ln f
t

DS=0
<->
C[f]=0
Collision integral is associated to entropy production but if a local equilibrium
is reached there are many collisions without dissipations!
Does such an approach can make sense for a quantum system?
One can account also for the quantum effect of Pauli-Blocking
in the collision integral
f1f 2  f1 f 2  f1f 2(1  f1 )(1  f 2 )  f1 f 2 (1  f1)(1  f 2)
does not allow scattering if the final momenta have occupation number =1
-> Boltzmann-Nordheim Collision integral
This can appear quite simplistic, but notice that C[f]=0 now is
1
f FD ( p) 
1  exp   ( p  u )   
So one gets the correct quantum equilbrium distribution, but what is
F(x,p) for a quantum system?
Quantum Transport Theory
In quantum theory the evolution of a system can be described in terms of
the density matrix operator:
 ˆ
̂   wi i i and any expectation value
O(t )  Tr  (t )O
can
be
calculated
as
i


For any operator one can define the Weyl transform of any operator:
dy ipy
~
A( x, p)  
e  x Aˆ x
2
x  x  y / 2
~
~
ˆ
ˆ


Tr AB   A( x, p) B ( x, p) dxdy
which has the property
(*)
The Weyl transform of the density operator is called Wigner function
ipy

dy  ipy
dy
f W ( x, p )  
e  x ˆ x  
e    ( x ) ( x )
2
2
and by (*)
 
~
ˆ
O  Tr ̂ O   fW ( x, p)O ( x, p) dxdp
fW plays in many respects
the same role of the distribution
function in statistical mechanics
Properties of the Wigner Function
 
~
O  Tr ̂ Oˆ   fW ( x, p)O ( x, p) dxdp

dp
f
(
x
,
p
)


( x) ( x)   ( x)
 W

dx
f
(
x
,
p
)


( p) ( p)   ( p)
 W
However for pure state fW can be negative so it cannot be a probability
On the other hand if we interpret its absolute value as a probabilty it does
not violate the uncertainty principle because one can show:
fW ( x , p ) 
1

1 dN
1

2 dxdp  
 DN 
1
2 DxDp  1

So if we go in a phase space smaller than
DxDp<h/2 one can never locate a particle
In agreement with the uncertainty principle
Quantum Transport Equation
One can Wigner transform this or the Schr. Equation

ˆ
 ˆ , Hˆ
t



ˆ
x
 ˆ , Hˆ x  0
t
dy i py
ˆ  pˆ 2 ˆ 
 2 e x t  ˆ , 2m  U  x  0
After some calculations one gets the following equation
 2  3 2 k 1
fW p fW
1



 xU ( x) p fW ( xU, (px))( x  px)U ( xf)w( x3p, fpW)(x,0 p)  ...  0
t m x k 0 (2k  1)!  2 
12
2k
This exactly equivalent to the Equation for the denity matrix or the Schr. Eq.
NO APPROXIMATION but allows an approximation where h does not appear
explicitly and still accounting for quantum evolution when the gradient
of the potential are not too strong :

This has the same form of the classical

fW p fW 

   xU   p f w  0 transport equation, but it is for example
t m x
exact for an harmonic potential
See : W.B. Case, Am. J. Phys. 76 (2008) 937
Transport Theory in Field Theory
One can extend the Wigner function (4x4 matrix):
d 4 y i py
F ( x, p)  
e :  ( x ) ( x ) :
4
(2)
, x  x  y / 2
It can be decomposed in 16 indipendent components (Clifford Algebra)
1
F  FS    FV   5 FP  i   5 F5    FT
2
For example the vector current
j       d 4 p Tr   F ( x, p)  4 d 4 p FV ( x, p)
In a similar way to what done in Quantum mechanics
one can start from the Dirac equation for the fermionic field
 d(iR e ggVV ()(xx )())M ((igM)ggV( x((x)x)))0 ((Mx ) g 0 ( x )  ( x )  0
 4
 (2 )
 ipRVV 

4



s

s
V 


See : Vasak-Gyulassy- Elze, Ann. Phys. 173(1987) 462
Elze and Heinz, Phys. Rep. 183 (1989) 81
Blaizot and Iancu, Phys. Rep. 359 (2002) 355


s



Just for simplicity lets consider the case with only a scalar field
i
d 4 R ipR


e
:  ( x ) ( x ) : :  ( x ) :  0
   p      m  F ( x, p)  
4
2
(2 )

 
This is the semiclassical approximation.
R x
 ( x )   ( x) 
  ( x)  ... If one include higher order derivatives gets
2
For the NJL
G 
an expansion in terms of higher order derivatives
of the field and of the Wigner function
The validity of such an expansion is based on the assumption ħ∂x∂pFW >>1
Again the point is to have not too large gradients:
X F  PW  1
XF typical length scale of the field
PW typical momentum scale of the system
A very rough estimate for the QGP
XF ~ RN ~ 4-5 fm , PW ~ T ~ 1-3 fm-1 -> XF·PW ~ 5-15 >> 1
better for larger and hotter systems
Substituting the semiclassical approximation one gets:
i 
i
p






p

[
m


(
x
)]



(
x
)


x
  FW ( x, p )  0
2
2

There is a real and an imaginary part




Which contains the
2
*2
p  M FS ( x, p)  0
in medium mass-shell
Including more terms in the gradient expansion would have brougth terms
breaking the mass-shell constraint
  p  M FˆW ( x, p)  0
*
    M ( x)  x M ( x) p FˆW ( x, p)  0
*

*
FW  FS    FV

Decomposing, using both real and imaginary part and taking the trace
Vlasov Transport Equation in QFT
p


 
   p F   m  m  p* FS ( x, p )  0
This substitute the force term mF(x) of classical transport
Quantum effects encoded in the fields while f(x,p) evolution appears as the classical one.
Transport solved on lattice
p   f  C 22  ...
Solved discretizing the
space in h, x, y cells
Rate of collisions
per unit of phase
space
D3x
Dt0
D3x0
Putting massless partons
at equilibrium in a box
than the collision rate is
See: Z. Xhu, C. Greiner, PRC71(04)
exact
solution
Approaching equilibrium in a box
Highly non-equilibrated distributions
where the temperature is
anisotropy in p-space
F.Scardina
Transport vs Viscous Hydrodynamics in 0+1D
Knudsen number-1
K
K
K0  2
Huovinen and Molnar, PRC79(2009)
L




1 T
5h /s

4h
 1.2 T0 0
s
Transport Theory
p


 
   p F   m  m  p* f ( x, p )  C22  C23  ...
 valid also at intermediate pT out of equilibrium:
region of modified hadronization at RHIC
 valid also at high h/s  LHC and/or hadronic phase
 Relevant at LHC due to large amount of minijet production
 Appropriate for heavy quark dynamics
 can follow exotic non-equilibrium phase CGC:
A unified framework against a separate modelling with
a wider range of validity in h, z, pT + microscopic level.
Applications of transport approach
to the QGP Physics
- Collective flows & shear viscosity
- dynamics of Heavy Quarks & Quarkonia
First stage of RHIC
Hydrodynamics
No microscopic details
(mean free path  -> 0, h=0)


T
( x)  0

 
+ EoS P(e)



j

  B ( x)  0
Parton cascade
Parton elastic 22 interactions
1/ - P=e/3)
p   f  C 22
v2 saturation pattern reproduced
Information from non-equilibrium: Elliptic Flow
z
y
v2/e measures the efficiency
of the convertion of the anisotropy
from Coordinate to
Momentum space
Fourier expansion in p-space
x
y x
2
e x
2
y2  x2
1 | viscosity
c2s=dP/de | EoS
Hydrodynamics
Parton Cascade
2v2/e
c2s= 0.6
Massless gas e=3P -> c2s=1/3
More generally one can distinguish:
=0
c2s= 1/3
dN
dN
1  2 v 2 cos(2 )  ... 

dpT d dpT
c2s= 0.1
Measure of
P gradients
time
-Short range: collisions -> viscosity
-Long range: field interaction -> e ≠ 3P
Bhalerao
D.
Molnaret&al.,
M.PLB627(2005)
Gyulassy, NPA 697 (02)
If v2 is very large
P. Kolb
More harmonics needed to describe
an elliptic deformation -> v4
v4  cos( 4 ) 
p x4  6 p x2 p y2  p y4
( p x2  p y2 ) 2
To balance the minimum v4 >0 require
v4 ~ 4% if v2= 20%
At RHIC a finite v4 observed
for the first time !
Viscosity cannot be neglected
Fx
v x
 h
Ayz
y


T   Tideal
  dissip
Relativistic Navier-Stokes
but it violates causality,
II0 order expansion needed -> Israel-Stewart
tensor based on entropy increase ∂ s 0
h,z two parameters appears +
f ~ feq reduce the pT validity range
P. Romatschke, PRL99 (07)
Transport approach
p


 
   p F   m  m  p* f ( x, p )  C22  C23  ...
Free streaming
Field Interaction -> e≠3P
Collisions -> h≠0
C23 better not to show…
Discriminate short and long range interaction:
Collisions (≠0) + Medium Interaction ( Ex. Chiral symmetry breaking)
,T
decrease
We simulate a constant shear viscosity
Cascade code
Relativistic Kinetic theory
h 4
p

 cost.
s 15  tr n(4   T )

 tr (  (r ), T )   tr , 
p
4
1
(*)
15 n 4   T  h / s
=cell index in the r-space
Time-Space dependent cross
section evaluated locally
The viscosity is kept
constant varying 
A rough estimate of (T)
Neglecting  and inserting in (*)
h
1

s 4
s  4n 
1
 tr  2
T
e P
T
2 2g 3

T
45
At T=200 MeV
tr10 mb
G. Ferini et al., PLB670 (09)
V. Greco at al., PPNP 62 (09)
Two kinetic freeze-out scheme
Finite lifetime for the QGP small h/s fluid!
a) collisions switched off
for e<ec=0.7 GeV/fm3
No f.o.
 tr 
b) h/s increases in the cross-over
region, faking the smooth
transition between the QGP and
the hadronic phase
1 p 1
15 n h / s
This gives also automatically
a kind of core-corona effect
At 4h/s ~ 8 viscous hydrodynamics is not applicable!
Role of ReCo for h/s estimate
Parton Cascade at fixed shear viscosity
Hadronic h/s included
 shape for v2(pT)
consistent with that needed
by coalescence
Agreement with Hydro at low pT
A quantitative estimate needs
an EoS with e≠ 3P :
cs2(T) < 1/3 -> v2 suppression ~ 30%
-> h/s ~ 0.1 may be in agreement
with coalescence
 4h/s >3  too low v2(pT) at pT1.5 GeV/c even with coalescence
 4h/s =1 not small enough to get the large v2(pT) at pT2 GeV/c
without coalescence
Short Reminder from coalescence…
 dNq

dNM
(
p
)
α
(
p
2
)
T
T
d 2 p

d 2 pT


T
2
 dNq

dN B
(
p
)

(
p
3
)
T
T
d 2 p

d 2 pT


T
dN q

dN q
1  2v
2q
Quark Number Scaling
3
cos( 2 )

pT dpT dφ pT dpT
Molnar and Voloshin, PRL91 (03)
Greco-Ko-Levai, PRC68 (03)
Fries-Nonaka-Muller-Bass, PRC68(03)
1  pT 
V2  
n  n 
I° Hot Quark
Is it reasonable the v2q ~0.08
needed by
Coalescence scaling ?
Is it compatible with a fluid
h/s ~ 0.1-0.2 ?
Effect of h/s of the hadronic phase
Hydro evolution at h/s(QGP)
down to thermal f.o.  ~50%
Error in the evaluation of h/s
Uncertain hadronic h/s
is less relevant
Effect of h/s of the hadronic phase at LHC
Suppression of v2 respect the ideal 4h/s=1
LHC – 4h/s=1 + f.o.
RHIC – 4h/s=1 + f.o.
RHIC – 4h/s=2 +No f.o.
At LHC the contamination of mixed and hadronic phase becomes negligible
Longer lifetime of QGP -> v2 completely developed in the QGP phase
S. Plumari, Scardina, Greco in preparation
Impact of the Mean Field and/or
of the Chiral phase transition
- Cascade  Boltzmann-Vlasov Transport
- Impact of an NJL mean field dynamics
- Toward a transport calculation with a lQCD-EoS
NJL Mean Field
free gas
scalar field interaction

d3p


M (T )  m  4 gN f N c M (T ) 
1

f
(
T
)

f
(T )
3
(2 ) E p
L

Associated
Gap Equation
Two effects:
 e ≠ 3p no longer a massless free gas, cs <1/3
 Chiral phase transition
gas
Boltzmann-Vlasov equation for the NJL
Numerical solution with -function test particles
Contribution of the NJL mean field
Test in a Box with equilibrium f distribution
Simulating a constant h/s with a NJL mean field
Massive gas in relaxation time approximation
=cell index in the r-space
M=0
h
4
 pn
15
Theory
Code
 =10 mb
The viscosity is kept modifying
locally the cross-section
Dynamical evolution with NJL
Au+Au @ 200 AGeV for central collision, b=0 fm.
Does the NJL chiral phase transition affect
the elliptic flow of a fluid at fixed h/s?
S. Plumari et al., PLB689(2010)
- NJL mean field reduce the v2 : attractive field
- If h/s is fixed effect of NJL compensated by cross section increase
- v2  h/s not modified by NJL mean field dynamics!
Next step - use a quasiparticle model
with a realistic EoS [vs(T)]
for a quantitative estimate of h/s
to compare with Hydro…
Using the QP-model of Heinz-Levai
U.Heinz and P. Levai, PRC (1998)
WB=0 guarantees
Thermodynamicaly consistency
M(T), B(T) fitted to lQCD [A. Bazavov et al. 0903.4379 ]data on e and P
e
NJL
P
° A. Bazavov et al. 0903.4379 hep-lat
QP
Summary for ligth QGP
Transport approach can pave the way for a consistency
among known v2,4 properties:
 breaking of v2(pT)/e & persistence of v2(pT)/<v2> scaling
 v2(pT), v4(pT) at h/s~0.1-0.2 can agree with what needed
by coalescence (QNS)
 NJL chiral phase transition do not modify v2  h/s
 Signature of h/s(T): large v4/(v2)2
Next Steps for a quantitative estimate:
 Include the effect of an EoS fitted to lQCD
 Implement a Coalescence + Fragmentation mechanism
A Nearly Perfect Fluid
 T ( x)  0
 
  jB ( x)  0

 
f eq ( x, p)  g  e

E  pu  
T
 g e

mT
T
*
Tf ~ 120 MeV
<T> ~ 0.5
1
T *  Tf  m v T2
2
No microscopic description (->0)
 Blue shift of dN/dpT hadron spectra
 Large v2/e
 Mass ordering of v2(pT)
For the first time very close
to ideal Hydrodynamics
Finite viscosity is not negligible
Jet Quenching
How much modification respect to pp?
Nuclear Modification Factor Jet triggered angular correl.
 Jet gluon radiation observed:
 all hadrons RAA <<1 and flat in pT
near
 photons not quenched
-> suppression due to QCD
Medium
away
Surprises…
Baryon/Mesons
Quenching
p+p
PHENIX, PRL89(2003)
In vacuum p/ ~ 0.3
due to Jet fragmentation
 Jet quenching should affect both
0 suppression: evidence of jet
quenching before fragmentation
Protons not suppressed
Hadronization has been modified
pT < 4-6GeV !?
Hadronization in Heavy-Ion Collisions
Initial state: no partons in the vacuum but a
thermal ensemble of partons -> Use in medium quarks
No direct QCD factorization scale for the bulk:
dynamics much less violent (t ~ 4 fm/c)
Parton spectrum
More easy to
produce baryons!
d 3NH
  f q  f q   M   f q  DqH
3
d P


Baryon
Fragmentation:
 energy needed to create quarks from vacuum
 hadrons from higher pT
Meson
H
Coalescence:
 partons are already there
$ to be close in phase space $
 ph= n pT ,, n = 2 , 3
baryons from lower momenta (denser)
V. Greco et al./ R.J. Fries et al., PRL 90(2003)
ReCo pushes out soft physics by factors x2 and x3 !
Hadronization Modified
Baryon/Mesons
Quark number scaling
p+p
v2q fitted from v2
GKL
PHENIX, PRL89(2003)
dN q
pT dpT dφ

dN q
pT dpT
1  2v 2 cos(2 )
 dN q

dN H
( pT )   2 ( pT n)
2
d pT
 d pT

n
Coalescence
scaling
Enhancement
of v2
v 2,M (p T1)  2v
 p2,Tq (p T /2)
V2  
v 2,B (p Tn)  3v
 n2,q (p T /3)
Dynamical quarks are visible
Collective flows
Heavy Quarks
 mc,b >> LQCD produced by pQCD processes (out of
equilibrium)
 eq > QGP with standard pQCD cross section (and also with
non standard pQCD)
Hydrodynamics does not apply to heavy quark dynamics (f≠feq)
Equilibration time
pQCD
QGP- RHIC
“D”npQCD
What about v4 ?
Relevance of time scale !
 v4 more sensitive to both h/s and f.o.
 v4(pT) at 4h/s12 could also be consistent with coalescence
 v4 generated later than v2 : more sensitive to properties at TTc
Effect of EOS on v2
Decrease in v2 of about 40%
H. Song and U.Heinz
Very Large v4/(v2)2 ratio
Same Hydro with
the good dN/dpT and v2
Ratio v4/v22 not very much depending on h/s
and not on the initial eccentricity
and not on particle species …
see also M. Luzum, C. Gombeaud, O. Ollitrault, arxiv:1004.2024
Effect of h/s(T) on the anisotropies
4h/s
V2 develops earlier at higher h/s
V4 develops later at lower h/s
-> v4/(v2)2 larger
2
1
QGP
T/Tc
2
1
Au+Au@200AGeV-b=8fm |y|<1
v4/(v2)2 ~ 0.8 signature of h/s
close to phase transition!
Effect of h/s(T) + f.o.
Hydrodynamics
Effect of finite h/s+f.o.
p


 
   p F   m  m  p* f S ( x, p )  C[ f s ]
If the system if very dense 1/3 one can derive and add the three-body
collision that make the transition from the dilute to the dense system:
See: Zhu and Greiner PRC71 (2004)