Hydrodynamic Instability in the Quark

Report
Hydrodynamic Instability
in the Quark-Gluon Plasma
Carlos E. Aguiar
Instituto de Física - UFRJ
C.E.A., E.S. Fraga, T. Kodama, nucl-th/0306041
Outline:
• Introduction; explosive hadronization
• Thermodynamics of the chiral phase transition
• Supercooling and spinodal decomposition
• Hydrodynamics of the chiral phase transition
• Fluid mechanical instability in the QGP
• Comments
Heavy Ion Collisions at High Energies
Au
Au
Heavy Ion Collisions at High Energies
t
hadrons
mixed
phase
10 fm/c
QGP
z
SPH calculation
Explosive Hadronization?
___ strong 1st order
...... weak 1st order
- - - crossover
D. Zschiesche et al.
Phys. Rev. C 65
(2002) 064902
RHIC
SPS
2
2
• HBT radii: Rout
 R2side  Vpair
()2
Rout / Rside  1
  0
sudden emission
The Phase Diagram of
Strongly Interacting Matter
T
crossover
150 MeV
QGP
supercooling
Hadrons
922 MeV

Lattice QCD and Freezeout States
RHIC
SPS
Z. Fodor and S. D. Katz,
Phys. Lett. B 534 (2002) 87,
JHEP 0203 (2002) 014
Thermodynamics of the
Chiral Phase Transition
Linear sigma model


 
1
L  q i     g   i  5     q         U0 ()
2


2 2
U0 () 
  2  v 2
4
h  fm ,

2
2
m

m
2   2  ,
2f
2
 h

q  (u, d) ,   (, )
2
2
m

3
m

v 2  f2 2
,
2
m  m 
m q  g f
f  93 MeV , m  138 MeV , m  600 MeV , g  3.3
Partition function

 
1/ T

3
0
ˆ
ˆ
Z  Tr exp  H   N T   D q DqD exp   d(i t )  d x L   q q  q 
 0

V
( q   / 3 )


Effective potential
T
U(T, )   ln Z
V
U   / V  P
• mean field approximation:   <>

d3p
U(T, , )  U0 ()   qT 
ln1  exp (E   q ) / T
3
( 2 )
E  p2  m2q

   (q  q )
m2q  g22  g2 2  2

Effective Potential
U (MeV/fm3)
-10
-20
-20
 = 800 MeV
 = 500 MeV
-40
T = 41 MeV
U (MeV/fm3)
0
49.9 MeV
57 MeV
-30
T = 105 MeV
-60
-80
118 MeV
63.5 MeV
-40
69 MeV
-100
130 MeV
-50
-120
-20 0 20 40 60 80 100 120
 (MeV)
1st order
-20 0 20 40 60 80 100 120
 (MeV)
crossover
Supercooling and Spinodal Decomposition
U (MeV/fm3)
-10
-20
-20
 = 800 MeV
 = 500 MeV
-40
T = 41 MeV
U (MeV/fm3)
0
49.9 MeV
57 MeV
-30
T = 105 MeV
-60
-80
118 MeV
63.5 MeV
-40
69 MeV
-100
130 MeV
-50
-120
-20 0 20 40 60 80 100 120
 (MeV)
1st order
-20 0 20 40 60 80 100 120
 (MeV)
crossover
Pressure and Chiral Field
First order
 = 800 MeV
100
40
sh
30
sc
20
40
sigma field (MeV)
pressure (MeV/fm3)
50
80
60
40
20
0
50
60
temperature (MeV)
70
0
40
80
120
temperature (MeV)
160
Mesons
2
2

U

U
2
2
m  2 , m   2


First order
800
800
600
600
m (MeV)
m (MeV)
 = 800 MeV
400
400
200
200
0
0
0
40
80
T (MeV)
120
160
0
40
80
T (MeV)
120
160
Pressure and Chiral Field
Crossover
 = 500 MeV
100
sigma field (MeV)
pressure (MeV/fm3)
600
400
200
80
60
40
20
0
0
0
40
80
120 160 200
temperature (MeV)
0
40
80
120 160 200
temperature (MeV)
Mesons
Crossover
800
800
600
600
m (MeV)
m (MeV)
 = 500 MeV
400
400
200
200
0
0
0
40
80
120 160 200
T (MeV)
0
40
80
120 160 200
T (MeV)
Chiral Phase Diagram
temperature (MeV)
120
chiral
symmetry
80
40
spinodal
line
broken chiral
symmetry
0
400
600
800
1000
chemical potential (MeV)
T-n Diagram
temperature (MeV)
120
broken
symmetry
chiral
symmetry
80
40
spinodal
0
0
0.1
0.2
baryon density (1/fm3)
0.3
Hydrodynamics of the
Chiral Phase Transition
Action:
1

A   d x        (n, s, )
2

4
(n, s, )  energy density  U  Ts   n
  (n u )  0 
constraints:   (s u )  0 
 u u  1 
 
baryon number conservation
entropy conservation
flow velocity normalization
Chiral Hydrodynamics
      R
  T   R   
  (n u )  0
  (s u )  0

R
U(T, , )

T   (  P) uu  Pg
Wave Motion
( x)  0  1 eiK x
Perturbation of
equilibrium:
Linearized
equations:
w 0  (  P)0
u ( x)  u0  u1 eiK x






1  0
k
2
 m2
2
k
2
 m2
2
 P k 2 v1   k R 1
2


K  (, k )

k
w 0 R v1
1



P   P(, s / n, )
 
0


v1 // k


R   R (, s / n, )
 
0
Chiral and Sound Modes
Dispersion relation

2


 P k 2 2  k 2  m2  w 0 R 2 k 2
Long wavelengths
sound waves:
chiral waves:
2


w
R
2s   P  0 2  k 2  
m 

2  m2  
Hydrodynamic Instability
If
2

w0 R
P  2  0
m
then s2 < 0, and the sound modes become
unstable, growing exponentially instead of
propagating. This instability occurs before
the chiral spinodal line (m2 = 0) is reached.
More importantly, the crossover region
(m2  0) is unstable.
Hydrodynamic Instability in the QGP
instability
line
temperature (MeV)
160
120
spinodal
80
40
0
0
200 400 600 800 1000
chemical potential (MeV)
Instability Line in the T-n Plane
temperature (MeV)
160
instability
line
120
80
40
0
0
0.1
0.2
baryon density (1/fm3)
0.3
In summary:
• The nonequilibrium chiral condensate changes
qualitatively the hydrodynamical behavior of
the QGP
• Explosive hadronization doesn’t need spinodal
decomposition, and can occur even in the
crossover region.
Final comments:
• This is a very general effect; it doesn’t depend
on specific aspects of the sigma model.
• The instability develops even for very slow
cooling, contrary to spinodal decomposition.
• Finite size effects may be important in nuclei:
min ~ 5 fm at the critical point
• Implications for the hadronization process in:
 heavy ion collisions (?)
 early universe (!)

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