Hydrodynamic Instability in the Quark
Download
Report
Transcript Hydrodynamic Instability in the Quark
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 fm ,
2
2
m
m
2 2 ,
2f
2
h
q (u, d) , (, )
2
2
m
3
m
v 2 f2 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
Z Tr exp Ĥ N̂ 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
ln1 exp (E q ) / T
3
( 2 )
E p2 m2q
(q q )
m2q g22 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) uu Pg
Wave Motion
( x) 0 1 eiK x
Perturbation of
equilibrium:
Linearized
equations:
w 0 ( P)0
u ( x) u0 u1 eiK 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 (m2 = 0) is reached.
More importantly, the crossover region
(m2 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 (!)