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Charge Fluctuations and transport coefficients near CEP
Charge fluctuations in the presence of spinodal instabilities and their scaling
Shear, and Bulk viscosities and its scaling near CEP
Krzysztof Redlich, Workshop on QCD-CP, INT
08
Use an effective chiral models and scaling theory to study:
Scaling properties:
1 1
2 | |TCPT T-
The strength of the singularity at TCP depends on direction in plane( , )BT m
1| |Cq T PT Tc --µ along 1st order line
any direction not parallel
along 2nd order line 11| |
2 Tq CPT Tc --µ
1/ 2| |Tq CPT Tc --µ
Going beyond the mean field:B.-J. Schaefer & J. Wambach
0.53( 0 0.78)| | mq TCPT Tc - ¹ =>µ -
Z(2) univer. class
1| |TCPT T --
11| |
2 TCPT T --
1/ 2| |TCPT T --
(2 ) 0| |ndcT T --
C.Sasaki, B. Friman & K.R.
FRG: Stokic, Friman & K.R
| |q TCPa b T T qc -= + -
See also Y. Hatta, T. Ikeda (0 ) ( 0.25 (2))( 0.3MF Za = - -
O(4) univer. class
Scaling properties: The strength of the singularity at TCP depends on direction in plane( , )BT m
1| |Cq T PT Tc --µ along 1st order line
any direction not parallel
along 2nd order line 11| |
2 Tq CPT Tc --µ
1/ 2| |Tq CPT Tc --µ
Going beyond the mean field:B.-J. Schaefer & J. Wambach
0.53( 0 0.78)| | mq TCPT Tc - ¹ =>µ -
Z(2) univer. class
C.Sasaki, B. Friman & K.R.
FRG: Stokic, Friman & K.R
| |q TCPa b T T qc -= + -
See also Y. Hatta, T. Ikeda (0 ) ( 0.25 (2))( 0.3MF Za = - -
O(4) univer. class
Quark and isovector fluctuations along the critical line
sensitive probes of TCP/CEPNon-monotonic behavior of the net
quark susceptibility as function of
in LGT or in HIC
NJL-model results: C. Sasaki, B. Friman, K.R.
[ ]cT MeV s:
( , )c cT m s
The nature of the 1st order chiral phase transition
instability of a system:
/ 0
0
/ 0
/
P
P
P
V
V
V¶ ¶¶
>¶
¶
<
=
¶ : stable : unstable : spinodal
A-B: supercooling (symmetric phase)B-C: non-equilibrium stateC-D: superheating (broken phase)
.symc -
brokenc -
Convex anomaly in thermodynamic pressure NJL model results
0qm = 0qm ¹
P- is differentiable at all values of m
P- has a cusp at the point where the dynamical quark mass vanishes
symmetric phase
broken phase
Dynamical Quark Mass and Instabilities
0qm = 0qm ¹
Maxwell constration : two degenerate minima in the thermodynamic potential
Spinodal Lines : no-unique solution of the gap equation for between spinodals
( , )M T fixedm =
m
equilibriumtransition
equilibriumtransition
Entropy per baryon and 1st order phase transition
Smooth evolution of entropy/baryon across the 1st order transition
0qm = 0qm ¹
Fluctuations in the chiral limit across spinodals
Non-singular behavior of fluctuations
when crossing spinodal line from the side of symmetric phase:
=> directly related with a cusp structure of the pressure
TCP
Critical exponents at 1st order line and CEP
1/ 2 (0.53) 2 /3 (0.70 1/
82 1 0 1
)/ 2 1{ , {
q q
TCm st m
P C PtE
sg g= ¹= =
| |cq
c
gm mc
m--: 1/ 2 (0.53) 2 /3 (0.7
0 1/8
2 1 0 1)
/ 2 1{ , {q q
TCm st m
P C PtE
sg g= ¹= =with
1st
1st
TCP
TCP
B. Friman, C. Sasaki & K.R. , Phys.Rev.D77:034024,2008.
Experimental Evidence for 1st order transition
2
2
2TT T q
P qP
q
S s sT TV
T n nC mc c c
é ù¶æ ö= = - +ê úç ÷¶è ø ê úë û
Specific heat for constant pressure: Low energy nuclear collisions
Net-quark fluctuations on spinodals
at any spinodal points:
21
|.T
q
qnPV
V compresc¶
- = =¶
Singularity at CEP arethe remnant of that alongthe spinodals
CEP
spinodals
spinodals
C. Sasaki, B. Friman & K.R., Phys.Rev.Lett.99:232301,2007.
Minimum of shear viscosity may indicate the location of : L. Csernai, J. Kapusta & L. McLerran (06)
Divergent bulk viscosity at CEP from QCD trace anomaly argument: D. Kharzeev & Tuchin 07
F. Karsch, D. Kharzeev & Tuchin 08
cT
Find viscosities in quasi particle models with dynamically generated mass ,
under relaxation time approximation Derive scaling behavior of viscosities in O(4) and Z(2)
universality class using scaling theory
Transport coefficients near CEP
Our goal: C. Sasaki & K.R. hep-ph 0806.4745
( , ) ( , )bareM T m f Tm m= +
Transport Coefficient near phase transition
L. Csernay, J. Kapusta & L. McLerran 06
R. Lacey et al. 07: data for shear viscosity Harvey B. Meyer 08 LGT results
for bulk viscosity, SU(3) gauge th.
0.134(33) 1.65
0.102(56) 1.24c
c
for T T
for T Ts
h ==
=
Shear viscosity to entropy ratio in LGT
Transport coefficients from kinetic theory:
Energy momentum tensor
Assume small deviations from equilibrium with , consequently
3
3
1[ ]
(2 )
d pT p p f f
Emn m n
p= +ò
22 2 ( , )E p M T m= +ur
0f f fd = -1 exp( ) 1f E pu m- = - ±
urrm
3
3
1[ ]
(2 )
d pT p p f f
Emn m nd d d
p= +ò
To get use Boltzmann equation under relaxation time approximation
with the collision time obtained from the particle
density and cross section
fd
0
0
[ ]pf ff
v f C ft
Ep f fm
m
t
dt
-¶+ Ñ = - -
¶
Þ ¶ = -
r ur;
1f f reln vt s- = < >
derivation of transport coefficients:
from Boltzmann equation: energy conservation: charge conservation: stationary condition : thermodynamics relation:
fd00
0 0T¶ =0
0 0j¶ =/ 0P M¶ ¶ =
/ /T P T P Pe m m= ¶ ¶ - + ¶ ¶
use:
get: ij kij k ijT u WVd hd= - ¶ -
bulk viscosity shear viscosity
10f E p fm
md t -= - ¶
shear viscosity is not modified by thermal change of quasi-particle energy
( , )Th m
43
0 0 0 03 2[ (1 ) (1 )]
15 (2 )
g d p pf f f f
T Et t
ph = ± + ±ò
ur
the above coincides with Hosoya & Kajantie (1985),
however, the bulk viscosity
is modified by thermal change of quasi-particle energy through: and( , )TV m
/E T¶ ¶ /E m¶ ¶
( )
( )
3 2
0 0 0 03
2
2
0 0 0 0
[ (1 ) (1 )3 (2 )
3
(1 ) (1 ) ]
n
E E
g d p Mf f f f
T E
p P PE T
E n
M Pf f f f
E
E
n
T e
e
V t tp
me
t t
m m
= - ± + ±
æ öæ ö¶ ¶æ ö æ öç ÷´ - - - +ç ÷ ç ÷ç ÷ç ÷¶ ¶è ø è ø¶ ¶ ¶¶è øè ø
¶æ ö- ± - ± ç ÷¶è
¶ ¶
ø
òur
for the above coincides with Hosoya & Kajantie (1985)
For the above coincides with Arnold, Dogan & Moore (06)/ 0E m¶ ¶ =/ ( ) 0E T m¶ ¶ ¶ =
Near phase transition the bulk viscosity can be singular through derivatives terms:
21
2
E M
x E x
¶ ¶=
¶ ¶
2
xy
P
x yc ¶
=¶ ¶
with , ( , )x y T m=
1| ( )n T
V
Ps n
C mm mmm
c ce c
¶= -
¶
1| ( ( ) )TT T
V
PnT n sT s
n Ce m mmmm
c m c mcc
¶= + - -
¶
2
| TV V TT
sC T T
Tm
mm
cc
c
æ ö¶æ ö= = -ç ÷ç ÷ ç ÷¶è ø è øwhich all can diverge at the critical points !
critical behavior of bulk viscosity: consider a system where dynamical mass acts as an order
parameter
Mean Field Scaling from Ginzburg-Landau potential
=> 2nd order: a=0 and b>0 TCP at a=b=0
with and from gap equation
2 40( , , ) ( , ) ( , )MT a TM M hM Mb Tm m m=W = W + + -
( , ) | | | |c ca T T Tm a b m m= - + -
132 14 20 0 0|singular
TCPV
M M MM t
C T sa cb m
V× -æ ö¶ ¶
- ´ ´ »ç ÷¶ ¶è ø: : :
sin
2
2| 0nd
gular MV =:
Conclusions: under MF approximation there is no singularity of bulk viscosity at the TCP and 2nd order phase transition. actually : in the chiral limit the bulk viscosity vanishes at the transition point
O(4) scaling of bulk viscosity: use the free energy
that gives
consequently
2( , ) ( )S SF T t f t ha bdm - -= 2, | | / , /c c ct t A t T T T Tm m mº + = - =
Ejiri, Karsch & K.R. 06; Hatta & Ikeda 04
1 , , ,TT Vt t C t M ta a a bmmc c- - -: : : :
3sin 4 1
V
gular M M
C Tta bV + -¶
¶: :
for => 0.24 , 0.38a b= - = sin 0.28 0gular tV + ®:Conclusions: bulk viscosity is non-singular at O(4) critical point
Z(2) scaling of bulk viscosity: use the free energy
that gives
consequently
11/( , ) ( )S SF T h f h t
dbddm
+-=
, , , | | /t h t h c ct A t B x x x xmº + = -
Stephanov, Rajagopal & Shuryak 1998
/ / 1/, , , ,T TT Vh C h M hg bd g bd d
mm mc - -: : :sin 4 1
3/ /gular
V
M Mh
C Tg bd dV + -¶
¶: :
for => 1.25 , 0.31 , 5.2g b d= = =sin 0.54 0gular hV + ®:
Conclusions: bulk viscosity is non-singular at Z(2) critical
point within the relaxation time approximation
Karsch et. al. (08)
Beyond the relaxation time approximation
modes with long wave-length could stay in non-equil.
modified by dynamic critical exponents “z” Onuki, “Phase transition dynamics”, Cambridge Univ. Press (02)
For O(4) :
For Z(2) :
V
sin gular zt n aV - +:
sin gular zt aV*- +:
Singularity in bulk viscosity along O(4) critical line and at the TCP
due to dynamic critical exponents
Large change of
bulk viscosity due to contribution from
temperature derivative of dynamical quark mass, /M T¶ ¶
Bulk Viscosity across the phase transition in the NJL model Cross over region CEP and 1st order line
Conclusions
A non-monotonic change of the net-quark susceptibility probes the existence of CEP However in non-equilibrium: due to spinodal instabilitie the charge fluctuations diverge at 1st order critical line
=> Large fluctuations signals 1st order transition Under relaxation time approximation the bulk
viscosity is finite at CEP and O(4) line Þ Divergence of bulk viscosity controlled by the dynamical, rather then static critical exponents