Transcript Document 7502419

Shear viscosity to entropy density ratio
below QCD critical temperature
- Checking the viscosity/entropy ratio bound conjectured by string theory-
Eiji Nakano
Dept. of Physics,
National Taiwan Univ.
Outline:
1) What is the shear viscosity?
2) Background and motivation
3) Shear viscosity/Entropy in Pionic gas
4) Summary and outlook
April/21th/2006 at IoP, AS
1) What is the shear viscosity?

Shear viscosity (coefficient) is one of transport coefficients
in macroscopic hydrodynamic equations
for non-equilibrium systems:
Basic equations:
1) Energy-momentum conservation:
2) Number conservation:

  T   0,

  nV   0
Where Local collective flow velocity: V ( x) 
Elementary volume:
V (x)
1,  V( x)
1 V2
T ( x)  T   T  ,
( 0)
The first term describes Perfect fluid dynamics (dissipationless) :
T
(0)
  ( x)  P( x)V ( x)V ( x)  P( x) 
 appears in spatial traceless part (dissipative):
 iV j ( x)   jVi ( x) 1

Tij   
  ij  V ( x) .
2
3


y
Vx ( y )
Stress pressure (friction) in shear flow
( :coefficient of frictional force)
x
Let’s remember,
1) Isotropic pressure
Unit cross-section
px
vx
The number of particle reflected by
the cross-section per second:
nvx / 2
Thus the isotropic pressure becomes
nv x
1
P   Pii  2 p x
 n BT
3 i 13
2
2) Anisotropic(stress) pressure
y
Vx (y )
vy
x
Momentum transfer of x comp. per sec.
across unit area normal to y direction: a frictional force facing -x direction
m BT Vx ( y )
Vx ( y )
p x
Txy  
 nv y
l
y
y

y

m BT

Maxwell formula
Scattering cross-section
Mean-free path: l 
1
n
Viscos dynamics
e.g., Diffusion equation for transverse momentum
  D  p
2
t
diffusion constant:
T
D
pT
0

 P
:
py

 1
sT
 relaxes transverse fluctuation, in other words,
diminishes the velocity gradient (shear flow).
pT
px
Hierarchy in theories for space-time scales,
theories
scales
micro
~1fm
Hamiltonian
r0
mesoscopic
~100fm
~10^4fm
Jeon-Yaffe
(1996)
Kinetic theories
l
macro
sw
• Liouville eq.
• Linear response theory
Fluid dynamics
• Boltzmann eq.
• GL eq.
• Langevin eq.
Our attempt (T<m_pi)
• Fluid eqs,
e.g. ,  in Navier-Stokes eq.
Basic properties of shear viscosity
Roughly speaking,  is proportional to the mean-free path
Maxwell formula: 

m BT
1
l
n
:

Scattering cross-section
This can be also seen from more microscopic theory, Kubo formula:
Auto correlation function of T ij :

1
lim  dx 4 e ipx T ij ( x)Tij (0)
20T p 0
3
3
LO
e. g ., 
4
T

Im
T

 
2
by S-G. Jeon (1995)
(One has to resum infinite number of diagrams to get LO result even for weak coupling theory).
Keep in mind that large cross section gives small viscosity.
2) Background and motivation
1) A perturbative gravity analysis with a black hole metric corresponding to
N=4 supersymmetric gauge field theory in strong coupling
(Ads/CFT correspondence) conjectures a lower bound (KSS bound):
Shear viscosity/entropy ratio :

s
 41  0.08
Kovtun, Son, Starinet, hep-th/0405231
2) Elliptic flow produced
just after non-central relativistic heavy ion collisions (RHIC),
Hadronic (chiral broken) phase
Quark-Gluon Plasma (QGP),
RHIC

suggests that the system is near perfect fluid (small viscosity:  ).
dN
 v0  v2 cos( 2 )    
d
~0
It implies that expected QGP is in strong coupling regime.
Directed flow
dN
 1  2 v1 cos(φ)  2 v2 cos( 2φ)  
dφ
Elliptic flow
v2  cos2φ 
 px
p    p y p 
2
2
y
x
QGP
Hadrons
QCD phase diagram on Density-Temp. plane
RHIC
~0
Tc
?
Karsch & Laermann, hep-lat/0305025
Recent trapped cold atom experiments give an opportunity to investigate
strong interacting matter via tunable Feshbach resonance.
This dilute and strongly-coupled system of Li6 also behaves hydrodynamically,
showing elliptic flow.
Time evolution after trap is turned off
O ’Hara et al., Science 298, 2179 (2002)
Small viscosity is common feature in strongly-coupled systems.
Motivation:
….We investigate how the shear viscosity of QCD (pionic gas) behaves
below Tc (chiral / deconfinement transition),
with special attentions:
a) How the viscosity  behaves in Hadronic phase
approaching Tc from below,
b) How about

? Small or Large?
s
taking the pionic gas….
3) Shear visc./entropy in pionic gas in Kinetic theory
T ( x)  

3
dp c p p
2 3 E p
dp3 c p p
2 3 E p
f ( x, p )
Local equilibrium distribution,
(Dissipationless process)
[ f ( 0) ( x, p )  f ( x, p )]
Small deviation
(Dissipative process)
 T (0)  T  
f
( 0)
1
( x, p ) 
e
Ep / T
Bose distribution function
1

at local rest frame: V ( x )  0
T ( 0)   ( x)  P( x)V ( x)V ( x)  P( x)  ,
  iV j ( x)   jVi ( x) 1

Tij   
  ij   V ( x) 
2
3



is given by as a functional of f ,
which we will obtain from Boltzmann eq. .
The distribution function is obtained from Boltzmann eq. for f p  f ( x, p) ,
df p
dt
 C[ f p ]
with collision integral



1
C[ f p ]   d f1 f 2  f 3  1 f p  1   f1  1 f 2  1 f 3 f p
2 123
k1
p
k2
k3

2
~Scattering cross-section
Strategy to obtain f(x,p) from Boltzmann eq.
( 0)
Step 1. Expand to the 1st order f ( x, p)  f ( x, p)  f ( x, p)
Step 2. parametrize
f ( x, p)  B( p) f (0) ( x)
Known (by symmetry)
Step 3. Substitute it into Boltzmann eq.
unknown
Step 4. Linearize the eq. in terms of B( p )
Step 5. Expand B( p ) using a set of specific polynomials
B( p)   br B ( r ) ( p)
r
p
A polynomial up to
r
br
Step 6. Linearized Bolzmann = Matrix eq. for
Finally, the viscosity is given by,
   B( p)   br  
B
(0)
| l.h.s
2
B ( 0 ) | C[ B ( 0 ) ]

1

Linearized Boltzmann equation for B(p);
Pion-Pion scattering
ChPT: effective theory on the basis of chiral symmetry
LO
| Τ ChPT |2 
 | Τ I |2 •
I  Isospin


1
4
4
M

p
 

4
9 F
Increase with collision energy!
23 M 4
| Τ ChPT | 
(low energy limit: Weinberg theorem)
3 F4
2
vanishes in massless limit!
Shear viscosity with const. coupling



(Low energy limit)
| Τ ChPT |2 
23  M 

2  F
T 1 / 2 at Low T
~ 3
T at High T
4
m  139 (MeV)
F  93 (MeV)
coincide with the behavior in
 4

4!
by Jeon,Yaffe, Heinz,Wang, etc…
m  139 (MeV)
F  93 (MeV)
Nonmonotonic!
TC  170 MeV
From very naïve dimensional analysis, we find a power law in T:
Universal behavior!
 χpt
T for Low T
~  1
T for high T
1/ 2
| TChP T |2  M 
4
| TChP T |2  p 4 ~ T 4
m  0
Intensive behavior at low T, divergent at T=0 !
But it seems to be typical for pure NG bosons with derivative couplings.
This aspect is also seen for CFL phonon by Manuel etal (2004).
~ 0.35
KSS_bound  0.08
2
2

S: statistical entropy ~ g
T3
45
4) Summary and Outlook
We have shown small ratio of the visc./entropy in Chpt
approaching Tc of QCD:

s
 0.64  0.32
for T  120  140 ( MeV).
So we conclude that the small viscosity/entropy ratio <1 is
not unique only above Tc, but below Tc.
But it suggests discontinuity at Tc (~2times larger than KSS bound).

s
QGP
Hadron
KSS
Tc  170MeV
T
As future works
We are interested in shear visc. behavior
in BCS-BEC crossover regime, above and below Tc.
Quasiparticle
with fluctuations
Superfluid phonon + ….
This work is close collaboration with Prof. J-W Chen at NTU.
Thank you for your attention…
Back up files
Hadronic gas at finite density 1-2 rho_0
Muroya and Sasaki, PRL(2005)
Applicability of ChPT
/s
Melting of Chiral cond.
qq  0
Hadrons
qq  0
QGP
qq  0
?
T ~ M   140
Data
TC ~ 170
T (MeV )
In 1st Chapman-Enskog expansion,
f p  f ( x, p )  f
(0)
g ( x, p) 

(0)
( x, p) 1  1  f ( x, p)
T 



with parametrization
  V ( x) 

g ( x, p)  A( p) V ( x)  B( p)  p  V ( x)  p
3 

Related to bulk viscosity
to shear viscosity
[CMF]
q
| Τ ChPT |2 
Scatt. Amp. of ChPT
  0,
 
,
4 2
 q


1
M 4  p 4    
4
F
q[MeV ]
S: statistical entropy
S
S
 (T LogZ)
T

1
with LogZ   dp Log 1 - e-E(p)/T 
2 2 3
~
T
45
T