Shear viscosity to entropy density ratio below QCD critical temperature

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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
• What is the shear viscosity?
• Background and motivation
• Shear viscosity/Entropy in Pionic gas
• Summary and outlook

April/21th/2006 at IoP, AS
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1) What is the shear viscosity?
Shear viscosity (coefficient) is one of
transport coefficients in macroscopic
hydrodynamic equations for non-equilibrium
systems
Basic equations
Energy-momentum conservation
1)
2) Number conservation
Where Local collective flow velocity
Elementary volume
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The first term describes Perfect fluid dynamics
(dissipationless)
appears in spatial traceless part (dissipative)

Stress pressure (friction) in shear flow (
coefficient of frictional force)
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Lets remember,
1) Isotropic pressure
Unit cross-section
The number of particle reflected by the
cross-section per second
Thus the isotropic pressure becomes
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2) Anisotropic(stress) pressure
Momentum transfer of x comp. per sec. across
unit area normal to y direction a frictional
force facing -x direction
Scattering cross-section
Mean-free path
Maxwell formula
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Viscos dynamics
e.g., Diffusion equation for transverse momentum

diffusion constant
relaxes transverse fluctuation, in other words,
diminishes the velocity gradient (shear flow).
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Hierarchy in theories for space-time scales,
theories
scales

• Liouville eq.
• Linear response theory

micro
Hamiltonian
1fm
Jeon-Yaffe (1996)

• Boltzmann eq.
• GL eq.
• Langevin eq.

mesoscopic
Kinetic theories
100fm
Our attempt (Tltm_pi)

• Fluid eqs,
• e.g. , in Navier-Stokes eq.

macro
Fluid dynamics
104fm
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Basic properties of shear viscosity
Maxwell formula
Scattering cross-section
This can be also seen from more microscopic
theory, Kubo formula Auto correlation function
of
LO
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.
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2) Background and motivation
1) A perturbative gravity analysis with a black
hole metric corresponding to N4
supersymmetric gauge field theory in strong
coupling (Ads/CFT correspondence)
conjectures a lower bound (KSS bound)
Shear viscosity/entropy ratio
Kovtun, Son, Starinet, hep-th/0405231
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2) Elliptic flow produced just after
non-central relativistic heavy ion collisions
(RHIC),
RHIC
suggests that the system is near perfect fluid
(small viscosity ). It implies that
expected QGP is in strong coupling regime.
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Directed flow
Elliptic flow
y
x
QGP
Hadrons
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QCD phase diagram on Density-Temp. plane
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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.
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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?
taking the pionic gas.
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3) Shear visc./entropy in pionic gas in Kinetic
theory
Local equilibrium distribution,
(Dissipationless process)
Small deviation
(Dissipative process)
Bose distribution function
at local rest frame
is given by as a functional of , which
we will obtain from Boltzmann eq. .
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The distribution function is obtained from
Boltzmann eq. for ,
with collision integral
Scattering cross-section
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Strategy to obtain f(x,p) from Boltzmann eq.

1. Expand to the 1st order
2. parametrize
3. Substitute it into Boltzmann eq.
4. Linearize the eq. in terms of
5. Expand using a set of specific
   polynomials
6. Linearized Bolzmann Matrix eq. for

Step
Step
Known (by symmetry)
Step
unknown
Step
Step
A polynomial up to
Step
Finally, the viscosity is given by,
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Linearized Boltzmann equation for B(p)
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Pion-Pion scattering
ChPT effective theory on the basis of chiral
symmetry
LO
Increase with collision energy!
(low energy limit Weinberg theorem)
vanishes in massless limit!
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(Low energy limit)
coincide with the behavior in
by Jeon,Yaffe, Heinz,Wang, etc
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Non- monotonic!
From very naïve dimensional analysis, we find a
power law in T
Universal behavior!
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Intensive behavior at low T, divergent at T0 !
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).
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S statistical entropy
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4) Summary and Outlook
We have shown small ratio of the visc./entropy
in Chpt approaching Tc of QCD
So we conclude that the small viscosity/entropy
ratio lt1 is not unique only above Tc, but below
Tc. But it suggests discontinuity at Tc (2times
larger than KSS bound).
QGP
Hadron
KSS
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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
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Back up files
Hadronic gas at finite density 1-2 rho_0
Muroya and Sasaki, PRL(2005)
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Applicability of ChPT
Melting of Chiral cond.
Hadrons
QGP
?
Data
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In 1st Chapman-Enskog expansion,
with parametrization
Related to bulk viscosity
to shear viscosity
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CMF
Scatt. Amp. of ChPT
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S statistical entropy
with