Nonequilibrium
Dynamics
in Astrophysics and Material Science
YITP, Kyoto, Japan, Oct. 31

Nov. 3, 2011
Tetsufumi
Hirano
Sophia Univ./the Univ. of Tokyo
1.
Introduction
2.
Physics of the quark gluon plasma
3.
Relativistic heavy ion collisions
•
Elliptic flow
•
“perfect fluidity”
•
Higher harmonics
4.
Some topics in relativistic hydrodynamics
•
“Condensed” matter physics for
elementary particles
•
Heavy ion collisions as a playground of
non

equilibrium physics in relativistic
system
•
Recent findings of the quark gluon plasma
and related topics
Confinement of
color charges
Asymptotic free
of QCD
coupling vs. energy scale
QCD: Theory of strong interaction
Hadron Gas
Quark Gluon Plasma (QGP)
Degree of freedom: Quarks (matter) and gluons (gauge)
Mechanics: Quantum
ChromoDynamics
(QCD)
Low
Temperature
High
Novel matter under extreme conditions
Suppose “transition” happens when
a pion (the lightest hadron) gas is
close

packed,
𝑛
𝜋
≈
1
4
𝜋
𝑟
𝜋
3
3
≈
3
.
6
𝑇
3
𝜋
≈
0
.
5
fm
𝑇
≈
100
MeV
≈
10
12
K
Note 1:
T
inside the Sun ~ 10
7
K
Note 2:
T
c
from the 1
st
principle calculation ~2x10
12
K
History of the Universe
~ History of form of matter
Micro seconds after Big Bang
Our Universe is filled
with the QGP!
Front View
Side View
Relativistic heavy ion collisions
Turn kinetic energy (
v
> 0.99
c
) into thermal energy
Figure
adapted
from
http://www

utap.phys.s.u

tokyo.ac.jp/~sato/index

j.htm
Big Bang
Little Bang
Time scale
10

5
sec >>
m.f.p
./c
10

23
sec ~ m.f.p./c
Expansion rate
10
5

6
/sec
10
22

23
/sec
Spectrum
Red shift (CMB)
Blue shift (hadrons)
m.f.p
. = Mean Free Path
Non

trivial issue on thermal equilibration
0
collision axis
time
Au
Au
3. QGP
fluid
4. hadron
gas
1. Entropy production
2. Local equilibration
3. Dissipative relativistic
fluids
4. Kinetic approach
for relativistic gases
1.
2.
quark gluon plasma
Time scale ~ 10

22
sec
http://youtu.be/p8_2TczsxjM
dN
/
d
f
f
0
2
p
2
v
2
~10
35
Pa
Elliptic flow (
Ollitrault
, ’92)
•
Momentum anisotropy as
a response to spatial
anisotropy
•
Known to be sensitive to
properties of the system
•
(Shear) viscosity
•
Equation of state
2
nd
harmonics (elliptic flow)
Indicator of
hydrodynamic behavior
Zhang

Gyulassy

Ko
(’99)
v
2
is
generated through secondary collisions
saturated in the early stage
sensitive to cross section (~1/m.f.p.~1/viscosity)
=
1
𝜎
∝
l
: mean free path
h
: shear viscosity
=
2
−
2
2
+
2
Eccentricity
x
y
2
=
Momentum
Anisotropy
Spatial
Anisotropy
Response of the system
reaches “hydrodynamic
limit”
𝑣
2
𝜀
vs. transverse
particle density
Au+Au
Cu+Cu
T.Hirano
et al. (in preparation)
2
vs. centrality
Ratio of shear viscosity to entropy density
QGP
=
0
.
08
~
0
.
16
ℏ
𝑘
≈
1
380
Water
≈
1
9
Liquid
He
𝑣
2
𝜀
vs. transverse particle density
H.Song
et al., PRL106, 192301 (2011)
𝑉
𝑉
=
div
≈
10
22
/
sec
L
arge expansion rate of the QGP in relativistic
heavy ion collisions
Tiny
v
iscosity when hydrodynamic description of
the QGP works in any ways
Manifestation of the strong coupling nature of
the QGP
Note: Underlying theory
Quantum
ChromoDynamics
(theory of strong interaction)
Figure adapted from talk
by
J.Jia
(ATLAS) at QM2011
Ideal, but unrealistic?
OK on average(?)
Actual collision?
Higher order
deformation
0
collision axis
time
Au
Au
QGP fluid
hadron gas
Relativistic Boltzmann
Relativistic Ideal Hydro
Monte Carlo I.C.
*
K.Murase
et al.
(in preparation)
𝑛
=
2
𝑖𝑛
𝜙
2
Sample of entropy
density profile in a
plane perpendicular
to collision axis
𝑛
vs. centrality (input)
𝑛
vs. centrality (output)
Response of the QGP to initial deformation
𝑛
roughly scales with
𝑛
Tendency similar to experimental data
Absolute value
Viscosity
Theory
Experiment
•
Most of people did not
believe hydro description
of the QGP (~ 1995)
•
Hydro at work to describe
elliptic flow (~ 2001)
•
Hydro at work (?) to
describe higher harmonics
(~ 2010)
≲
5
fm
coarse
graining
size
i
nitial
profile
≲
1
fm
Photon spectra in relativistic
heavy ion collisions
Blue shifted spectra with
T~200

300 MeV~(2

3)x10
12
K
1.
Non conservation of particle
number nor mass
2.
Choice of local rest frame
3.
Relaxation beyond Fourier, Fick and
Newton laws
𝜕
𝜕
𝑡
+
𝜵
∙
+
𝑃
𝒗
=
0
,
=
+
𝑃
2
−
𝑃
Energy conservation
Momentum conservation
𝜕
𝜕
𝑡
𝑖
+
𝜵
∙
𝑖
𝒗
=
−
𝛻
𝑖
𝑃
,
𝑖
=
+
𝑃
2
𝑖
Charge conservation (net baryon number in QCD)
𝜕
𝜕
𝑡
+
𝜵
∙
𝒗
=
0
,
=
𝑛
−
𝑛
𝜕
𝑇
=
0
,
𝜕
=
0
1.
Non conservation of particle
number nor mass
2.
Choice of local rest frame
3.
Relaxation beyond Fourier, Fick and
Newton laws
1.
C
harge flow
=
1
𝑖
𝑛
𝑖
𝑖
2. Energy flow
(Eigenvector of energy

momentum tensor)
𝑇
=
Charge diffusion
vanishes on average
No heat flow!
heat flow
charge
diffusion
See also talk by
Kunihiro
*Energy flow is relevant
in heavy ion collisions
1.
Non conservation of particle
number nor mass
2.
Choice of local rest frame
3.
Relaxation beyond Fourier, Fick and
Newton laws
Constitutive equations
at
Navier

Stokes leve
l
=
2
𝜕
<
>
,
Π
=
−
𝜍
𝜕
thermodynamics force
0
or
𝑅
/
Realistic response
Instantaneous response
violates causality
Critical issue in
relativistic theory
Relaxation plays an
essential role
See also talk by
Kunihiro
𝜏
𝑅
=
′
𝑅
,
′
′
Within linear response
Suppose
𝑅
,
′
=
𝜏
exp
−
−
′
𝜏
(
−
′
)
one obtains differential form
𝑅
=
−
𝑅
−
𝜏
,
Maxwell

Cattaneo
Eq.
signal
=
𝜏
<
𝑅
=
4
′
,
′
′
+
𝑅
(
)
𝑅
(
)
𝑅
(
′
)
≈
(
,
′
)
Thermal fluctuation in event

by

event simulations
d
issipative current
thermal noise
thermodynamic force
* In non

relativistic cases, see Landau

Lifshtiz
, Fluid Mechanics
** Similar to glassy system, polymers,
etc
?
Fluctuation
Dissipation
𝑅
′
𝑅
𝑅
=
𝜏
−
𝑡
𝜏
𝑅
≈
′
coarse
graining
???
Existence of upper bound in coarse

graining time
(or lower bound of frequency) in relativistic theory???
Navier
Stokes
causality
Non

Markovian
Markovian
Maxwell

Cattaneo
→
′
𝜉
∝
1
∆
𝑉
F
luctuation
Local volume.
•
Information about coarse

grained size?
•
Fluctuation term ~ average value?
Non

equilibrium small system?
Fluctuation would play a crucial role.
•
Need to consider (?) finite size effects in
equation of state and transport coefficients
Physics of the quark gluon plasma
•
Strong coupling nature
•
Small viscosity
Physics of relativistic heavy ion collisions
•
Playground of relativistic non

equilibrium system
•
Relativistic dissipative hydrodynamics
•
Relativistic kinetic theory
•
Non

equilibrium field theory
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