Sophia Univ./the Univ. of Tokyo

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