DYNAMICS OF RELATIVISTIC HEAVY ION COLLISIONS AND THE QUARK GLUON PLASMA

Nonequilibrium Dynamicsin Astrophysics and Material ScienceYITP, Kyoto, Japan, Oct. 31-Nov. 3, 2011

Tetsufumi HiranoSophia Univ./the Univ. of Tokyo

Outline1. Introduction2. Physics of the quark gluon plasma3. Relativistic heavy ion collisions• Elliptic flow• “perfect fluidity”• Higher harmonics

4. Some topics in relativistic hydrodynamics

Introduction• “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

PHYSICS OF THE QUARK GLUON PLASMA

Two Aspects of Quantum ChromoDynamics

Confinement ofcolor charges

Asymptotic freeof QCD

coupling vs. energy scale

QCD: Theory of strong interaction

What is the Quark Gluon Plasma?

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

How High?Suppose “transition” happens whena pion (the lightest hadron) gas is close-packed,𝑛𝜋 ≈

14 𝜋𝑟 𝜋

3

3

≈3.6𝑇 3𝑟 𝜋 ≈0.5 fm

𝑇 𝐶≈100MeV ≈1012K

Note 1: T inside the Sun ~ 107 KNote 2: Tc from the 1st principle calculation ~2x1012 K

Where/When was the QGP?

History of the Universe~ History of form of matter

Micro seconds after Big Bang

Our Universe is filledwith the QGP!

RELATIVISTIC HEAVY ION COLLISIONS

The QGP on the Earth

Front View Side ViewRelativistic heavy ion collisionsTurn kinetic energy (v > 0.99c) into thermal energy

Big Bang vs. Little Bang

Figure adapted fromhttp://www-utap.phys.s.u-tokyo.ac.jp/~sato/index-j.htm

Big Bang vs. Little Bang (contd.)

Big Bang Little BangTime scale 10-5 sec >>

m.f.p./c10-23 sec ~

m.f.p./cExpansion

rate 105-6/sec 1022-23/sec

Spectrum Red shift (CMB) Blue shift (hadrons)

m.f.p. = Mean Free PathNon-trivial issue on thermal equilibration

Non-Equilibrium Aspects of Relativistic Heavy Ion Collisions

0collision axis

time

Au Au

3. QGP fluid

4. hadron gas

1. Entropy production2. Local equilibration

3. Dissipative relativisticfluids

4. Kinetic approachfor relativistic gases

1.2.

Hydrodynamic Simulation of a Au+Au Collision

quark gluon plasma

Time scale ~ 10-22 sec

http://youtu.be/p8_2TczsxjM

Elliptic FlowdN

/df

f0 2p

2v2

~1035Pa 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

2nd harmonics (elliptic flow) Indicator of hydrodynamic behavior

Relativistic Boltzmann Simulations

Zhang-Gyulassy-Ko(’99)

v2 is generated through secondary collisions saturated in the early stage sensitive to cross section (~1/m.f.p.~1/viscosity)

𝜆=1𝜎𝜌 ∝𝜂

l: mean free pathh: shear viscosity

“Hydrodynamic Limit”

𝜀= ⟨ 𝑦2−𝑥2 ⟩⟨𝑦 2+𝑥2 ⟩

Eccentricity xy

𝑣2𝜀 =Momentum Anisotropy

Spatial Anisotropy

Response of the systemreaches “hydrodynamic limit”

vs. transverse particle density

Ideal Hydrodynamics at Work

Au+Au Cu+CuT.Hirano et al. (in preparation)

vs. centrality

Viscous Fluid Simulations

Ratio of shear viscosity to entropy density

(𝜂𝑠 )QGP

=(0.08 0.16 ) ℏ𝑘𝐵≈ 1380 (𝜂𝑠 )

Water≈ 19 (𝜂𝑠 )

Liquid He

vs. transverse particle density

H.Song et al., PRL106, 192301 (2011)

Strong Coupling Nature of the QGP

Large expansion rate of the QGP in relativistic heavy ion collisions

Tiny viscosity when hydrodynamic description of the QGP works in any ways Manifestation of the strong coupling

nature of the QGPNote: Underlying theory Quantum ChromoDynamics (theory of strong interaction)

Event-by-event Fluctuation

Figure adapted from talkby J.Jia (ATLAS) at QM2011

Ideal, but unrealistic?OK on average(?)

Actual collision? Higher

order deformation

Deformation at Higher Order

Dynamical Modeling of Relativistic Heavy Ion Collisions*

0collision axis

time

Au Au

QGP fluid

hadron gas

Relativistic Boltzmann

Relativistic Ideal Hydro

Monte Carlo I.C.

*K.Murase et al. (in preparation)

Deformation in Model Calculations

𝜀𝑛=|⟨𝑟 2𝑒𝑖𝑛 𝜙 ⟩|

⟨𝑟2 ⟩

Sample of entropydensity profile in aplane perpendicularto collision axis

Higher Harmonics from Ideal QGP Fluids

vs. centrality (input) vs. centrality (output)

Response of the QGP to initial deformation roughly scales with

Comparison of vn with Data

Tendency similar to experimental dataAbsolute value Viscosity

Theory Experiment

Impact of Finite Higher Harmonics• 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)

𝑑≲5 fm

coarsegraining

sizeinitialprofile

𝑑≲1 fm

Thermal Radiation from the QGP

Photon spectra in relativisticheavy ion collisions

Blue shifted spectra with T~200-300 MeV~(2-3)x1012K

RELATIVISTIC HYDRODYNAMICS

Several Non-Trivial Aspects of Relativistic Hydrodynamics

1. Non conservation of particle number nor mass

2. Choice of local rest frame3. Relaxation beyond Fourier,

Fick and Newton laws

Hydrodynamic Equations (Ideal Hydro Case)

, Energy conservation

Momentum conservation,

Charge conservation (net baryon number in QCD),

𝜕𝜇𝑇 𝜇𝜈=0 ,𝜕𝜇𝑁𝜇=0

Several Non-Trivial Aspects of Relativistic Hydrodynamics

1. Non conservation of particle number nor mass

2. Choice of local rest frame3. Relaxation beyond Fourier,

Fick and Newton laws

Choice of Local Rest Frame in Dissipative Fluids1. Charge flow

𝑢𝜇= 1𝑁∑

𝑖

𝑁 𝑖𝜇

𝑛𝑖2. Energy flow (Eigenvector of energy-momentum tensor)

𝑇 𝜇𝜈𝑢𝜈=𝑒𝑢𝜇

Charge diffusion vanishes on

average

No heat flow!

heat flow

chargediffusion

See also talk by Kunihiro

*Energy flow is relevantin heavy ion collisions

Several Non-Trivial Aspects of Relativistic Hydrodynamics

1. Non conservation of particle number nor mass

2. Choice of local rest frame3. Relaxation beyond Fourier,

Fick and Newton laws

Relaxation and CausalityConstitutive equations at Navier-Stokes level

𝑡

thermodynamics force

𝑡 0

𝐹∨𝑅

/𝜅

Realistic response

Instantaneous response violates causality Critical issue in

relativistic theory Relaxation plays an

essential role

See also talk by Kunihiro

𝜏

Causal Hydrodynamics

𝑅 (𝑡 )=∫𝑑𝑡′𝐺𝑅 (𝑡 , 𝑡 ′ )𝐹 ( 𝑡′ )Within linear response

Suppose𝐺𝑅 (𝑡 ,𝑡 ′ )= 𝜅

𝜏 exp(− 𝑡− 𝑡′

𝜏 )𝜃 (𝑡−𝑡 ′)

one obtains differential form�̇� (𝑡 )=− 𝑅 (𝑡 )−𝜅 𝐹 (𝑡 )

𝜏 ,

Maxwell-Cattaneo Eq.

𝑣 signal=√ 𝜅𝜏 <𝑐

Relativistic Fluctuating Hydrodynamics (RFH)

)⟨𝛿𝑅 (𝑥)𝛿 𝑅(𝑥 ′) ⟩≈𝐺(𝑥 ,𝑥 ′)

Thermal fluctuation in event-by-event simulations

dissipative current

thermal noise

thermodynamic force

* In non-relativistic cases, see Landau-Lifshtiz, Fluid Mechanics** Similar to glassy system, polymers, etc?

Fluctuation Dissipation

Coarse-Graining in Time

𝑡

𝐺𝑅

𝑡 ′

𝐺𝑅

𝐺𝑅=𝜅𝜏 𝑒

− 𝑡𝜏 𝐺𝑅≈ 𝜅𝛿 (𝑡 ′ )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

𝑡→ 𝑡 ′

Finite Size Effect

𝜉 (𝑥 )∝ 1√∆𝑉

Fluctuation 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

ConclusionPhysics of the quark gluon plasma• Strong coupling nature• Small viscosityPhysics of relativistic heavy ion collisions• Playground of relativistic non-equilibrium system• Relativistic dissipative hydrodynamics• Relativistic kinetic theory• Non-equilibrium field theory