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Evaluation of Israel-Stewart parameters in lattice gauge theory. KEK 理論センター研究会 『 原子核・ハドロン物理 』 Aug 11-13, 2009. Yasuhiro Kohno (Osaka University) M. Asakawa 1 , M. Kitazawa 1 , C. Nonaka 2 , S. Pratt 3 1 Osaka Univ. 2 Nagoya Univ. 3 Michigan State Univ. Contents. 1. Introduction

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evaluation of israel stewart parameters in lattice gauge theory

Evaluation of Israel-Stewart parameters in lattice gauge theory

KEK理論センター研究会

『原子核・ハドロン物理』

Aug 11-13, 2009

Yasuhiro Kohno

(Osaka University)

M. Asakawa1, M. Kitazawa1, C. Nonaka2, S. Pratt3

1Osaka Univ. 2Nagoya Univ. 3Michigan State Univ.

contents
Contents

1. Introduction

2. Strategy

3. Numerical Results

4. Summary

contents1
Contents

1. Introduction

2. Strategy

3. Numerical Results

4. Summary

slide4

クォーク・グルーオン・プラズマ(QGP)の

物性・時空発展および非平衡現象

  • クォーク・ハドロンの世界
slide5

RHIC Scientists Serve Up “Perfect” Liquid

New state of matter more remarkable than predicted

-- raising many new questions April 18,2005

  • 重イオン衝突実験@RHIC

QGP(near TC)≒完全流体(?)

QGPの時空発展は相対論的流体力学で記述できる

強結合QGP

強相関QGP

Lattice QCD ○

摂動論 ×

輸送係数(粘性係数etc.)に着目

Lattice QCDで輸送係数を数値計算

slide6

相対論的流体力学

  • 1st order theories for dissipative fluid (by Eckart or Landau & Lifshitz) ⇒散逸の効果を1次まで取り入れる

entropy current

s: entropy density , uμ: 4-velocity , T: temprature

散逸量の輸送方程式⇒因果律 ×

輸送係数:ζ(bulk viscosity),κ(heat conductivity),

η(shear viscosity)

散逸項(qμ: 熱流)

C. Eckart, Phys. Rev. 58, 919 (1940)

L. D .Landau and E. M. Lifshitz, Fluid Mechanics (1959)

slide7

相対論的流体力学

  • 2nd order theory for dissipative fluid (by Muller or Israel & Stewart) ⇒1st order theoryに緩和時間を導入

entropy current

緩和時間(τi→0で2nd order→1st order)

散逸量の輸送方程式⇒ 因果律○(ただし例外有り)

輸送係数 : ζ, κ, η, α0, α1, β0, β1, β2

散逸項

I.Muller, Z. Phys. 198, 329 (1967)

W.Israel and J.M.Stewart, Ann. Phys. (N.Y.) 118, 341 (1979)

slide8

先行研究

  • Using Kubo formula with ansatz for spectral function.

But the validity remains questionable.

Analytic continuation

Lattice QCD

Viscosities

Kubo formula

Imaginary time correlator

Real time correlator

?

F. Karsch and H. W. Wyld, Phys. Rev. D35, 2518(1987)

A. Nakamura and S. Sakai, Phys. Rev. Lett. 94, 072305(2005)

H. B. Meyer, Phys. Rev. D76, 101701(2007)

slide9

Evaluation of the ratios of the viscosities to the relaxation times of Israel-Stewart (IS) theory in SU(3) lattice QCD.

  • 研究方針

Reduce the number of IS parameters

We try to obtainsecond order coefficients β0&β2.

contents2
Contents

1. Introduction

2. Strategy

3. Numerical Results

4. Summary

slide11

Israel-Stewart entropy

usingthese relations

and

・・・(1)

  • Israel-Stewart entropy

uμ:4-velocity of particles

Seq: equilibrium entropy , qμ: heat flux

Π : bulk viscous pressure , πμν: shear viscous pressure

slide12

 平衡状態におけるゆらぎの確率分布はBoltzmann-Einsteinの原理に従う 平衡状態におけるゆらぎの確率分布はBoltzmann-Einsteinの原理に従う

 状態変数a=a0の状態が実現される確率は

Equation (1) と (2)より

  • Boltzmann-Einsteinの原理

c.f. S=logW

・・・(2)

A. Muronga, Eur. Phys. J. ST 155:107-113(2008)

S. Pratt, Phys. Rev. C77, 024910(2008)

slide13

期待される分布

  • Lattice QCDでやること
  • π13のゆらぎの確率分布を数値計算する
  • π13の分布とequation (3)を比較してβ2を得る

・・・(3)

π13

BE principle

Probability of fluctuations

The ratios between IS parameters

IS entropy

Lattice QCD

slide14

イメージ

  • 4次元Euclid空間の格子

・・・

Configuration = 微視状態

・・・確率 1/6

π13

・・・確率∝ exp[-Vβ2π132/2T]

contents3
Contents

1. Introduction

2. Strategy

3. Numerical Results

4. Summary

slide16

Lattice parameters

  • SU(3) pure gauge theory (gluon only)
  • 3 isotropic lattice boxes
  • 10,000 configurations for each box
  • Blue Gene @ KEK

β = 2NC/g2

a: lattice spacing

TC: critical temperature (~300MeV)

Nτ: number of sites in spatial direction

NS: number of sites in temporal direction

slide19

Result (The ratio β2)

  • Our present result with box1
  • Characteristic velocity of dissipative flow

From AdS/CFT

From our result

R. Baier, P. Robatschke, D. T. Son, A. O. Starinets

and M. A. Stephanov, JHEP 0804:100 (2008)

⇒因果律○

ε : energy density

P : pressure

⇒因果律 ×

contents4
Contents

1. Introduction

2. Strategy

3. Numerical Results

4. Summary

slide21

Summary

  • Lattice QCDによる散逸量(π13)のゆらぎの確率分布の数値計算を行った。
  • Boltzmann-Einsteinの原理に基づき、Israel-Stewart (2nd order)理論の枠組み内で粘性係数と緩和時間の比(β2)を導出した。
  • Lattice QCDからの結果からは、Israel-Stewart(2nd order)理論は因果律を破る(?)

⇒AdS/CFTからの結果と矛盾…

  • Future plan
  • β0=τΠ/Πの導出(box1)
  • その他のLattice(box2,box3)のデータの解析
  • AdS/CFTとの矛盾を議論
slide24

Result (Spatial correlation)

Lattice spacing dependence of π13

slide25

Shear viscosity from perturbation theory

In high temperature region

P. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0011 001 (2000)