Transport coefficients of Relativistic Causal

2008年12月6日Heavy Ion Café
三題噺
QGP，流体モデル，巨視と微視
より
Shin MUROYA
Dept. of C.M.
Matsumoto Univ.
原子核どうしをぶつけています
URASiMAによる原子核衝突シミュレーション
(左)陽子・中性子の粒子数
（右）生成された中間子の粒子数
微視的モデルによる計算で十分か？
粒子の散乱描像でいいのか？
•強い相互作用は相互作用が強い
•massless グルーオンは長距離相関？
•粒子数が多い・・・
波動関数は重なっていないのか？
•反応の途中で“粒子”？？？漸近場？
•ＱＧＰとハドロンの間をどうする
•見たいのはＱＧＰ“相”
相補的な
モデル
流体モデル
定散
義乱
でを
き微
る視
の的
かに
流体モデルのパラメーター
状態方程式
ＱＧＰ相 u,d,s（m=0） + g
相転移(１次，その他)
ハドロン相 π, p, n, K,ρΔ，，
＋特殊効果
物性論的な
情報
local rest system
巨視的
微視的
平均化
完全流体モデル = エントロピー保存
K. Morita Symposium and Workshop on the Quark-Gluon Plasma and Heavy-Ion Physics at RHIC and LHC, 25 Jul 2003
Initial Conditions at t = 1.0 fm/c
• Longitudinal Flow : vz = z/t (YL=h )
(Scaling ansatz : Lorentz invariant solution – Only as
an initial condition!)
• Transverse Flow: neglected
E, nB Distributions
1.2A1/3
So tuned as to reproduce the experimental results.
Space-time evolutions of
Temperature
and
chemical Potential
by Ishii and Muroya PRC (’92)
Pressure distribution
by Nonaka, Honda and Muroya
Eur. Phys. J. C17(00)
Pressure Gradient
Expansion of the Fluid
横flow分布
K. Morita et al
Energy Density (Left:(x,y)-space, Right:(ux,uy)-space)
http://tkynt2.phys.s.u-tokyo.ac.jp/~hirano/
The above figures show, respectively, the hydrodynamic evolution of energy density in the transverse
(x,y) plane (left) and the energy in flow space (right) in Au+Au collisions with a finite impact
parameter (b=7.2fm) at √sNN = 200 GeV. The collision axis (z-axis) is assumed to be perpendicular
to the screen. Initial transverse flow is taken to be vanishing.
Please note that the dominant expansion of the system is directed to z-axis in relativistic heavy ion
collisions. This is the reason why the contours of the energy density apparently move inward.
体積要素の状態変化を追跡する
M. Asakawa and C. Nonaka
Nucl.Phys.A774:753-756,2006.
freeze-out
hyper surface
流体から粒子へ
保存流が連続になるように
Cooper-Frye formula
全粒子に共通
統計分布＋
流れ
local rest system
巨視的
微視的
平均化
完全流体モデル = エントロピー保存
Relativistic Hydrodynamical Equation
巨視的
where
Phenomenological equations,
These equations are parabolic.
Are these consistent with relativity?
高エネルギー極限で
Balk viscosity =0
かどうかに重要
Relativistic Causal Hydrodynamics
Relativistic Causal Hydrodynamics
Developed by Israel & Stewart
Israel, Ann of Phys. 100, 310(76)
Israel and Stewart, Ann of Phys. 118, 341(79)
Extended Navier-Stokes Equation
where
Hyperbolic equation
Higher order additional terms
If you want to use causal hydro.,
in addition to ordinary h and k,
new coefficients are needed!!
If ai and bi are local quantities,
how can we calculate them
by using microscopic dynamics?
輸送係数を決めたい
基本的な指針
•微分方程式の係数なので流れの境界条件や解によらない
•“流体モデル”なので熱力学量を通じてのみ時空性を持つ
かつ，時空座標の各点ごとに決まる局所的な量である
平衡系の統計力学的な計算で
Nakajima,
Kubo,Yokota,Nakajima
求められるはずである
Zubarev,
Entropy Production
Current
Jm
Thermo-dynamical force
Xm
s= p-ps
Thermo-dynamical forces
we may replace
these terms iteratively
higher derivative terms of Um and T
線形応答を使った輸送係数の計算の復習
非平衡密度行列の方法
局所平衡部分
平衡からのずれ
テンソルに分解
局所平衡分布
による期待値は等方的
平衡からのずれ部分
体積項へ
where
計算用メモ
に
をかける
体積項へ
Linear response theory for non-equilibrium (Kubo et al; Zubarev)
where
Phenomenological equations,
Linear response theory for non-equilibrium (Kubo et al; Zubarev)
微視的なサイズの相関距離・相関時間
微視的な統計力学の計算結果
If the change of thermo-dynamical force
is slow, we may move it outside of the integral.
If the correlation
does not vanish only in the small region.
local quantities
巨視的サイズの
変動
Linear response theory for non-equilibrium (Kubo et al; Zubarev)
Expand thermo-dynamical forces as,
Changes
slowly
k, h(v), h(s)
New coefficients, ai, bi
Local hyperbolic equation
微視的な相関距離・時間 vs 巨視的な変化の様子
流体モデルの診断
Then, we can obtain usual formulae
for local heat conductivity and
local viscosity
Coefficients
related to the
heat flow
relaxation of thermo-dynamical
current of hadronic gas by URASiMA
tau (pp) /fm
tk [fm]
Shear Viscosity
5
4
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
3
2
1
0
80
100
120
140
160
Temperature T /MeV
180
ハドロンの相関時間
～２－３ｆｍ
/ fm
ts [fm]
heat conductivity
5
4
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
3
2
1
0
80
100
120
140
160
Temperature T / MeV
180
QGPの相関は短いか？
k, h(v), h(s)
New coefficients, ai, bi
Local hyperbolic equation
微視的な相関距離・時間 vs 巨視的な変化の様子
流体モデルの診断
phenomenological estimation, suppose
in local rest frame
if
etc.
Then
relaxation time of
the current
relaxation time of the currents has already
appeared in the calculation of transport
coefficients as follows:
Hadro-Molecular
Dynamic Simulation
by using Event
Generator
URASiMA
URASiMA
1.
(Ultra-Relativistic AA collision Simulator based on Multiple Scattering Algorithm)
box with periodic
boundary condition
hadronic picture
stationary
state
relativistic
URASiMA 2.
common slope parameter
for the distribution
function of
different particles
“temperature”
2-flavor low energy version
Baryons, mesons and their resonances
included in the present URASiMA.
Strangeness and anti-baryon is neglected
Equation of the state (ε- P relation)
Linear
Response
Theory
transport coefficients of hadron gas
Shear Viscosity
η/s
heat conductivity
5
k /fm
tau t
(pp)
4
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
3
2
1
0
80
100
120
140
160
Temperature T /MeV
180
5
/ fm
3
ts
4
2
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
1
0
80
100
120
140
160
Temperature T / MeV
180
<q<mpp
qm>>
150
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
100
50
0
80
100
120
140
Temperature T /MeV
160
180
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
<p<xypippixy>>
20
10
0
120
140
Temperature
160
T/MeV
180
tau (pp) /fm
tk [fm]
Shear Viscosity
5
4
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
3
2
1
0
80
100
120
140
160
Temperature T /MeV
180
heat conductivity
/ fm
ts [fm]
5
4
NB=1.0*NB0
NB=1.5*NB0
NB=2.0*NB0
3
2
1
0
80
100
120
140
160
Temperature T / MeV
180
bi / (1/P)
Summary
• Causal Relativistic Hydrodynamics needs
8 transport coefficients.
– We give “Kubo-formula” for all coefficients
–
are calculated by Hadro-Molecular
calculation with URASiMA
2 fm
How can we calculate
these quantities for QGP ?

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