A Calculation of Coefficients in a Relativistic Causal Hydrodynamics

1
A Calculation of Coefficients in a Relativistic
Causal Hydrodynamics

• Shin MUROYA
• Dept. of C.M.
• Matsumoto Univ.

2
Relativistic Hydrodynamical Equation
where
3
Phenomenological equations,
These equations are parabolic. Are these
consistent with relativity?
Relativistic Causal Hydrodynamics
4
Analyses using a Relativistic Causal
Hydrodynamics, I found before

• Second Order Dissipative Fluid Dynamics for
  Ultra-Relativistic Nuclear Collisions, Azwinndini
  Muronga, Phys.Rev.Lett. 88 (2002) 062302
  Erratum-ibid. 89 (2002) 159901 nucl-th/0104064
• Causal Theories of Dissipative Relativistic Fluid
  Dynamics for Nuclear Collisions, Azwinndini
  Muronga, Phys.Rev. C69 (2004) 034903
  nucl-th/0309055
• Title Evolution of Hot, Dissipative Quark Matter
  in Relativistic Nuclear Collisions, Azwinndini
  Muronga, Dirk H. Rischke, nucl-th/0407114
• Hydrodynamical evolution of dissipative QGP
  fluids, A. K. Chaudhuri, U. Heinz
  nucl-th/0504022
• Dissipative hydrodynamics for viscous
  relativistic fluids, Ulrich W. Heinz, Huichao
  Song, Asis K. Chaudhuri, Phys.Rev. C73 (2006)
  034904 nucl-th/0510014
• Dissipative Hydrodynamics and Heavy Ion
  Collisions, Rudolf Baier, Paul Romatschke, Urs
  Achim Wiedemann, Phys.Rev. C73 (2006) 064903
  hep-ph/0602249
• There may exist more recent ones.

5
Relativistic Causal HydrodynamicsDeveloped by
Israel Stewart
Israel, Ann of Phys. 100, 310(76) Israel and
Stewart, Ann of Phys. 118, 341(79)
Extended Navier-Stokes Equation
Hyperbolic equation
where
Higher order additional terms
6
If ai and bi are local quantities,how can we
calculate themby using microscopic dynamics?
If you want to use causal hydro., in addition to
ordinary h and k, new coefficients are needed!!
7
Coefficietns in causal hydro. eq.
must be

• Independent of boundary condition or solution.
• Local quantities as a function of local
  temperature, density, etc,
• because it is Hydro-model

We must be able to evaluate them based on
statistical Mechanics
Nakajima, Kubo,Yokota,Nakajima Zubarev,
8
Entropy Production
s p-ps
Thermo-dynamical force
Current
Jm
Xm
9
Linear response theory for non-equilibrium (Kubo
et al Zubarev)
where
Phenomenological equations,
10
Linear response theory for non-equilibrium (Kubo
et al Zubarev)
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
11
Linear response theory for non-equilibrium (Kubo
et al Zubarev)
Changes slowly
Expand thermo-dynamical forces as,

k, h(v), h(s)
New coefficients, ai, bi
Local hyperbolic equation
12
Then, we can obtain usual formulae
for local heat conductivity and local
heat viscosity
13
Coefficients related to the heat flow
14
phenomenological estimation, suppose
in local rest frame
if
etc.
Then
relaxation time of the current
15
relaxation time of the currents has already
appeared in the calculation of transport
coefficients as follows
16
relaxation of thermo-dynamical current of
hadronic gas by URASiMA
17
Hadro-Molecular Dynamic Simulationby using Event
GeneratorURASiMA
18
URASiMA 1.
(Ultra-Relativistic AA collision Simulator based
on Multiple Scattering Algorithm)
box with periodic boundary condition
hadronic picture
stationary state
relativistic
19
URASiMA 2.
common slope parameter for the distribution
function of different particles temperature
20
2-flavor low energy version
Baryons, mesons and their resonances included in
the present URASiMA.
Strangeness and anti-baryon is neglected
21
Equation of the state (e- P relation)
22
Linear Response Theory
23
transport coefficients of hadron gas
Shear Viscosity
24
?/s
25
heat conductivity
26
tk
27
ts
28
ltqmqmgt
29
ltpxy pxygt
30
heat conductivity
Shear Viscosity
tk fm
ts fm
31
bi / (1/P)
32
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
?