スライド 1 - University of Tokyo
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AM and T. Hirano, Nucl. Phys. A 847 (2010) 283
Viscous Hydrodynamics
Akihiko Monnai
Department of Physics, The University of Tokyo
Collaborator: Tetsufumi Hirano
Heavy Ion Meeting 2010-12
December 10th 2010, Yonsei University, Korea
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Outline
1.
Introduction
Relativistic hydrodynamics and heavy ion collisions
2.
Relativistic Dissipative Hydrodynamics
Extended Israel-Stewart theory from law of increasing entropy
3.
Results and Discussion
Constitutive equations in multi-component/conserved current systems
4.
Summary
Summary and Outlook
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Quark-gluon plasma (QGP) at relativistic heavy ion collisions
Hadron phase
(crossover)
sQGP
QGP phase
(wQGP?)
RHIC experiments (2000-)
Discovery of QGP as nearly perfect fluid
Thermodynamics is at work in QGP at
RHIC energies
Non-equilibrium effects need investigation
for quantitative understandings
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Hydrodynamic modeling of RHIC
particles
t
t
Freezeout
Hadronic cascade picture
Hydro to particles
hadronic
phase
QGP phase
Hydrodynamic picture
Preequilibrium
z
•
•
Initial condition
CGC/glasma picture?
Intermediate stage (~1-10 fm) is described by hydrodynamics
Results are dependent on the inputs:
Equation of state, Transport coefficients
Initial conditions
Hydrodynamic equations
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
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Output
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Properties of QCD fluid
Equation of state: relation among thermodynamic variables
sensitive to degrees of freedom in the system
Transport coefficients: responses to thermodynamic forces
sensitive to interaction in the system
Naïve interpretation of dissipative processes
Shear viscosity
= response to
deformation
Bulk viscosity
= response to
expansion
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Energy dissipation
= response to
thermal gradient
Next slide:
Charge dissipation
= response to
chemical gradients
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Viscosity
Initial cond.
Ideal hydro
Glauber
Eq. of state
1st order
theoretical prediction ~ experimental data
Ideal hydro works well
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Viscosity
Initial cond.
Ideal hydro
Glauber
Eq. of state
Lattice-based
1st order
theoretical prediction > experimental data
Ideal hydro shows slight overshooting
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Viscosity
Initial cond.
Ideal hydro
Color glass
condensate
Glauber
Eq. of state
Lattice-based
*Gluons in fast moving nuclei are
saturated to CGC
theoretical prediction > experimental data
Viscosity in QGP phase plays important role in reducing v2
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Why viscous hydrodynamic models?
RHIC experiments (2000-)
Success of ideal hydro
Necessity of viscous hydro
for improved inputs to the hydrodynamic models
LHC experiments (2010-)
Asymptotic freedom in QCD
Viscous hydro?
QGP might become less-strongly coupled
CERN Press release, November 26, 2010:
“… confirms that the much hotter plasma produced at
the LHC behaves as a very low viscosity liquid”
Viscous hydro is likely to work also at the LHC energies
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Viscous hydrodynamics needs improvement
1. Form of dissipative hydro equations
Song & Heinz (‘08)
Fixing the equations is essential in finetuning viscosity from experimental data
2.
Treatment of conserved currents
Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR)
Only 1 conserved current can be treated
3.
Treatment of multi-component systems
# of conserved currents
baryon number, strangeness, etc.
# of particle species
pion, proton, quarks, gluons, etc.
We need to construct a firm framework of viscous hydro
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Introduction
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Categorization of relativistic hydrodynamic formalisms
Types of interactions
Number of components
Single component with
binary collisions
Multi-components with
binary collisions
Israel & Stewart (‘79), etc…
Prakash et al. (‘91)
Single component with
inelastic scatterings
Multi-components with
inelastic scatterings
(-)
(-)
Required for QGP/hadron gas at heavy ion collisions
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Overview
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Introduction
Categorization of relativistic hydrodynamic formalisms
Types of interactions
Number of components
Single component with
binary collisions
Multi-components with
binary collisions
Israel & Stewart (‘79), etc…
Prakash et al. (‘91)
Single component with
inelastic scatterings
Multi-components with
inelastic scatterings
(-)
Monnai & Hirano (‘10)
Required for QGP/hadron gas at heavy ion collisions
In this work we formulate relativistic dissipative hydro
for multi-component + multi-conserved current systems
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Overview
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Overview
Formulation of relativistic dissipative hydrodynamics
START
Energy-momentum conservation
Charge conservations
Law of increasing entropy
Moment eqs.
,
Generalized moment method
GOAL
EoM for dissipative currents
,
,
,
Onsager reciprocal relations satisfied
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
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Thermodynamics Quantities
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Thermodynamic Quantities
Tensor decompositions by flow
where
is the projection operator
2+N equilibrium quantities
Energy density:
Hydrostatic pressure:
J-th charge density:
*Stability conditions
should be considered afterward
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
10+4N dissipative currents
Energy density deviation:
Bulk pressure:
Energy current:
Shear stress tensor:
J-th charge density dev.:
J-th charge current:
Next slide:
Thermodynamic Stability
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Thermodynamic Stability
Maximum entropy state condition
- Stability condition (1st order)
- Stability condition (2nd order)
automatically satisfied in kinetic theory
*Stability conditions are NOT the same as the law of increasing entropy
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Relativistic Hydrodynamics
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Relativistic Hydrodynamics
Ideal hydrodynamics
Unknowns (5+N)
,
,
,
Conservation laws (4+N) + EoS(1)
,
,
Dissipative hydrodynamics (“perturbation” from equilibrium)
Additional unknowns (10+4N)
,
,
,
,
,
Moment equations (10+4N)
!?,
Defined in relativistic kinetic theory as
New equations
in our work
Estimated from the law of increasing entropy
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
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Moment Equations
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Moment Equations
Introduce distortion of distribution
*Grad’s moment method extended to multi-conserved current systems
10+4N unknowns
,
are determined in self-consistency conditions
The entropy production is expressed in terms of
Dissipative currents
,
,
,
,
Viscous distortion
tensor & vector
and
Moment equations
,
,
Matching matrices
for dfi
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Semi-positive
definite condition
Next slide:
Constitutive Equations
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Constitutive Equations
Bulk Pressure
response to expansion
cross terms (linear)
2nd order corrections
relaxation term
Cross terms appear (reciprocal relations)
2nd order terms in full form (multi-conserved currents)
Relaxation term appears (causality is preserved)
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Constitutive Equations
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Constitutive Equations
Energy current
response to temperature gradient
cross terms (linear)
2nd order corrections
relaxation term
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
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Constitutive Equations
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Constitutive Equations
Charge currents
response to chemical gradient (+ cross terms)
cross terms (linear)
2nd order corrections
relaxation terms
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
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Constitutive Equations
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Constitutive Equations
Shear stress tensor
response to deformation
2nd order corrections
relaxation term
Discussion - Relaxation term
Linear response
Hiscock & Lindblom (‘85)
Acausal and unstable in relativistic systems
Relaxation effect to limit the propagation faster
than the light speed
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Discussion – Cross terms
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Discussion - Cross terms
Coupling of thermodynamic forces in the dissipative currents
ex.
Onsager reciprocal relations (
)is satisfied
Cf: “Cooling” process for cooking tasty oden (Japanese soup)
ingreadients
potato
soup
thermal gradient
Chemical diffusion via thermal gradient
Soret effect
It should play an important role in our “quark soup”
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Discussion – 2nd order terms
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Discussion – 2nd order terms
Comparison with AdS/CFT+phenomenological approach
Baier et al. (‘08)
• Our approach goes beyond the limit of conformal theory
• Vorticity-vorticity terms do not appear in kinetic theory
Comparison with Renormalization group approach
Tsumura & Kunihiro (‘09)
• Consistent, as vorticity terms are added in their recent revision
• Frame-dependent equations
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Discussion – 2nd order terms
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Discussion – 2nd order terms
Comparison with Grad’s 14-moment approach
Betz et al. (‘09)
• The form of their equations are consistent with that of ours
• Multiple conserved currents are not supported in 14-moment method
Consistencies suggest we have successfully extended 2nd order
theory to multi-component + conserved current systems
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Summary and Outlook
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
Summary and Outlook
We formulated generalized 2nd order dissipative hydro from the
entropy production w/o violating causality
1. Multi-component systems with multiple conserved currents
Inelastic scattering (e.g. pair creation/annihilation) included
2.
1st order cross terms are present
Onsager reciprocal relations are satisfied
3.
Frame independent
Independent equations for energy and charge currents
Future prospects include applications to…
• Numerical estimation of viscous hydrodynamic models for
AM & T. Hirano, in preparation
relativistic heavy ion collisions
• Cosmological fluid, and more
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Viscous Hydrodynamics
Akihiko Monnai (The University of Tokyo)
The End
Thank you for listening!
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Next slide:
Appendices