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Akihiko MonnaiDepartment of Physics, The University of Tokyo
Collaborator: Tetsufumi Hirano
Viscous Hydrodynamics
Heavy Ion Meeting 2010-12December 10th 2010, Yonsei University, Korea
AM and T. Hirano, Nucl. Phys. A 847 (2010) 283
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Outline1. Introduction
Relativistic hydrodynamics and heavy ion collisions
2. Relativistic Dissipative HydrodynamicsExtended Israel-Stewart theory from law of increasing entropy
3. Results and DiscussionConstitutive equations in multi-component/conserved current systems
4. SummarySummary and Outlook
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Quark-gluon plasma (QGP) at relativistic heavy ion collisions
Hadron phase QGP phase
(crossover) sQGP (wQGP?)
Discovery of QGP as nearly perfect fluid
RHIC experiments (2000-)
Thermodynamics is at work in QGP at RHIC energies
Non-equilibrium effects need investigation for quantitative understandings
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Hydrodynamic modeling of RHIC
particles
hadronic phase
QGP phase
Freezeout
Pre-equilibrium
Hydrodynamic picture
CGC/glasma picture?
Hadronic cascade picture
Initial condition
Hydro to particles
t
z
t
• Intermediate stage (~1-10 fm) is described by hydrodynamics• Results are dependent on the inputs:
Hydrodynamic equationsInitial conditions Output
Equation of state, Transport coefficients
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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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
Shear viscosity = response to deformation
Bulk viscosity= response to
expansion
Energy dissipation= response to
thermal gradient
Charge dissipation= response to
chemical gradients
Naïve interpretation of dissipative processes
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Ideal hydro Glauber1st order
Viscosity
Eq. of state
Initial cond.
theoretical prediction ~ experimental dataIdeal hydro works well
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Ideal hydro Glauber1st order
Viscosity
Eq. of state
Initial cond.
theoretical prediction > experimental data
Lattice-based
Ideal hydro shows slight overshooting
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Elliptic flow coefficients from RHIC data
Viscosity in QGP phase plays important role in reducing v2
Hirano et al. (‘09)
Ideal hydro Glauber
Viscosity
Eq. of state
Initial cond.
theoretical prediction > experimental data
Color glass condensate
Lattice-based
*Gluons in fast moving nuclei are saturated to CGC
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Why viscous hydrodynamic models?
RHIC experiments (2000-)Success of ideal hydro
for improved inputs to the hydrodynamic models
Necessity of viscous hydro
QGP might become less-strongly coupled
LHC experiments (2010-)Asymptotic freedom in QCD Viscous hydro?
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
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Viscous hydrodynamics needs improvement
1. Form of dissipative hydro equations
2. Treatment of conserved currents
3. Treatment of multi-component systems
Fixing the equations is essential in fine-tuning viscosity from experimental data
# of conserved currents # of particle speciesbaryon number, strangeness, etc. pion, proton, quarks, gluons,
etc.
Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR)
Only 1 conserved current can be treated
We need to construct a firm framework of viscous hydro
Song & Heinz (‘08)
Introduction
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Categorization of relativistic hydrodynamic formalisms
Number of components
Types of interactions
Single component with binary collisions
Israel & Stewart (‘79), etc…
Multi-components with binary collisionsPrakash et al. (‘91)
Single component with inelastic scatterings
(-)
Multi-components with inelastic scatterings
(-)
Required for QGP/hadron gas at heavy ion collisions
Overview
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Introduction Categorization of relativistic hydrodynamic formalisms
Number of components
Types of interactions
Single component with binary collisions
Israel & Stewart (‘79), etc…
Multi-components with binary collisionsPrakash et al. (‘91)
Single component with inelastic scatterings
(-)
Multi-components with inelastic scatteringsMonnai & 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
Overview
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Overview Formulation of relativistic dissipative hydrodynamics
Energy-momentum conservationCharge conservationsLaw of increasing entropy
START
GOAL
Onsager reciprocal relations satisfied
Moment eqs.,
Generalized moment method
EoM for dissipative currents , , ,
Thermodynamics Quantities
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Thermodynamic Quantities Tensor decompositions by flow
where is the projection operator
10+4N dissipative currents2+N equilibrium quantities
*Stability conditions should be considered afterward
Energy density deviation:Bulk pressure:
Energy current:Shear stress tensor:
J-th charge density dev.:J-th charge current:
Energy density:Hydrostatic pressure:
J-th charge density:
Thermodynamic Stability
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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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
Relativistic Hydrodynamics
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Relativistic Hydrodynamics Ideal hydrodynamics
Dissipative hydrodynamics (“perturbation” from equilibrium)
, , ,Conservation laws (4+N) + EoS(1)Unknowns (5+N)
, ,
Defined in relativistic kinetic theory as
Additional unknowns (10+4N) , , , , , !?Moment equations (10+4N)
,
Estimated from the law of increasing entropy
New equations in our work
Moment Equations
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Moment Equations Introduce distortion of distribution
*Grad’s moment method extended to multi-conserved current systems
Dissipative currents Viscous distortion tensor & vector
Moment equations, ,, ,
,,
Semi-positive definite condition
Matching matrices for dfi
10+4N unknowns , are determined in self-consistency conditions
The entropy production is expressed in terms of and
Constitutive Equations
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Constitutive Equations Bulk Pressure
response to expansion cross terms (linear)
relaxation term
2nd order corrections
Cross terms appear (reciprocal relations)
Relaxation term appears (causality is preserved)2nd order terms in full form (multi-conserved currents)
Constitutive Equations
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Constitutive Equations Energy current
response to temperature gradient cross terms (linear)
relaxation term
2nd order corrections
Constitutive Equations
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Constitutive Equations Charge currents
response to chemical gradient (+ cross terms) cross terms (linear)
relaxation terms
2nd order corrections
Constitutive Equations
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Constitutive Equations Shear stress tensor
Discussion - Relaxation term
response to deformation
relaxation term
2nd order corrections
Linear response Acausal and unstable in relativistic systemsHiscock & Lindblom (‘85)
Relaxation effect to limit the propagation faster than the light speed
Discussion – Cross terms
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Discussion - Cross terms Coupling of thermodynamic forces in the dissipative currents
potato
thermal gradient
ingreadients
soup
Chemical diffusion via thermal gradient
Cf: “Cooling” process for cooking tasty oden (Japanese soup)
It should play an important role in our “quark soup”
Soret effect
ex.
Onsager reciprocal relations ( ) is satisfied
Discussion – 2nd order terms
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Discussion – 2nd order terms Comparison with AdS/CFT+phenomenological approach
Comparison with Renormalization group approach
• Our approach goes beyond the limit of conformal theory • Vorticity-vorticity terms do not appear in kinetic theory
Baier et al. (‘08)
Tsumura & Kunihiro (‘09)
• Consistent, as vorticity terms are added in their recent revision• Frame-dependent equations
Discussion – 2nd order terms
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Discussion – 2nd order terms Comparison with Grad’s 14-moment approach
• The form of their equations are consistent with that of ours• Multiple conserved currents are not supported in 14-moment method
Betz et al. (‘09)
Consistencies suggest we have successfully extended 2nd order theory to multi-component + conserved current systems
Summary and Outlook
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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Summary and Outlook We formulated generalized 2nd order dissipative hydro from the
entropy production w/o violating causality1. Multi-component systems with multiple conserved currents
Inelastic scattering (e.g. pair creation/annihilation) included
2. 1st order cross terms are presentOnsager reciprocal relations are satisfied
3. Frame independentIndependent equations for energy and charge currents
Future prospects include applications to…• Numerical estimation of viscous hydrodynamic models for
relativistic heavy ion collisions• Cosmological fluid, and more
AM & T. Hirano, in preparation
Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea
Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics
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The End Thank you for listening!
Appendices