Transcript スライド 1 - University of Tokyo

Reference: AM and T. Hirano, arXiv:1003:3087
Relativistic Viscous Hydrodynamics for
Multi-Component Systems with
Multiple Conserved Currents
Akihiko Monnai
Department of Physics, The University of Tokyo
Collaborator: Tetsufumi Hirano
Hot Quarks 2010
June 25th 2010, La Londe-Les-Maures, France
Outline
1.
Introduction
Relativistic hydrodynamics and Heavy ion collisions
2.
Formulation of Viscous Hydro
Israel-Stewart theory for multi-component/conserved current systems
3.
Results and Discussion
Constitutive equations and their implications
4.
Summary
Summary and Outlook
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Quark-Gluon Plasma (QGP) at Relativistic Heavy Ion Collisions
Hadron phase
QGP phase
T (GeV)
Tc ~0.2
•
RHIC experiments (2000-)
Well-described in relativistic ideal hydrodynamic models
“Small” discrepancies; non-equilibrium effects?
•
LHC experiments (2009-)
Asymptotic freedom -> Less strongly-coupled QGP?
Relativistic viscous hydrodynamic models are the key
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Hydrodynamic modeling of heavy ion collisions for RHIC
particles
ｔ
t
Freezeout surface Σ
Hadronic cascade picture
Hydro to particles
hadronic
phase
QGP phase
Hydrodynamic picture
Preequilibrium
z
Initial condition
CGC/glasma picture?
Hydrodynamics works at the intermediate stage (~1-10 fm/c)
Purpose of Viscous Hydro:
1. Explaining the space-time evolution of the QGP
2. Constraining the parameters (viscosities, etc.) from experimental data
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Viscosity
theoretical prediction
Initial condition
Ideal hydro
Glauber
Eq. of state
1st order
experimental data
Ideal hydro works well
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Viscosity
Initial condition
Ideal hydro
Glauber
Eq. of state
Lattice-based
1st order
*EoS based on lattice QCD results
theoretical prediction
experimental data
Ideal hydro works… maybe
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Elliptic flow coefficients from RHIC data
Hirano et al. (‘09)
Viscosity
Initial condition
Ideal hydro
CGC
Glauber
Eq. of state
Lattice-based
*Gluons in fast nuclei may form
color glass condensate (CGC)
theoretical prediction
experimental data
Viscous hydro in QGP plays important role in reducing v2
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Formalism of viscous hydro is not settled yet:
Song & Heinz (‘08)
1. Form of viscous hydro equations
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)
Multiple conserved currents?
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
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Categorization of relativistic systems
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)
QGP/hadronic gas at heavy ion collisions
Cf.
etc.
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Overview
START
Energy-momentum conservation
Charge conservations
Law of increasing entropy
Moment equations
,
Generalized Grad’s moment method
GOAL (constitutive eqs.)
,
,
,
Onsager reciprocal relations: satisfied
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Thermodynamic Quantities

Tensor decompositions by flow
where
2+N equilibrium quantities
Energy density:
Hydrostatic pressure:
J-th charge density:
*Stability conditions
should be considered afterward
is the projection operator
10+4N dissipative currents
Energy density deviation:
Bulk pressure:
Energy current:
Shear stress tensor:
J-th charge density dev.:
J-th charge current:
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Relativistic Hydrodynamics

Ideal hydrodynamics
Unknowns（5+N）
,
,
,

Conservation laws（4+N）+ EoS(1)
,
,
Viscous hydrodynamics
Additional unknowns（10+4N）
,
,
,
,
,
Constitutive
!? equations
We derive the equations from the law of increasing entropy
“perturbation” from equilibrium
0th order theory
1st order theory
2nd order theory
ideal; no entropy production
linear response; acausal
relaxation effects; causal
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Second Order Theory

Kinetic expressions with distribution function
:
: degeneracy
: conserved charge number

Conventional formalism
Dissipative currents (14)
(9)
,
,
,
,
,
frame fixing, stability conditions
Israel & Stewart (‘79)
?
Moment equations (10)
(9)
one-component, elastic scattering
Not extendable for multi-component/conserved current systems
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Extended Second Order Theory

Moment equations
New eqs.
introduced
Unknowns (10+4N)
Moment eqs. (10+4N)
,
,
All viscous quantities determined in arbitrary frame

Expressions of
and
Determined through the 2nd law of thermodynamics
where
Off-equilibrium distribution is needed
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Extended Second Order Theory

Moment expansion
*Grad’s 14-moment method extended for multi-conserved current systems
so that it is consistent with Onsager reciprocal relations
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
Semi-positive
definite condition
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Results

2nd order constitutive equations for systems with
multi-components and multi-conserved currents
Bulk pressure
1st order terms
2nd order terms
relaxation
: relaxation times
, : 1st, 2nd order transport coefficients
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Results

(Cont’d)
Energy current
1st order terms
2nd order terms
relaxation
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Results

(Cont’d)
J-th charge current
1st order terms
2nd order terms
relaxation
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Onsager Cross Effects

1st order terms
Dufour effect
Vector
Soret effect

Cool down once – for cooking tasty oden (Japanese pot-au-feu)
Permeation of ingredients
potato
soup
Thermal gradient
Chemical diffusion caused by thermal gradient (Soret effect)
It may well be important in our relativistic soup of quarks and gluons
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Results

(Cont’d)
Shear stress tensor
1st order terms
2nd order terms
relaxation

Our results in the limit of single component/conserved current
Consistent with other results based on
Baier et al. (‘08)
AdS/CFT approach
Renormalization group method Tsumura and Kunihiro (‘09)
Betz et al. (‘09)
Grad’s 14-moment method
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Discussion

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, but vorticity terms need further checking as some are
added in their recent revision
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Discussion

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
Consistency with other approaches suggest our multicomponent/conserved current formalism is a natural extension
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Summary and Outlook

We formulated generalized 2nd order theory from the entropy
production w/o violating causality
1. Multi-component systems with multiple conserved currents
Inelastic scattering (e.g. pair creation/annihilation) implied
2.
Frame independent
Independent equations for energy and charge currents
3.
Onsager reciprocal relations (
1st order theory)
Justifies the moment expansion

Future prospects include applications to…
• Hydrodynamic modeling of Quark-gluon plasma at relativistic
heavy ion collisions
• Cosmological fluid etc…
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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The End

Thank you for listening!
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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The Law of Increasing Entropy

Linear response theory
: dissipative current
: thermodynamic force
: transport coefficient matrix (symmetric; semi-positive definite)

Entropy production
Theorem: Symmetric matrices
orthogonal matrix
can be diagonalized with
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Thermodynamic Stability

Maximum entropy state condition
- Stability condition (1st order)
- Stability condition (2nd order)
Preserved for any
*Stability conditions are NOT the same as the law of increasing entropy
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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First Order Limits

2nd order constitutive equations
transport coefficients thermodynamics forces
(symmetric) (2nd order)
Equilibrium limit
thermodynamic forces
(Navier-Stokes)

Onsager reciprocal relations are satisfied

1st order theory is recovered in the equilibrium limit
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Hydrodynamic modeling of heavy ion collisions for RHIC
particles
ｔ
t
Freezeout surface Σ
Hadronic cascade picture
Hydro to particles
hadronic
phase
QGP phase
Hydrodynamic picture
Preequilibrium
z
Initial condition
CGC/glasma picture?
Hydrodynamic model requires
Initial condition
Equation of state
Glauber, color glass condensate, etc.
1st order phase transition, crossover(lattice), etc.
Viscosity
Ideal hydro, viscous hydro
Hadronization
JAM, UrQMD, etc.
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Introduction

Relativistic hydrodynamics
Macroscopic theory defined on (3+1)-D spacetime
Flow
(vector field)
Temperature
(scalar field)
Chemical potentials
(scalar fields)
Physics should be described by the macroscopic fields only
Gradient in the fields: thermodynamic force
Response to the gradients: dissipative current
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Distortion of distribution

Express
in terms of dissipative currents
10+4N (macroscopic) self-consistency conditions
,
Fix
and
Moment expansion with 10+4N unknowns
through matching
,
*Grad’s 14-moment method extended for multi-conserved current systems
(Consistent with Onsager reciprocal relations)
Dissipative currents
,
,
,
,
Viscous distortion
tensor & vector
,
,
: Matching matrices
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
A power point template created by Akihiko Monnai
Second Order Equations

Entropy production
Semi-positive definite condition
Viscous distortion
tensor & vector
Moment equations
,
: symmetric, semi-positive definite matrices

Constitutive equations
Dissipative currents
,
,
,
,
Viscous distortion
tensor & vector
Moment equations
,
,
Matching matrices
for dfi
Semi-positive
definite condition
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
A power point template created by Akihiko Monnai
Discussion

Comparison with AdS/CFT+phenomenological approach
Shear stress tensor
in conformal limit, no charge current
Baier et al. (‘08)
(Our equations)
Mostly consistent w ideal hydro relation
• Our approach goes beyond the limit of conformal theory
• Vorticity-vorticity terms do not appear in kinetic theory
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
A power point template created by Akihiko Monnai
Discussion

Comparison with Renormalization group approach
in energy frame, in single component/conserved current system
Tsumura & Kunihiro (‘09)
(Our equations)
Form of the equations agrees with our equations in the
single component & conserved current limit w/o vorticity
Note: Vorticity terms added to their equations in recent revision
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
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Discussion

Comparison with Grad’s 14-momemt approach
in energy frame, in single component/conserved current system
Betz et al. (‘09)
*Ideal hydro relations in use for comparison
equations)
Form of the equations agrees with ours in the(Our
single
component & conserved current limit
Consistency with other approaches suggest our multicomponent/conserved current formalism is a natural extension
Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 2010
A power point template created by Akihiko Monnai