Viscous Hydrodynamics

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



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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



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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



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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



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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



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Ideal hydro Glauber1st order

Viscosity

Eq. of state

Initial cond.

theoretical prediction > experimental data

Lattice-based

Ideal hydro shows slight overshooting

Introduction



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Viscosity in QGP phase plays important role in reducing v2


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



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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



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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



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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



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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



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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



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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



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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



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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



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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



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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)




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Constitutive Equations Energy current

response to temperature gradient cross terms (linear)

relaxation term





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Constitutive Equations Charge currents

response to chemical gradient (+ cross terms) cross terms (linear)

relaxation terms





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Constitutive Equations Shear stress tensor

Discussion - Relaxation term

response to deformation

relaxation term


Linear response Acausal and unstable in relativistic systemsHiscock & Lindblom (‘85)

Relaxation effect to limit the propagation faster than the light speed

Discussion – Cross terms



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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



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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



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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



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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



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The End Thank you for listening!

Appendices

Documents

Viscous Hydrodynamics