Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano

Akihiko MonnaiDepartment of Physics, The University of Tokyo

Collaborator: Tetsufumi Hirano

Causal Viscous Hydrodynamics for 　Relativistic Systems with Multi-Components

and Multi-Conserved Currents

Strong and Electroweak Matter 2010June 29th 2010, McGill University, Montreal, Canada

Reference: AM and T. Hirano, arXiv:1003:3087

A power point template created by Akihiko MonnaiAkihiko Monnai (The University of Tokyo) , Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29th 2010

Outline1. Introduction

Relativistic hydrodynamics and Heavy ion collisions

2. Formulation of Viscous HydroIsrael-Stewart theory for multi-component/conserved current systems

3. Results and DiscussionConstitutive equations and their implications

4. SummarySummary and Outlook


Introduction Quark-Gluon Plasma (QGP) at Relativistic Heavy Ion Collisions

• RHIC experiments (2000-)

• LHC experiments (2009-)

T (GeV)Tc ~0.2

Hadron phase QGP phase

“Small” discrepancies; non-equilibrium effects?

Relativistic viscous hydrodynamic models are the key

Well-described in relativistic ideal hydrodynamic models

Asymptotic freedom -> Less strongly-coupled QGP?


Introduction Hydrodynamic modeling of heavy ion collisions

particles

hadronic phase

QGP phase

Freezeout surface Σ

Pre-equilibrium

Hydrodynamic picture

CGC/glasma picture?

Hadronic cascade picture

Initial condition

Hydro to particles

ｔ

z

t

Viscous hydro helps us …1. Explain the space-time evolution of the QGP2. Extract viscosities of the QGP from experimental data

Hydrodynamics works at the intermediate stage (~1-10 fm/c)


Introduction Fourier analyses of particle spectra from RHIC data

Hirano et al. (‘09)

Ideal hydro Ideal hydro

GlauberGlauber

1st order 1st order

Viscosity

Eq. of state

theoretical prediction experimental data

Ideal hydro works well

Initial condition





GlauberGlauber

1st order 1st order

Viscosity

Eq. of state


Lattice-based Lattice-based

Ideal hydro works… maybe

Initial condition

*EoS based on lattice QCD results



Viscous hydro in QGP plays important role in reducing v2



GlauberGlauber

Viscosity

Eq. of state

Initial condition


CGCCGC

Lattice-basedLattice-based

*Gluons in fast nuclei may form color glass condensate (CGC)


Introduction Formalism of viscous hydro is not settled yet:

1. Form of viscous hydro equations

2. Treatment of conserved currents

3. Treatment of multi-component systems

Important 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)

Multiple conserved currents?

We need to construct a firm framework of viscous hydro


Introduction Categorization of relativistic systems

QGP/hadronic gas at relativistic heavy ion collisions

Cf.

etc.


Introduction Relativistic hydrodynamics

Macroscopic theory defined on (3+1)-D spacetime

Flow (vector field)

Temperature (scalar field)

Chemical potentials (scalar fields)

and are described by the macroscopic fields

Gradient in the fields: thermodynamic forceResponse to the gradients: dissipative current


Overview

Energy-momentum conservationCharge conservationsLaw of increasing entropy

START

GOAL (constitutive eqs.)

Onsager reciprocal relations: satisfied

Moment equations,

Generalized Grad’s moment method

, , ,


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:


Relativistic Hydrodynamics Ideal hydrodynamics

Viscous hydrodynamics

　　 , , ,Conservation laws (4+N) + EoS (1)Unknowns (5+N)

, ,

We derive the equations from the law of increasing entropy

0th order theory1st order theory2nd order theory

ideal; no entropy production linear response; acausalrelaxation effects; causal

Additional unknowns (10+4N)　　 , , , , , !?Constitutive equations

“perturbation” from equilibrium


First Order Theory Kinetic expressions with distribution function :

The law of increasing entropy (1st order)

: degeneracy: conserved charge number


First Order Theory Linear response theory

The cross terms are symmetric due to Onsager reciprocal relations

Scalar

Vector

Tensor

conventional terms cross terms


First Order Theory Linear response theory

Vector

Dufour effect

Soret effect

potato

Thermal gradient

Permeation of ingredients

soup

Chemical diffusion caused by thermal gradient (Soret effect)

Cool　down　once – for cooking tasty oden (Japanese pot-au-feu)


Second Order Theory Causality Issues

Conventional formalism

Not extendable for multi-component/conserved current systems

one-component, elastic scattering

Israel & Stewart (‘79)

Dissipative currents (14)　　 , , , , ,

Moment equations (10)

frame fixing, stability conditions

(9) (9)?

Linear response theory implies instantaneous propagationLinear response theory implies instantaneous propagation

Relaxation effects are necessary for causalityRelaxation effects are necessary for causality


Extended Second Order Theory Moment equations

Expressions of andDetermined through the 2nd law of thermodynamics

Unknowns (10+4N)

　　 ,

Moment eqs. (10+4N)

,

New eqs. introduced

All viscous quantities determined in arbitrary frame

where

Off-equilibrium distribution is needed


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

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


Results 2nd order constitutive equations for systems with

multi-components and multi-conserved currentsBulk pressure

1st order terms

: relaxation times, : 1st, 2nd order transport coefficients

2nd order terms

relaxation


Results (Cont’d)Energy current

1st order terms

2nd order terms

Dufour effect

relaxation


Results (Cont’d)J-th charge current

1st order terms

2nd order terms

Soret effect

relaxation


Results (Cont’d)Shear stress tensor

Our results in the limit of single component/conserved current

1st order terms2nd order terms

Consistent with other results based onAdS/CFT approachRenormalization group methodGrad’s 14-moment method

Betz et al. (‘09)

Baier et al. (‘08)

Tsumura and Kunihiro (‘09)

relaxation


Discussion 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, but vorticity terms need further checking as some are added in their recent revision


Discussion Comparison with Grad’s 14-moment approach

• The form of their equations are consistent with that of ours• Our method can deal with multiple conserved currents while Grad’s 14-moment method does not

Betz et al. (‘09)

Consistency with other approaches suggest our multi-component/conserved current formalism is a natural extension


Summary and Outlook We formulated generalized 2nd order theory from the entropy

production w/o violating causality1. Multi-component systems with multiple conserved currents

Inelastic scattering (e.g. pair creation/annihilation) implied

2. Frame independentIndependent 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 the Quark-gluon plasma at

heavy ion collisions• Cosmological fluid etc…


The End Thank you for listening!


The Law of Increasing Entropy Linear response theory

Entropy production

: transport coefficient matrix (symmetric; semi-positive definite)

: dissipative current : thermodynamic force

Theorem: Symmetric matrices can be diagonalized with orthogonal matrix


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


First Order Limits 2nd order constitutive equations

transport coefficients(symmetric)

thermodynamics forces(2nd order)

1st order theory is recovered in the equilibrium limit

Onsager reciprocal relations are satisfied

Equilibrium limit

thermodynamic forces(Navier-Stokes)


Distortion of distribution Express in terms of dissipative currents

*Grad’s 14-moment method extended for multi-conserved current systems (Consistent with Onsager reciprocal relations)

Fix 　　　 and through matching

Viscous distortion tensor & vector

Dissipative currents, ,, ,

,,

: Matching matrices

Moment expansion with 10+4N unknowns ,

10+4N (macroscopic) self-consistency conditions

　　 ,


Second Order Equations Entropy production

Constitutive equations


: symmetric, semi-positive definite matrices

Dissipative currents Viscous distortion tensor & vector

Moment equations, ,, ,

,,


Matching matrices for dfi

Viscous distortion tensor & vector

Moment equations

,


Expanding universe

Einstein equation

with Robertson-Walker metric in flat space

because the R-W metric assumes isotropic and homogeneous expansion.

“Bulk viscous cosmological model”

Discussion

Only “ideal hydrodynamic” energy-momentum tensor is allowed:

Viscous correction to the pressure (= bulk pressure ) is allowed

Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano

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