akihiko monnai department of physics, the university of tokyo collaborator: tetsufumi hirano

Akihiko MonnaiDepartment of Physics, The University of Tokyo

Collaborator: Tetsufumi Hirano

Viscous Hydrodynamics for Relativistic Systems with Multi-Components and

Multiple Conserved Currents

Berkeley School of Collective Dynamics in High Energy CollisionsJune 10th 2010, Lawrence Berkeley National Laboratory, USA

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

A power point template created by Akihiko MonnaiAkihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10th 2010

Outline1. Introduction

Relativistic hydrodynamics and Heavy ion collisions

2. Relativistic Viscous HydrodynamicsExtended Israel-Stewart theory and Distortion of distribution

3. Results and DiscussionConstitutive equations in multi-component/conserved current systems

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 Elliptic flow coefficients from RHIC data

Hirano et al. (‘06)

Ideal hydro + CGC initial condition > experimental data

Viscous hydro in QGP plays important role in reducing v2

Hirano et al. (‘09)

Ideal hydro + lattice EoS > experimental data


Introduction Formalism of viscous hydro is not complete yet:

1. Form of viscous hydro equations

2. Treatment of conserved currents

3. Treatment of multi-component systems

We need to construct a firm framework of viscous hydro

Israel & Stewart (‘79) Muronga (‘02) Betz et al. (‘09)e.g. …

Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR)Multi-conserved current systems are not supported

# of conserved currents # of particle speciesbaryon number, strangeness, etc. pion, proton, quarks, gluons,

etc.


Introduction Categorization of relativistic systems

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)

QGP/hadronic gas at heavy ion collisions

Cf.

etc.


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 (“perturbation” from equilibrium)

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

Additional unknowns （ 10+4N ） :　　 , , , , ,

Constitutive equations are necessary

the law of increasing entropyIrreversible processes

0th order theory1st order theory2nd order theory

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


First Order Theory Kinetic expressions with distribution :

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 -> 9 constitutive eqs.frame fixing, stability conditions -> 9 unknowns

Israel & Stewart (‘79)

Linear response theory implies instantaneous propagation

Relaxation 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


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-consistent conditions

　　 ,


Second Order Equations Entropy production

Constitutive equations

Semi-positive definite condition

: symmetric, semi-positive definite matrices

Dissipative currents Viscous distortion tensor & vector

Moment equations, ,, ,

,,

Semi-positive definite condition

Matching matrices for dfi

Viscous distortion tensor & vector

Moment equations

,


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

• Our approach goes beyond the limit of conformal theory • Vorticity-vorticity terms do not appear in kinetic theory

Shear stress tensor in conformal limit, no charge current

Mostly consistent w ideal hydro relation

Baier et al. (‘08)

(Our equations)


Discussion Comparison with Renormalization group approach

in energy frame, in single component/conserved current system

Form of the equations agrees with our equations in the single component & conserved current limit w/o vorticity

Tsumura & Kunihiro (‘09)

Note: Vorticity terms added to their equations in recent revision

(Our equations)


Discussion Comparison with Grad’s 14-momemt approach

Form of the equations agrees with ours in the single component & conserved current limit

Betz et al. (‘09)

*Ideal hydro relations in use for comparison

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

in energy frame, in single component/conserved current system

(Our equations)


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 Quark-gluon plasma at relativistic

heavy ion collisions• Cosmological fluid etc…


The End Thank you for listening!

akihiko monnai department of physics, the university of tokyo collaborator: tetsufumi hirano

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