relativistic viscous hydrodynamics for multi-component systems with multiple conserved currents

Akihiko MonnaiDepartment of Physics, The University of Tokyo

Collaborator: Tetsufumi Hirano

Relativistic Viscous Hydrodynamics for Multi-Component Systems with

Multiple Conserved Currents

Hot Quarks 2010June 25th 2010, La Londe-Les-Maures, France

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

A power point template created by Akihiko MonnaiAkihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th 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 for RHIC

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

Purpose of Viscous Hydro:1. Explaining the space-time evolution of the QGP2. Constraining the parameters (viscosities, etc.) from experimental data

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


Introduction Elliptic flow coefficients from RHIC data

Hirano et al. (‘09)

Ideal hydro Glauber1st order

Viscosity

Eq. of state

theoretical prediction experimental data

Ideal hydro works well

Initial condition




Ideal hydro Glauber1st order

Viscosity

Eq. of state


Lattice-based

Ideal hydro works… maybe

Initial condition

*EoS based on lattice QCD results



Viscous hydro in QGP plays important role in reducing v2


Ideal hydro Glauber

Viscosity

Eq. of state

Initial condition


CGCLattice-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

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)

Multiple conserved currents?

We need to construct a firm framework of viscous hydro

Song & Heinz (‘08)


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

　　 , , ,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


Second Order Theory Kinetic expressions with distribution function :

Conventional formalism

Not extendable for multi-component/conserved current systems

one-component, elastic scattering

Israel & Stewart (‘79)

: degeneracy: conserved charge number

Dissipative currents (14)　　 , , , , ,Moment equations (10)

frame fixing, stability conditions

(9) (9)?


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

relaxation


Results (Cont’d)J-th charge current

1st order terms

2nd order terms

relaxation


Onsager Cross Effects 1st order terms

VectorDufour 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)

It may well be important in our relativistic soup of quarks and gluons


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• Multiple conserved currents are not supported in 14-moment method

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 Quark-gluon plasma at relativistic

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)


Introduction Hydrodynamic modeling of heavy ion collisions for RHIC

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

Initial condition Hydrodynamic model requires

Equation of stateViscosityHadronization

1st order phase transition, crossover(lattice), etc. Ideal hydro, viscous hydro JAM, UrQMD, etc.

Glauber, color glass condensate, etc.


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 forceResponse to the gradients: dissipative current


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

,


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)

relativistic viscous hydrodynamics for multi-component systems with multiple conserved currents

Documents