Relativistic Ideal and Relativistic Ideal and Viscous HydrodynamicsViscous Hydrodynamics

Tetsufumi HiranoTetsufumi HiranoDepartment of PhysicsDepartment of Physics

The University of TokyoThe University of Tokyo

Intensive LectureIntensive LectureYITP, December 11th, 2008YITP, December 11th, 2008

Intensive LectureIntensive LectureYITP, December 11th, 2008YITP, December 11th, 2008

TH, N. van der Kolk, A. Bilandzic, arXiv:0808.2684[nucl-th];to be published in “Springer Lecture Note in Physics”.

Plan of this Lecture1st Day• Hydrodynamics in Heavy Ion Collisions• Collective flow • Dynamical Modeling of heavy ion collisions (sem

inar)2nd Day• Formalism of relativistic ideal/viscous hydrodyn

amics• Bjorken’s scaling solution with viscosity• Effect of viscosity on particle spectra (discussio

n)

PART 3

Formalism ofrelativistic

ideal/viscoushydrodynamics

Relativistic Hydrodynamics

Energy-momentum conservation

Current conservation

Energy-Momentum tensor

The i-th conserved current

In H.I.C., Ni = NB

(net baryon current)

Equations of motion in relativistic hydrodynamics

Tensor/Vector Decomposition

Tensor decomposition with a given time-likeand normalized four-vector u

where,

“Projection” Tensor/Vector

•u is perpendicular to .

•u is local four flow velocity. More precisemeaning will be given later.

•Naively speaking, u () picks up time-(space-)like component(s).

•Local rest frame (LRF):

Intuitive Picture of Projection

time likeflow vectorfield

Decomposition of T

:Energy density

:(Hydrostatic+bulk) pressure P = Ps +

:Energy (Heat) current

:Shear stress tensor

<…>: Symmetric, traceless and transverse to u & u

Decomposition of N

:charge density

:charge current

Q. Count the number of unknownsin the above decomposition and confirm

that it is 10(T)+4k(Ni).

Here k is the number of independent currents.Note: If you consider u as independent variables,

you need additional constraint for them. If you also consider Ps as an independent

variable, you need the equation of state Ps=Ps(e,n).

Ideal and Dissipative PartsEnergy Momentum tensor

Charge current

Ideal part

Dissipative part

Meaning of u

u is four-velocity of “flow”. What kind of flow?Two major definitions of flow are

2. Flow of conserved charge (Eckart)

1. Flow of energy (Landau)

Meaning of u (contd.)Landau

(W=0, uLV=0)

Eckart(V=0,uE

W=0)

uL

V

uE

W

Just a choice of local reference frame.Landau frame might be relevant in H.I.C.

Relation btw. Landau and Eckart

Relation btw. Landau and Eckart (contd.)

Entropy Conservationin Ideal Hydrodynamics

Neglect “dissipative part” of energy momentumtensor to obtain “ideal hydrodynamics”.

Q. Derive the above equation.

Therefore,

Assumption (1st order theory):Non-equilibrium entropy current vector haslinear dissipative term(s) constructed from(V, , , (u).

Entropy Current

Thus, = 0 since N = 0, W = 0 since consideringthe Landau frame, and = 0 since u S shouldbe maximum in equilibrium (stability condition).

(Practical) Assumption:•Landau frame (omitting subscript “L”).•No charge in the system.

The 2nd Law of Thermodynamics and Constitutive Equations

The 2nd thermodynamic law tells us

Q. Check the above calculation.

Constitutive Equations (contd.)

Thermodynamic force

Transport

coefficient“Current”

tensor shear

scalar bulk

Newton

Stokes

Equation of Motion

: Expansion scalar (Divergence)

: Lagrange (substantial) derivative

Equation of Motion (contd.)

Q. Derive the above equations of motion from energy-momentum conservation.

Note: We have not used the constitutive equations to obtain the equations of motion.

Intuitive Interpretation of EoM

Work done by pressure

Production of entropy

Change of volume•Dilution•Compression

Conserved Current Case

Lessons from (Non-Relativistic) Navier-Stokes Equation

Assuming incompressible fluids such that , Navier-Stokes eq. becomes

Final flow velocity comes from interplay betweenthese two effects.

Source of flow (pressure gradient)

Diffusion of flow(Kinematic viscosity, /,plays a role of diffusionconstant.)

Generation of Flow

x

P

ExpandExpand

Pressure gradient

Source of flow

Flow phenomenaare important in H.I.Cto understand EOS

Diffusion of Flow

Heat equation(: heat conductivity~diffusion constant)

For illustrative purpose, one discretizes the equation in (2+1)D space:

Diffusion ~ Averaging ~ Smoothing

R.H.S. of descretized heat/diffusion eq.

i

j

i

j

x x

y y

subtract

Suppose i,j is larger (smaller) than an average value around the site, R.H.S. becomes negative (positive).2nd derivative w.r.t. coordinates Smoothing

Shear Viscosity Reduces Flow Difference

Shear flow(gradient of flow) Smoothing of flow

Next time stepMicroscopic interpretation can be made.Net momentum flow in space-like direction. Towards entropy maximum state.

Necessity of Relaxation TimeNon-relativistic case (Cattaneo(1948))

Fourier’s law

: “relaxation time”

Parabolic equation (heat equation) ACAUSAL! Finite Hyperbolic equation (telegraph equation)

Balance eq.:

Constitutive eq.:

Heat Kernel

x x

causalityperturbationon top of

background

Heattransportation

Instability

• The 1st order equation is not only acausal but also unstable under small perturbation on a moving back-ground. W.A.Hiscock and L.Lindblom, PRD31,725(1985).

• For particle frame with new EoM, see K.Tsumura and T.Kunihiro, PLB668, 425(2008).

• For a possible relation btw. stability and causality, see G.S.Danicol et al., J.Phys.G35, 115102(2008).

Entropy Current (2nd)Assumption (2nd order theory):Non-equilibrium entropy current vector haslinear + quadratic dissipative term(s)constructed from (V, , , (u)).

Stability condition O.K.

The 2nd Law of Thermodynamics: 2nd order case

Same equation, but different definition of and .

Sometimesomitted, but needed.

Generalization of thermodynamic force!?

Summary:Constitutive Equations

•Relaxation terms appear ( and are relaxation time).•No longer algebraic equations! Dissipative currents become dynamical quantities like thermodynamic variables.•Employed in recent viscous fluid simulations. (Sometimes the last term is neglected.)

: vorticity

Plan of this Lecture1st Day• Hydrodynamics in Heavy Ion Collisions• Collective flow • Dynamical Modeling of heavy ion collisions (sem

inar)2nd Day• Formalism of relativistic ideal/viscous hydrodyn

amics• Bjorken’s scaling solution with viscosity• Effect of viscosity on particle spectra (discussio

n)

PART 4

Bjorken’s Scaling Solution with Viscosity

“Bjorken” Coordinate

0z

t

Boost parallel shiftBoost invariant Independent of s

Bjorken’s Scaling Solution

Hydrodynamic equation for perfect fluids with a simple EoS,

Assuming boost invariance for thermodynamic variables P=P() and 1D Hubble-like flow

Conserved and Non-Conserved Quantity in Scaling Solution

expansion

pdV work

Bjorken’s Equation in the 1st Order Theory

(Bjorken’s solution) = (1D Hubble flow)

Q. Derive the above equation.

Viscous Correction

Correction from shear viscosity(in compressible fluids)

Correction frombulk viscosity

If these corrections vanish, the above equationreduces to the famous Bjorken equation.Expansion scalar = theta = 1/tau in scaling solution

• is obtained from Super Yang-Mills theory.

• is obtained from lattice.

•Bulk viscosity has a prominent peak around Tc.

Recent Topics on Transport Coefficients

Need microscopic theory (e.g., Boltzmann eq.)to obtain transport coefficients.

Kovtun, Son, Starinet,…

Nakamura, Sakai,…

Kharzeev, Tuchin, Karsch, Meyer…

Bjorken’s Equationin the 2nd Order Theory

New terms appear in the 2nd order theory. Coupled differential equationsSometimes, the last terms are neglected.Importance of these terms see Natsuume and Okamura, 0712.2917[hep-th].

where

Why only 00-zz?In EoM of energy density,

appears in spite of constitutive equations.According to the Bjorken solution,

Relaxation Equation?

Digression: Full 2nd order equation?

Beyond I-S equation, see R.Baier et al., JHEP 0804,100 (2008); Tsumura-Kunihiro?; D. Rischke, talk at SQM 2008. According to Rischke’s talk, constitutive equations with vanishing heat flow are

Digression (contd.): Bjorken’s Equationin the “full” 2nd order theory

See also, R.Fries et al.,PRC78,034913(2008). Note that the equation for shear is valid only for conformal EOS andthat no 2nd and 3er terms for bulk.

Model EoS (crossover)

Crossover EoS: Tc = 0.17GeV = Tc/50dH = 3, dQ = 37

Energy-Momentum Tensorat 0 in Comoving Frame

In what follows, bulk viscosity is neglected.

Numerical Results (Temperature)

Same initial condition(Energy momentumtensor is isotropic)

T0 = 0.22 GeVt0 = 1 fm/c

/s = 1/4 = 3/4p

Numerical code (C++) is available upon request.

Numerical Results (Temperature)

Same initial condition(Energy momentumtensor is anisotropic)

T0 = 0.22 GeVt0 = 1 fm/c

/s = 1/4 = 3/4p

Numerical code (C++) is available upon request.

Numerical Results (Temperature)

Numerical code (C++) is available upon request.

Numerical Results (Entropy)

T0 = 0.22 GeVt0 = 1 fm/c

/s = 1/4 = 3/4p

Same initial condition(Energy momentumtensor is isotropic)

Numerical code (C++) is available upon request.

Numerical Results (Entropy)

T0 = 0.22 GeVt0 = 1 fm/c

/s = 1/4 = 3/4p

Same initial condition(Energy momentumtensor is anisotropic)

Numerical code (C++) is available upon request.

Numerical Results (Entropy)

Numerical code (C++) is available upon request.

Numerical Results(Shear Viscosity)

Numerical code (C++) is available upon request.

Numerical Results(Initial Condition Dependence

in the 2nd order theory)

Numerical code (C++) is available upon request.

Numerical Results(Relaxation Time dependence)

Saturated valuesnon-trivial

Relaxation time largerMaximum is smallerRelaxation time smallerSuddenly relaxes to 1st order theory

Remarks

• Sometimes results from ideal hydro are compared with the ones from 1st order theory. But initial conditions must be different.

• Be careful what is attributed for the difference between two results.

• Sensitive to initial conditions and new parameters (relaxation time for stress tensor)

Plan of this Lecture1st Day• Hydrodynamics in Heavy Ion Collisions• Collective flow • Dynamical Modeling of heavy ion collisions (sem

inar)2nd Day• Formalism of relativistic ideal/viscous hydrodyn

amics• Bjorken’s scaling solution with viscosity• Effect of viscosity on particle spectra (discussio

n)

PART 5

Effect of Viscosity on Particle Spectra

Particle Spectra in Hydrodynamic Model

• How to compare with experimental data (particle spectra)?

• Free particles (/L>>1) eventually stream to detectors.

• Need prescription to convert hydrodynamic (thermodynamic) fields (/L<<1) into particle picture.

• Need kinetic (or microscopic) interpretation of hydrodynamic behavior.

Microscopic Interpretation

Single particle phase space density in localthermal equilibrium:

Kinetic definition of current and energy momentumtensor are

Matter in (Kinetic) Equilibrium

u

Kinetically equilibratedmatter at rest

Kinetically equilibratedmatter at finite velocity

px

py

px

py

Isotropic distribution Lorentz-boosted distribution

Cooper-Frye Formula

•No dynamics of evaporation.•Just counting the net number of particles (out-going particles) - (in-coming particles) through hypersurface •Negative contribution can appear at some space-like hyper surface elements.

1st Moment

u is normalized, so we can always choose a such that

1st Moment (contd.)

Vanishing for = i due to odd function in integrant.

Q. Go through all steps in the above derivation.

2nd Moment

where,

Deviation from Equilibrium Distribution

1 4 9

Grad’s 14 momentsEoM for epsilons can be also obtained from BE.

Neglecting anti-particles,

Taylor Expansion around Equilibrium Distribution

Taylor Expansion around Equilibrium Distribution (contd.)

14 Conditions

“Landau” conditions (2)

Viscosities (12) Epsilons can beexpressed bydissipative currents

Relation btw. Coefficients and Dissipative Currents

For details, see Sec.6 and Appendix C inIsrael-Stewart paper. (Ann.Phys.118,341(1979))

Finally,

Remarks

• Boltzmann equation gives a microscopic interpretation of hydrodynamics.

• However, hydrodynamics may be applied for gas/liquid where the Boltzmann equation can not be applied.

• Deviation from equilibrium can be incorporated into phase space distribution.

ReferencesFar from Complete List…

• General– L.D.Landau, E.M.Lifshitz, Fluid Mechanics, Section 133-136– L.P.Csernai, Introduction to Relativistic Heavy Ion Collisions– D.H.Rischke, nucl-th/9809044. – J.-Y.Ollitrault, 0708.2433[nucl-th].

• Viscous hydro + transport coefficient– C.Eckart, Phys.Rev.15,919(1940).– M.Namiki, C.Iso,Prog.Theor.Phys.18,591(1957)– C.Iso, K.Mori,M.Namiki, Prog.Theor.Phys.22,403(1959)– I.Mueller, Z. Phys. 198, 329 (1967) – W.Israel, Ann.Phys.100,310(1976)– W.Israel. J.M.Stewart, Ann.Phys.118,341(1979)– A.Hosoya, K.Kajantie, Nucl.Phys.B250, 666(1985)– P.Danielewicz,M.Gyulassy, Phys.Rev.D31,53(1985) – I.Muller, Liv.Rev.Rel 1999-1.

References for Astrophysics/Cosmology

• N.Andersson, G.L.Comer, Liv.Rev.Rel.,2007-1• R.Maartens, astro-ph/9609119.

Disclaimer: I’m not familiar with these kinds of review…