PART 2

Formalism ofrelativistic ideal/viscous

hydrodynamics

Advertisement: Lecture NotesChapter 4Chapter 4

Hydrodynamics

• Framework to describe space-time evolution of thermodynamic variables

• Balance equations (equations of motion, conservation law)

+ equation of state (matter property) + constitutive equations (phenomenology)

Relativistic Hydrodynamics

Energy-momentum conservation

Current conservationEnergy-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 currentQ. Count the number of unknowns

in the above decomposition and confirmthat 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 can 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 interplaybetween these 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

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

PART 3

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

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; A.Monnai and TH, in preparation. 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 conformalEOS and that no 2nd and 3rd terms for bulk.

Model EoS (crossover)

Crossover EoS: Tc = 0.17GeV = Tc/50gH = 3, gQ = 37

Relativistic Ideal Gas

Thermodynamic potential for relativistic ideal gases

Energy-Momentum Tensorat 0 in Comoving Frame

In what follows, bulk viscosity is omitted.

Numerical Results (Temperature)

Same initial condition(Energy momentumtensor is isotropic)

T0 = 0.22 GeV0 = 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 GeV0 = 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 GeV0 = 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 GeV0 = 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)

PART 4

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.

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.

Matter in (Kinetic) Equilibrium

u

Kinetically equilibratedmatter at rest

Kinetically equilibratedmatter at finite velocity

px

py

px

py

Isotropic distribution Lorentz-boosted distribution

Relativistic Transport Theory

Boltzmann equationTime evolution of phase space dist. for a rarefied gas:

One-particle phase space distributionCollision termLorentz invariant transition rate

gain loss

Free Streaming, Gain and Loss

x

p

Note: p is mass-on-shell.

gain

loss

loss

gain

H-theorem“Entropy current”:

H-theorem (contd.)

H-theorem (cond.2)

H-theorem (contd.3)

=0

H-theorem (contd.4)

1

1

0

One can identifys with entropy current

Collision InvariantBinary collisions satisfy energy-momentum consevation

So

where(collision invariant)

Collision Invariant (contd.)

Q. Check this identity

Conservation law

5 conservedcurrents(quantities)

Thus should be collision invariant

Equilibrium Distribution

and can depend on position xand are to be /T and u/T

Quantum StatisticsBose enhancement or Pauli blocking

Collision invariant

Some Remarks

• Collision term vanishes if f0 is plugged in.

• If and is constant globally, f0 can be a solution of Boltzmann equation.

• Local equilibrium distribution is NOT a solution of Boltzmann equation.

• Deviation from local equilibrium distribution can be obtained from Boltzmann equation.

Microscopic Interpretation

Single particle phase space density in local equilibrium (no entropy production in Boltzmann eq.):

Kinetic definition of current and energy momentumtensor are

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

2nd Moment (contd.)

where

Deviation from Equilibrium Distribution

trace part scalar

Unknowns: 4 + 10 = 14

Important in a multi-component gas(A.Monnai and TH, (2009))

Taylor Expansion around Equilibrium Distribution

Taylor Expansion around Equilibrium Distribution (contd.)

14 Conditions

Stability conditions (2)

Viscosities (12) Epsilons can beexpressed bydissipative currents

Stability Condition

Q. Check the above derivation.

Stability Condition (contd.)

Relation btw. Coefficients and Dissipative Currents

Moments

Bosons do not contribute…

Solutions

Solutions (contd.)

Epsilons are expressed using hydrodynamic variablesAble to calculate particle spectra

Cooper-Frye Formulain Viscous Case

Transverse Momentum Spectra for Pions

Bulk pressure Isotropic = -Shear stress tensor Traceless = 4/3

Summary

• Effect of viscosity on particle spectra can be calculated using hydrodynamic variables.

• Important to constrain equation of state and transport coefficients from experimental data