from the boltzmann equation to fluid dynamics and its ... · from the boltzmann equation to uid...

Introduction Perfect Fluids Dissipative Fluids Fluid dynamics from kinetic theory Outlook

From the Boltzmann equation to fluid dynamics and itsapplications

Etele Molnar

MTA-DE Particle Physics Research Group

09.10.2012

in collaboration with:I. Bouras, G. Denicol, P. Huovinen, H. Niemi, D. H. Rischke

Etele Molnar From the Boltzmann equation to fluid dynamics and its applications


Stages of evolution in heavy-ion collisions I.

Initial stage (Belensky and Landau 1955)

When two nucle(i)ons collide, a compound systemis formed, and energy is released in a small volumeV subject to a Lorentz contraction in thelongitudinal direction.At the instant of the collision,a large number of ”particles” are formed: the”mean free path” (m.f.p.) in the resulting system issmall compared with its dimension, andstatistical equilibrium is set up.

Initial particle production τ � 1 fm/c:Two nuclei fly through each other (Bjorken1976, 1983), producing highly excited matter.

Non-equilibrium evolution of the matter(thermalization) τ . 1 fm/c.



Stages of evolution in heavy-ion collisions II.

Fluid dynamical evolution of theQGP τ ∼ 5 fm/c

Transition back to hadronic matter(QCD phase transition)

Evolution of the hadron gas

Hydrodynamical stage (Belensky and Landau 1955)

The second stage of the collision consists in theexpansion of the system. Here thehydrodynamical approach must be used, and theexpansion may beregarded as the motion of an ideal fluid (zeroviscosity and zero thermal conductivity). During theprocess of expansion the m.f.p., remains small incomparisons with the dimensions of the system, andthis justifies the use of hydrodynamics.Since the velocities in the system arecomparable with that of light, we must use notordinary but relativistic hydrodynamics. Particlesare formed and absorbed in the system throughoutthe first and second stages of collision. The highdensity of energy in the system is of importancehere. In this case, the number of particles is not anintegral of the system, on account of the stronginteraction between the individual particles.



Stages of evolution in heavy-ion collisions III.

Transition to free particles(Cooper-Frye 1976)

Freeze-out stage (Belensky and Landau 1955)

As the system expands,the interaction becomes weaker and the mean freepath becomes larger. The number of particlesappears as the physical characteristic when theinteraction is sufficiently weak. When the mean freepath becomes comparable with the lineardimensions of the system, the laterbreaks up into individual particles. This may becalled the ”break-up” stage. It occurs with atemperature of the system of the order T ≈ mπc2,where mπ is the mass of the pion.



Stages of evolution in heavy-ion collisions III+.

The space-time picture of evolution.dV /V = dτ/τ in 1+1D case.



Perfect Liquid



Notation

For simplicity we choose the flat space-time, the metric isgµν ≡ gµν = diag(1,−1,−1,−1).

The normalized 4-flow of matter is denoted by uµ(t, x), whereuµuµ = 1, (c2 = 1).

The local rest frame (LRF) is defined as, uµLRF = (1, 0, 0, 0).

The projection tensor is defined perpendicular to the 4-flow of matter,∆µν = gµν − uµuν , where ∆µνuµ = 0, and ∆µν∆µν = 3.

The comoving time-derivative or proper-time derivative in LRF of an arbitrary4-vector Aµ is denoted by, Aµ = uα∂αAµ while, the comoving spatial-derivativeor gradient is denoted by, ∇µ = ∆µν∂ν .

The orthogonal projection of Aµ ≡ ∆µνAν = A〈µ〉.

The symmetric, traceless and orthogonal part of any tensor is denoted by,

A〈µν〉 ≡ ∆µναβA

αβ =[

12

(∆µα∆ν

β + ∆να∆µ

β

)− 1

3∆µν∆αβ

]Aαβ .



Perfect Fluids I.

Conservation laws for a simple (single component) perfect fluid (no dissipation)

∂µNµ0 = 0 charge conservation ⇒ 1 eq.

∂µTµν0 = 0 energy-momentum conservation ⇒ 4 eqs.

Perfect fluid decomposition with respect to uµ

Nµ0 = n0uµ

Tµν0 = e0uµuν − p0∆µν

n0 = Nµ0 uµ (net)charge density

e0 = Tµν0 uµuν energy density

p0 = −1

3∆µνT

µν0 isotropic pressure

We have 5 equations for 6 unknowns not closed: n0(1), e0(1), p0(1) and uµ(3).These equations are postulated!



Perfect Fluids II.

The assumption of local thermal equilibrium! Provides closure:

Equation of State (EoS)

p0 = p0(e0, n0) EoS⇒ 1 eq.

and/or p(T , µ) or s = s(e, n).

Auxiliary, Sµ0 = s0uµ, where s0 = Sµ0 uµ, and for continuous solutions

∂µSµ0 = 0

entropy is maximum in local thermal equilibrium

Thermodynamics

Ts = e + p − µnT s = e − µnp = sT + nµ

The fundamental thermodynamic relations are derived from,T∂µ(suµ) = ∂µ(euµ) + p(∂µuµ)− µ∂µ(nuµ)



Dissipative Fluids I.

Conservation laws for a simple (single component) dissipative fluid

∂µNµ = 0 charge conservation ⇒ 1 eq.

∂µTµν = 0 energy-momentum conservation ⇒ 4 eqs.

General decomposition with uµ

Nµ = nuµ + Vµ

Tµν = euµuν − (p + Π)∆µν + Wµuν + W νuµ + πµν

n = Nµuµ charge density

e = Tµνuµuν energy density in LRF

p + Π = −1

3∆µνT

µν isotropic (equilibrium + bulk viscous) pressure

Vµ = ∆µαNα charge flow

Wµ = ∆µαTαβuβ energy-momentum flow

qµ = Wµ −e + p

nVµ heat flow ⇒ 1 eq.

πµν = T 〈µν〉 stress tensor

We only have 5 equations for 18 unknowns, n(1), e(1), p(1), uµ(3) andΠ(1),Vµ(3),Wµ(3), πµν(5).



Dissipative Fluids II.

Generally, Nµ = Nµ0 + δNµ and Tµν = Tµν0 + δTµν

Simplifications (I): Matching to equilibrium and the EOS

n = n0 + δn

e = e0 + δe

p(e, n) = p0(e0, n0) + δp

Convenient choice, δn = 0, δe = 0, δp = Π hence, n = n0, e = e0,p0 = p0(e0, n0), such that p(e, n) = p0 + Π, while T = T0 and µ = µ0, buts = s0 + δs!

Still with 14 + 3 unknowns! n(1), e(1),Π(1), Vµ(3),Wµ(3), πµν(5) and uµ(3).

Simplifications (II): Fixing the Local Rest Frame

uµE = Nµ/n ⇔ Vµ = 0 ⇒ qµ = Wµ Eckart

uµL = TµνuLν/e ⇔Wµ = 0 ⇒ qµ = −e + p

nVµ Landau & Lifsitz

We eliminated (3) unknowns, such that uµL = uµE + δuµ.

We are left with 14 unknowns! n(1), e(1), uµ(3) and p(1), qµ(3), πµν(5).



Dissipative Fluids III. - The relativistic Navier-Stokes theory

The entropy current is also modified Sµ ≡ Sµ0 + δSµ = (s0 + δs)uµ + Φµ, wheres ≡ Sµuµ = (s0 + δs) and Φµ = ∆µνSν .

2nd law of thermodynamics (Eckart’s frame)

∂µSµ = ∂µ

[Φµ −

qµ

T

]−

qµ

T

(1

T∂µT − uµ

)−

Π

T∂µu

µ +πµν

T∂µuν ≥ 0

Assuming that s = s0, i.e., δs = 0, while Φµ = qµ/T

For small gradients, linear relations between thermodynamical forces and fluxes!

The relativistic Navier-Stokes relations

ΠNS = −ζ∇µuµ

πµνNS = 2 η∇〈µuν〉

qµNS = −κTT n

e + p∇µ

( µT

)ζ ≥ 0, η ≥ 0 bulk and shear viscosity coefficients, κ ≥ 0 coefficient of thermalconductivity.

Now, the equations of fluid dynamics are closed, but the relativistic Navier-Stokestheory leads to accausal signal propagation and stability issues.



Dissipative Fluids III+.

(Gas)Kinetic theory

πxzNS = − η

∂vz (x)

∂xη ' 1.2T/σ = 1.2Tλmfpn

η/s ≈ Tλmfp

(Dilute) Gases may have λ→∞ orσ → 0 hence η →∞ ideal gas!ηAir ≈ 1.8× 10−5Pa.s (T ∼ 20Co)Fluids (liquids), λ→ 0 or σ →∞ henceη → 0 ideal fluid (liquid)!ηWater ≈ 50× ηAir = 9× 10−4Pa.sηgas(T ) increases with increasing Tηliquid (T ) decreases with increasing T

Ref. Csernai et.al. PRL (2006)(ηs

)π

=15

16π

f 4π

T 4(ηs

)QCD

=5.12

g4 ln(2.42g)> 1(η

s

)ADS/CFD

=1

4π

fπ ≈ 130MeV pion decay constantg QCD coupling constant



Dissipative Fluids IV.

2nd order theories of Muller (1967), Israel (1976) and Stewart (1971, 1977)The previously mentioned issues are ”cured” if:

Entropy current generalization (Eckart’s frame)

Sµ ≡ s0uµ +

qµ

T−(β0Π2 − β1q

µqµ + β2πµνπµν

) uµ

2T

−α0Πqµ

T+α1π

µνqν

T+ O3

where the newly introduced coefficients β0, β1, β2, α0, α1 are related to therelaxation time/length of bulk viscosity, heat conductivity and shear viscosity

τΠ = ζβ0

τq = κTβ1

τπ = 2ηβ2

lΠq = ζα0

lqΠ = κTα0

lqπ = κTα1

lπq = 2ηα1

The β0, β1, β2, α0, α1 coefficients are frame dependent and remain undeterminedin phenomenological theories!



Dissipative Fluids V.

2nd order equations from entropy production

Relaxation equations in Eckart’s frame (Israel 1976)

τΠΠ + Π = ΠNS + lΠq∇µqµ

τq∆µαq

α + qµ = qµNS + lqΠ∇µΠ− lqπ∆µα∂νπ

αν

τππ〈µν〉 + πµν = πµνNS + lπq∇〈µqν〉

The equations determine the time evolution of Π, qµ, and πµν

The Navier-Stokes theory appears if the relaxation times and length scalesτi → 0, li → 0 with ζ, η and κq fixed

Later O2 corrections (∼ ∇µαi ,∼ ∇µβi , . . ., etc.) were added by Israel andStewart (1977-1979), Hiscock and Lindblom (1983), Relativistic ExtendedThermodynamics of Liu, Muller and Ruggeri (1983) etc.

We need kinetic theory to motivate and determine the above introducedphenomenological equations self-consistently!



The relativistic Boltzmann equation and conservation equations

In a dilute gas, the space-time evolution of the single-particle distribution functionfk = f (t, x, k0, k) due to particle motion and binary collisions is given by the

The relativistic Boltzmann equation

kµ∂µfk ≡ C [fk] =1

2

∫dK ′dPdP′Wkk′→pp′

(fpfp′ fk fk′ − fkfk′ fp fp′

)where kµ = (k0, k) is the four-momenta of particles with mass m and energyk0 =

√k2 + m2. fk = 1− afk, with a = 0 for Boltzmann, a = 1 for Fermi, a = −1 for

Bose statistics. The inv. phase-space element is, dK = gd3 k/[(2π)3k0

]. The

transition rate W is invariant, to final state momenta Wkk′→pp′ = Wkk′→p′p, andsatisfies detailed balance, Wkk′→pp′ = Wpp′→kk′.

Conservation laws from the Boltzmann equation

∂µNµ ≡

∫dK kµ∂µfk =

∫dK C [fk] = 0 charge cons.

∂µTµν ≡

∫dK kνkµ∂µfk =

∫dK kνC [fk] = 0 energy-momentum cons.

Conservation laws are obtained, 5-collisional invariants, but we still need the solutionof the Boltzmann equation fk!



The hierarchy of moments and balance equations

Moments of the single-particle distribution function and collision integral

Nµ(t, x) ≡∫

dK kµ fk = 〈kµ〉 charge current

Tµν(t, x) ≡∫

dK kµkν fk = 〈kµkν〉 energy-momentum tensor

Fµ1...µn (t, x) ≡∫

dK kµ1 . . . kµn fk = 〈kµ1 . . . kµn 〉 n-rank moment

Pµ1...µn (t, x) ≡∫

dK kµ1 . . . kµn C [fk] = 〈Cµ1...µn 〉 n-rank production term

where 〈pµ1pµ2 . . . pµn−1pµn 〉 ⇒∫dK pµ1 . . . pµn fk.

The mass-shell condition kµkµ = m2 gives

Fµ1...µnλλ = m2Fµ1...µn

Pµ1...µnλλ = m2Pµ1...µn

∂λFµ1...µnλ = Pµ1...µn balance of fluxes

These balance equations are never closed but the solution of the Boltzmann equationis also a solution of the hierarchy!



Local thermal equilibrium

In local thermal equilibrium ∂Sµ0 = 0, hence fk → f0k, where f0k = f (α0, β0,Ek)

f0k ≡ [exp (−α0 + β0Ek) + a]−1 Juttner distribution

where α0 = µ0/T0 is the chemical potential, β0 = 1/T0 is the inverse temperature,Ek = kµuµ, and a = 0 for Boltzmann, a = 1 for Fermi, a = −1 for Bose statistics.

Moments of the equilibrium distribution function

Nµ0 (t, x) ≡∫

dK kµ f0k = 〈kµ〉0 charge current

Tµν0 (t, x) ≡∫

dK kµkν f0k = 〈kµkν〉0 energy-momentum tensor

Sµ0 (t, x) ≡∫

dK kµ f0k (ln f0k − 1) = 〈kµΨ〉0 entropy current

Conservation eqs. for and ideal gas

∂µNµ0 = 0, ∂µT

µν0 = 0, ∂µS

µ0 ≥ 0

(a = 0) hence p0 = n0T0 ideal gas law!



General kinetic decompositions and definitions

Decompose the momenta, kµ = Ekuµ + k〈µ〉, where Ek = kµuµ and k〈µ〉 = ∆µ

ν kν

Nµ = 〈Ek〉uµ + 〈k〈µ〉〉

Tµν = 〈E2k 〉u

µuν +1

3∆µν〈∆αβkαkβ〉+ 〈Ekk

〈µ〉〉uν + 〈Ekk〈ν〉〉uµ + 〈k〈µkν〉〉

where k〈µkν〉 = ∆µναβk

αkβ and 〈kµ1kµ2 . . . kµn 〉 ⇒∫dK kµ1 . . . kµn fk for any fk

n ≡ 〈Ek〉 e ≡ 〈E2k 〉 p ≡ −

1

3〈∆αβkαkβ〉

Vµ ≡ 〈k〈µ〉〉 Wµ ≡ 〈Ekk〈µ〉〉 πµν ≡ 〈k〈µkν〉〉

In equilibrium 〈kµ1kµ2 . . . kµn 〉0 ⇒∫dK kµ1 . . . kµn f0k, hence

n0 = 〈Ek〉0 e0 = 〈E2k 〉0 p0 = −

1

3〈∆αβkαkβ〉0

Vµ ≡ 〈k〈µ〉〉0 = 0 Wµ ≡ 〈Ekk〈µ〉〉0 = 0 πµν ≡ 〈k〈µkν〉〉0 = 0



Out of equilibrium

For a system out of equilibrium

fk = f0k + δfk

Let us define the following irreducible moment

ρµ1...µ`(r)

≡⟨

(Ek)r k〈µ1 . . . k µ`〉⟩δ

where 〈. . .〉δ ≡ 〈. . .〉 − 〈. . .〉0 =∫dK . . . fk −

∫dK . . . f0k =

∫dK . . . δfk

δn = ρ(1) δe = ρ(2) Π = −m2

3ρ(0)

Vµ = ρµ(0)

Wµ = ρµ(1)

πµν = ρµν(0)

Matching: Non-equilibrium to equilibrium

δn ≡ 〈Ek〉δ = 0 ⇒ n = n0

δe ≡ 〈E2k 〉δ = 0 ⇒ e = e0

Π = −1

3〈∆αβkαkβ〉δ ⇒ p(e, n) = p0(e0, n0) + Π



General equations of motion I.

Equations of motion for the irreducible moments

ρ〈µ1...µ`〉(r)

= ∆µ1...µ`ν1...ν`

d

dτ

∫dK (Ek)r k〈ν1 . . . k ν`〉δfk

where A ≡ uµ∂µA ≡ dA/dτ and ρ〈µ1...µ`〉(r)

≡ ∆µ1...µ`ν1...ν` ρ

ν1...ν`(r)

and

k〈µ1 · · · k µm〉 = ∆µ1···µmν1···νmkν1 · · · kνm

Now, using the Boltzmann equation kµ∂µfk = C [f ] in the following form

δfk = −f0k − E−1k kν∇ν f0k − E−1

k kν∇νδfk + E−1k C [f ]

where ∇µ = ∆νµ∂ν , we obtain the exact equations for the comoving derivatives of the

irreducible moments, ρµ1...µ`(r)



General equations of motion II.

Scalar

ρr = Cr−1 + α(0)r θ +

(rρµr−1 +

G2r

D20Wµ

)uµ

−∇µρµr−1 +G3r

D20∂µV

µ −G2r

D20∂µW

µ

+1

3

[(r − 1) m2

ρr−2 − (r + 2) ρr − 3G2r

D20Π

]θ

+

[(r − 1) ρµνr−2 +

G2r

D20πµν

]σµν .

Inq =(−1)q

(2q + 1)!!

∫dKE n−2q

k

(∆αβkαkβ

)qf0k,

Jnq =(−1)q

(2q + 1)!!

∫dKE n−2q

k

(∆αβkαkβ

)qf0k f0k,

Gnm = Jn0Jm0 − Jn−1,0Jm+1,0,

Dnq = Jn+1,qJn−1,q − (Jnq)2.

Vector

ρ〈µ〉r = C

〈µ〉r−1 + α

(1)r ∇

µα0 − αh

r Wµ + rρµνr−1uν

+1

3

[rm2

ρr−1 − (r + 3) ρr+1 − 3αhr Π]uµ

−1

3∇µ

(m2ρr−1 − ρr+1

)+ α

hr∇

µΠ

−∆µν

(∇λρνλr−1 + α

hr ∂λπ

νλ)

+1

3

[(r − 1) m2

ρµr−2 − (r + 3) ρµr − 4αh

r Wµ]θ

+1

5

[(2r − 2) m2

ρνr−2 − (2r + 3) ρνr − 5αh

r Wν]σµν

+(ρνr + α

hr W

ν)ωµν + (r − 1) ρµνλr−2 σνλ.

C〈µ1···µ`〉r ≡ ∆

µ1···µ`ν1···ν` C

ν1...ν`r

= ∆µ1···µ`ν1···ν`

∫dKE r

kkµ1 · · · kµ`C [f ] .



General equations of motion III.

Tensor

ρ〈µν〉r = C

〈µν〉r−1 + 2α(2)

r σµν +

2

15

[(r − 1) m4

ρr−2

− (2r + 3) m2ρr + (r + 4)ρr+2

]σµν

+2

5

[rm2

ρ〈µr−1 − (r + 5) ρ

〈µr+1

]u ν〉

+ rρµνλr−1 uλ −2

5∇〈µ

(m2ρν〉r−1 − ρ

ν〉r+1

)+

1

3

[(r − 1) m2

ρµνr−2 − (r + 4) ρµνr

]θ

+2

7

[(2r − 2) m2

ρλ〈µr−2 − (2r + 5) ρλ〈µr

]σν〉λ

+ 2ρλ〈µr ων〉λ −∆µναβ∇λρ

αβλr−1

+ (r − 1)ρµνλκr−2 σλκ.

α0 =1

D20

[−J30

(n0θ + ∂µVµ

)+ J20

(ε0 + P0 + Π

)θ

+J20

(∂µWµ − Wµ uµ − π

µνσµν

)],

β0 =1

D20

[−J20

(n0θ + ∂µVµ

)+ J10

(ε0 + P0 + Π

)θ

+J10

(∂µWµ − Wµ uµ − π

µνσµν

)],

uµ = β−10

(h−10∇µα0 −∇

µβ0

)−

h−10

n0

(Πuµ −∇µΠ

)

−h−10

n0

[ 4

3Wµθ + Wν

(σµν − ωµν

)+ Wµ + ∆µν ∂λπ

νλ],

α(0)r = (1 − r) Ir1 − Ir0 −

n0

D20

(h0G2r − G3r

),

α(1)r = Jr+1,1 − h

−10

Jr+2,1,

α(2)r = Ir+2,1 + (r − 1) Ir+2,2,

αhr = −

β0

ε0 + P0

Jr+2,1,



General equations of motion IV.

Polynomial expansion

fk ≡ f0k + δfk = f0k

(1 + f0kφk

),

φk =∞∑`=0

λ〈µ1···µ`〉k k〈µ1

· · · kµ`〉,

λ〈µ1···µ`〉k =

N∑n=0

c〈µ1···µ`〉n P

(`)kn ,

P(`)kn =

n∑r=0

a(`)nr E

rk , orthogonal poly

c〈µ1···µ`〉n ≡

N (`)

`!

⟨P

(`)kn k〈µ1 · · · k µ`〉

⟩δ

=N (`)

`!

n∑r=0

ρµ1···µ`r a(`)

nr .

fk = f0k

1 + f0k

∞∑`=0

N∑n=0

H(`)kn ρ

µ1···µ`n k〈µ1

· · · kµ`〉

,

H(`)kn =

N (`)

`!

N∑m=n

a(`)mnP

(`)km .



The 14-moment method

Truncation

λk ≡N0∑n=0

cnP(0)kn ' c0P

(0)k0 + c1P

(0)k1 + c2P

(0)k2 ,

λ〈µ〉k ≡

N1∑n=0

c〈µ〉n P(1)kn ' c

〈µ〉0 P

(1)k0 + c

〈µ〉1 P

(1)k1 ,

λ〈µν〉k ≡

N2∑n=0

c〈µν〉n P(2)kn ' c

〈µν〉0 P

(2)k0 ,

Matching

ρr ≡N0∑n=0

n∑m=0

cna(0)nm Jr+m,0 = γ

Πr Π,

ρµr ≡

N1∑n=0

n∑m=0

c〈µ〉n a(1)nm Jr+m+2,1 = γ

Vr Vµ + γ

Wr Wµ

,

ρµνr ≡

N2∑n=0

n∑m=0

c〈µν〉n a(2)nm Jr+m+4,2 = γ

πr π

µν.



The 14-moment method - Collision integral in a Boltzmann gas

Collision integral for Boltzmann particles

Cµ1...µ`r =

1

2

∫dKdK ′dPdP′Wkk′→pp′f0kf0k′ (Ek)r kµ1 . . . kµ`

×[(φp + φp′ − φk − φk′

)+(φpφp′ − φkφk′

)]Linearized collision integral

Cr−1 ≡ uµ1· · · uµ`C

µ1···µ`r−`−1 = CΠχΠXr−3,1

C〈α〉r−1 ≡ ∆αµ1

uµ2· · · uµ`C

µ1···µ`r−` = BVV

µXr−2,3 + BWWµXr−2,3,

C〈αβ〉r−1 ≡ ∆αβµ1µ2

uµ3· · · uµ`C

µ1···µ`r−`+1 = AππµνXr−1,4.

Collision integral

Xµναβr ≡1

ν

∫dKdK ′dPdP′Wkk′→pp′f0kf0k′ f0p f0p′E

rkkµkν

(pαpβ + p′αp′β − kαkβ − k′αk′β

)=(Xr,1u

µuν + Xr,2∆µν)(

uαuβ −1

3∆αβ

)+ 4Xr,3u

(µ ∆ ν)(α u β) + Xr,4∆µναβ .



The 14-moment method - (r ?)

Scalar

Π = −Π

τ rΠ

−ζr

τ rΠ

θ + τrΠWWµu

µ + τrΠVVµu

µ

− `rΠW ∂µWµ − `rΠV ∂µV

µ − δrΠΠΠθ

+ λrΠWWµ∇µα0 + λ

rΠVV

µ∇µα0 + λrΠππ

µνσµν .

Tensor

π〈µν〉 = −

πµν

τ rπ

+2ηr

τ rπ

σµν + 2λr

πΠΠσµν

+ 2τ rπWW 〈µ u ν〉 + 2τ r

πVV〈µ u ν〉

+ 2`rπW∇〈µW ν〉 + 2`rπV∇

〈µ V ν〉

− 2λrπWW 〈µ∇ ν〉

α0 − 2λrπVV

〈µ∇ ν〉α0

− 2λrπππ

〈µα σ

ν〉α + 2π〈µα ων〉α − 2δrπππ

µνθ.

Vector

Vµ + ψWr Wµ = −

Vµ

τ rV

− ψWr

Wµ

τ rW

+κrq

τ rVβ

20h

20

∇µα0

+ ψWr Wνω

µν + Vνωµν

− ψWr λ

rWWWνσ

µν + λrVVVνσ

µν

− ψWr δ

rWWWµ

θ + δrVVV

µθ

− τ rqΠΠuµ − τ r

qππµν uν

+ `rqΠ∇

µΠ− `rqπ∆µα∂νπαν

+ λrqΠΠ∇µα0 + λ

rqππ

µν∇να0,



The ”exact” relaxation equations with a linearized collision integral

The original Israel-Stewart derivation neglected several terms; corrections added later.More terms from the non-linearized collision integral!

τΠ Π + Π = ΠNS + τΠq q · u − `Πq ∂ · q − ζ δ0 Π θ

+ λΠq q · ∇α + λΠπ πµνσµν

τq ∆µν qν + qµ = qµNS − τqΠ Π uµ − τqπ πµν uν

+ `qΠ∇µΠ − `qπ ∆µν ∂λπνλ + τq ωµν qν −κ

βδ1 qµ θ

− λqq σµν qν + λqΠ Π∇µα + λqπ πµν ∇να

τπ π<µν> + πµν = πµνNS + 2 τπq q<µuν>

+ 2 `πq∇<µqν> + 2 τπ π<µ

λ ων>λ− 2 η δ2 πµν θ

− 2 τπ π<µ

λ σν>λ − 2λπq q<µ∇ν>α + 2λπΠ Πσµν

W. Israel, J.M. Stewart, Ann. Phys. 118 (1979) 341W. Israel, J.M. Stewart, Ann. Phys. 118 (1979) 341A. Muronga, PRC 76 (2007) 014909; A. Muronga, ibid.B. Betz, D. Henkel, D. H. Rischke, Prog. Part. Nucl. Phys. 62:556 (2009); B. Betzet.al. EPJ Web of Conf (2011).



So Many Jokes, So Little Time ...

Seriously...

The new method leads to equations and coefficients that are in better agreementwith the non-equilibrium solution of the Boltzmann equation, (than the othersout there)!

Hard to see the fine details, though !

Different choices (of methods, moments, coefficients etc.) lead to the sameasymptotic (Navier-Stokes) solutions, (but not the other way around)!

New terms from the second order collision integral,(φpφp′ − φkφk′

)(to be

published soon)

References

B. Betz, G. S. Denicol, T. Koide, E. Molnar, H. Niemi and D. H. Rischke,“Second order dissipative fluid dynamics from kinetic theory,”EPJ Web Conf. 13, 07005 (2011).

G. S. Denicol, H. Niemi, E. Molnar and D. H. Rischke, “Derivation of transientrelativistic fluid dynamics from the Boltzmann equation,”Phys. Rev. D 85, 114047 (2012).

G. S. Denicol, E. Molnar, H. Niemi and D. H. Rischke, “Derivation of fluiddynamics from kinetic theory with the 14–moment approximation,”Eur. Phys. J. A (2012).



Backup slides

Backup



Approximate solutions: Grad’s method of moments

The previous definitions from kinetic theory are only useful if we can specify fk or δfk

fk = f0k + δfk ' f0k (1 + φk)

where φk = δfk/f0k � 1 close to equilibrium, f0k ≡ [exp (−α0 + β0Ek) + a]−1, whereα0 = µ0/T0 is the chemical potential, β0 = 1/T0 is the inverse temperature,Ek = kµuµ, and a = 0 for Boltzmann, a = 1 for Fermi, a = −1 for Bose statistics.

Relativistic Grad’s approach: Stewart (1972), Israel and Stewart (1977-1979)

φk ≡∞∑l=0

εµ1...µl(l)

kµ1 ...kµl = ε(0) + εµ(1)

kµ + εµν(2)

kµkν + . . .

Assuming that in φk only l = 0, 1, 2-rank tensors appear λ(1), λµ(4) and λµν(9)where λµµ = 0, we provide a closure for the hierarchy (truncation) and be able todetermine the 14 moments of dissipative fluid dynamics!



Applying the 14-method

Non-equilibrium 14-moment ansatz

fk ≡ f0k

[1 + ε(0) + Ekε

µ(1)

uµ + ε〈µ〉(1)

k〈µ〉 + E2k εµν(2)

uµuν +1

3

(∆αβkαkβ

)∆µνε

µν(2)

+ Ekε〈µ〉ν(2)

k〈µ〉uν + Ekε〈µ〉ν(2)

k〈ν〉uµ + ε〈µν〉(2)

k〈µkν〉

]which is basically an expansion in En

k and k〈µ1. . . kµn〉

Decomposition and matching

ε(0) − α0 = A0Π

εµ(1)

uµ − β0 = A1Π

εµν(2)

uµuν = A2Π

ε〈µ〉(1)

= B0qµ + B1V

µ

ε〈µ〉β(2)

uβ = C0qµ + C1V

µ

εµν(2)

= A3(3uµuν −∆µν)Π + 2C0u(µqν) + 2C1u

(µV ν) + D0πµν

ε〈µν〉(2)

= D0πµν

The coefficients ε(0), εµ(1), εµν

(2)expressed in terms of dissipative fields, Π, qµ(Vµ), πµν



The relaxation equations

Using the 14-moment ansatz, the 3rd moment, 〈kµkνkλ〉 = F (e, n, p, uµ,Π, qµ, πµν),therefore, using the Boltzmann transport eq., ∂λ〈kµkνkλ〉 = 〈Cµν〉, we get:

Israel-Stewart (1979)

uµuν∂λ〈kµkνkλ〉 = uµuν〈Cµν〉 Bulk eq.

∆αµuν∂λ〈kµkνkλ〉 = ∆α

µuν〈Cµν〉 Heat-flow eq.

∆αβµν ∂λ〈kµkνkλ〉 = ∆αβ

µν 〈Cµν〉 Shear viscosity eq.

The linearised collision integral

uµuν〈Cµν〉 = CΠΠ

∆αµuν〈Cµν〉 = Cqq

α

∆αβµν 〈Cµν〉 = Cππ

αβ

where CΠ,Cq ,Cπ ∼ 〈σ〉



Further corrections

Take into account 2nd order corrections in the collision integral, i.e.,C [f0 + δf ] ∼ δf + (δf )2

Full collision integral: Rischke et.al. (2010)

uµuν〈Cµν〉 ∼ CΠΠ + A0Π2 + A1Vµqµ + A2q

µqµ + A3πµνπµν

∆αµuν〈Cµν〉 ∼ Cqq

α + B0Πqα + B1ΠVα + B2qµπµα + B3Vµπ

µα

∆αβµν 〈Cµν〉 ∼ Cππ

αβ + C0Ππαβ + C1q〈αV β〉 + C2q

〈αqβ〉 + C3παλπ

βλ

some terms also by Moore (2009)

Problem with the traditional moment method

uµ1 . . . uµn∂λ〈kµ1 . . . kµnkλ〉 = uµ1 . . . uµn 〈Cµ1...µn 〉

∆αµ1

uµ2 . . . uµn∂λ〈kµ1 . . . kµnkλ〉 = ∆αµ1

uµ2 . . . uµn 〈Cµ1...µn 〉

∆αβµ1µ2

uµ3 . . . uµn∂λ〈kµ1 . . . kµnkλ〉 = ∆αβµ1µ2

uµ3 . . . uµn 〈Cµ1...µn 〉

Parts of higher order moments may contribute for Π, qµ, πµν , so why stop ?


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