From kinetic theory to dissipative fluid dynamics

Lecture at USTC, Hefei, China, October 6, 28 1 From kinetic theory to dissipative fluid dynamics Dirk H. Rischke Institut für Theoretische Physik and Frankfurt Institute for Advanced Studies with: Barbara Betz, Daniel Henkel thanks to: Ulrich W. Heinz, Giorgio Torrieri, Urs A. Wiedemann

Lecture at USTC, Hefei, China, October 6, 28 2 Preliminaries (I) Tensor decomposition of net charge current and energy-momentum tensor: 1. Net charge current: N µ = n u µ + ν µ u µ fluid 4-velocity, u µ u µ = u µ g µν u ν = 1 g µν diag(+,,, ) (West coast!!) metric tensor, n u µ N µ net charge density in fluid rest frame ν µ µν N ν diffusion current (flow of net charge relative to u µ ), ν µ u µ = µν = g µν u µ u ν projector onto 3-space orthogonal to u µ, µν u ν = 2. Energy-momentum tensor: T µν = ǫ u µ u ν (p + Π) µν + 2 q (µ u ν) + π µν ǫ u µ T µν u ν p energy density in fluid rest frame pressure in fluid rest frame Π bulk viscous pressure, p + Π 1 3 µν T µν q µ µν T νλ u λ heat flux current (flow of energy relative to u µ ), q µ u µ = π µν T <µν> shear stress tensor, π µν u µ = π µν u ν =, π µ µ = a (µν) 1 2 (aµν + a νµ ) symmetrized tensor ( a <µν> α (µ ν) β 1 ) 3 µν αβ a αβ symmetrized, traceless spatial projection

Lecture at USTC, Hefei, China, October 6, 28 3 Preliminaries (II) Fluid dynamical equations: 1. Net charge conservation: µ N µ = ṅ + n θ + ν = ȧ u µ µ a convective (comoving) derivative (time derivative in fluid rest frame, ȧ RF t a) θ µ u µ expansion scalar 2. Energy-momentum conservation: µ T µν = energy conservation: u ν µ T µν = ǫ + (ǫ + p + Π) θ + q q u π µν µ u ν = acceleration equation: µν λ T νλ = (ǫ+p) u µ = µ (p+π) Π u µ µν q ν q µ θ q ν ν u µ µν λ π νλ µ µν ν 3-gradient, (spatial gradient in fluid rest frame, u µ RF (1,,, ))

Lecture at USTC, Hefei, China, October 6, 28 4 Preliminaries (III) Problem: 5 equations, but 15 unknowns (for given u µ ): ǫ, p, n, Π, ν µ (3), q µ (3), π µν (5) Solution: 1. clever choice of frame (Landau, Eckart,...): eliminate ν µ or q µ = does not help! Promotes u µ to dynamical variable! 2. ideal fluid limit: all dissipative terms vanish, Π = ν µ = q µ = π µν = = 6 unknowns: ǫ, p, n, u µ (3) (not quite there yet...) = fluid is in local thermodynamical equilibrium = provide equation of state (EOS) p(ǫ, n) to close system of equations 3. provide additional equations for dissipative quantities = dissipative relativistic fluid dynamics (a) First-order theories: e.g. generalization of Navier-Stokes (NS) equations to the relativistic case (Landau, Lifshitz) (b) Second-order theories: e.g. Israel-Stewart (IS) equations

Lecture at USTC, Hefei, China, October 6, 28 5 Preliminaries (IV) Navier-Stokes (NS) equations: 1. bulk viscous pressure: Π NS = ζ θ ζ bulk viscosity 2. heat flux current: q µ NS = κ β n β(ǫ + p) µ α β 1/T 3. shear stress tensor: π µν NS = 2 η σ µν inverse temperature, α β µ, µ chemical potential, κ η thermal conductivity shear viscosity, σ µν = <µ u ν> shear tensor = algebraic expressions in terms of thermodynamic and fluid variables = simple... but: unstable and acausal equations of motion!!

Lecture at USTC, Hefei, China, October 6, 28 6 Motivation (I) v 2 (percent) 25 2 15 1 ideal η/s=.3 η/s=.8 η/s=.16 STAR P. Romatschke, U. Romatschke, PRL 99 (27) 17231 Au+Au @ s = 2 AGeV charged particles, min. bias 5 1 2 3 4 p T [GeV] H. Song, U.W. Heinz, PRC 78 (28) 2492 v 2.2 ideal hydro viscous hydro: simplified I-S eqn. Cu+Cu, b=7 fm SM-EOS Q.5 1 1.5 p T (GeV) (a) ideal hydro (b) ideal hydro viscous hydro: simplified I-S eqn. viscous hydro: simplified I-S eqn. (c) viscous hydro: full I-S eqn. η/s=.8, τ π =3η/sT η/s=.8, τ π =3η/sT e =3 GeV/fm 3 e =3 GeV/fm 3.1 τ =.6fm/c τ =.6fm/c T dec =13 MeV T dec =13 MeV T dec =13 MeV Au+Au, b=7 fm SM-EOS Q.5 1 1.5 p T (GeV) η/s=.8, τ π =3η/sT e =3 GeV/fm 3 τ =.6fm/c Au+Au, b=7 fm EOS L.5 1 1.5 2 p T (GeV)

Lecture at USTC, Hefei, China, October 6, 28 7 Motivation (II) Israel-Stewart (IS) equations: second-order, dissipative relativistic fluid dynamics W. Israel, J.M. Stewart, Ann. Phys. 118 (1979) 341 Simplified IS equations: e.g. shear stress tensor τ π π <µν> + π µν = π µν NS = dynamical (instead of algebraic) equations for dissipative terms! = π µν relaxes to its NS value π µν NS on the time scale τ π = stable and causal fluid dynamical equations of motion! Full IS equations: τ π π <µν> + π µν = π µν NS η 2β πµν λ τ π η β uλ + 2 τ π π <µ λ ω ν>λ ω µν 1 2 µα νβ ( α u β β u α ) vorticity

Lecture at USTC, Hefei, China, October 6, 28 8 Motivation (III) Note: η 2β πµν λ τ π η β uλ = τ π 2 πµν θ η 2β πµν τ π η β 2 η ˆδ 2 π µν θ Proof: define ϕ(α, β) τ π η β net charge conservation: ṅ n α energy conservation: ǫ ǫ α α + n β β n θ ǫ α + β (ǫ + p) θ β (Note: ideal fluid limit O.K., as ϕ is multiplied with small quantity π µν ) = linear system of eqs. for α and β = α = A(α, β) θ, β = B(α, β) θ = ϕ ϕ α = ˆδ 2 1 4 β ϕ α + β β = ϕ(α, β) + ϕ α ϕ ϕ A(α, β) + B(α, β) θ β ϕ A(α, β) + B(α, β), q.e.d. β α

Lecture at USTC, Hefei, China, October 6, 28 9 Motivation (IV) = Difference between simplified and full IS equations: the latter include higher-order terms? For instance, if π µν ǫ δ 1, τ π θ δ 1 = τ π θ πµν ǫ δ 2 = Goals: 1. What are the correct equations of motion for the dissipative quantities? = develop consistent power counting scheme 2. Generalization to µ (relevant for FAIR physics!) = include heat flux q µ 3. Generalization to non-conformal fluids (relevant near T c!) = include bulk viscous pressure Π

Lecture at USTC, Hefei, China, October 6, 28 1 Results (I) Power counting: 3 length scales: 2 microscopic, 1 macroscopic thermal wavelength λ th β 1/T mean free path l mfp ( σ n) 1 σ averaged cross section, n T 3 = β 3 λ 3 th length scale over which macroscopic fluid fields vary L hydro, µ L 1 hydro Note: since η ( σ λ th ) 1 = l mfp λ th 1 σ n 1 λ th λ3 th σ λ th λ3 th σ λ th η s s entropy density, s n T 3 = β 3 λ 3 th = η s solely determined by the 2 microscopic length scales! Note: similar argument holds for ζ s, κ β s

Lecture at USTC, Hefei, China, October 6, 28 11 Results (II) 3 regimes: dilute gas limit l mfp λ th η s 1 σ λ2 th = weak-coupling limit viscous fluids l mfp λ th η s 1 σ λ2 th interactions happen on the scale λ th = moderate coupling ideal fluid limit l mfp λ th η s 1 σ λ2 th = strong-coupling limit gradient (derivative) expansion: l mfp µ l mfp L hydro δ 1 = equivalent to k l mfp 1, k typical momentum scale R. Baier, P. Romatschke, D.T. Son, A.O. Starinets, M.A. Stephanov, arxiv:712.2451 = separation of macroscopic fluid dynamics (large scale L hydro ) from microscopic particle dynamics (small scale l mfp )

Lecture at USTC, Hefei, China, October 6, 28 12 Results (III) Primary quantities: ǫ, p, n, s Dissipative quantities: Π, q µ, π µν If l mfp µ δ 1, then Π ǫ qµ ǫ πµν ǫ δ 1 Dissipative quantities are small compared to primary quantities = small deviations from local thermodynamical equilibrium! Note: statement independent of value of ζ s, κ β s, η s! Proof: Gibbs relation: ǫ + p = Ts + µn = β ǫ s! Estimate dissipative terms by their Navier-Stokes values: Π Π NS = ζ θ, q µ q µ NS = κ n β β(ǫ + p) µ α, π µν π µν NS = 2 η σµν = Π ǫ ζ β ǫ β θ ζ β λ th l mfp θ l mfp µ u µ δ, s λ th l mfp q µ ǫ κ 1 n β β ǫβ(ǫ + p) β µ α κ β λ th l mfp µ α l mfp µ α δ, β s λ th l mfp π µν 2 η ǫ β ǫ β σµν 2 η β λ th l mfp σ µν l mfp <µ u ν> δ, q.e.d. s λ th l mfp

Lecture at USTC, Hefei, China, October 6, 28 13 Results (IV) IS equations (derived by Grad s 14 moment method, cf. back-up slides): τ Π Π + Π = Π NS + τ Πq q u l Πq q ζ ˆδ Π θ + λ Πq q α + λ Ππ π µν σ µν τ q µν q ν + q µ = q µ NS τ qπ Π u µ τ qπ π µν u ν + l qπ µ Π l qπ µν λ π νλ + τ q ω µν q ν κ β ˆδ 1 q µ θ λ qq σ µν q ν + λ qπ Π µ α + λ qπ π µν ν α τ π π <µν> + π µν = π µν NS + 2 τ πq q <µ u ν> + 2 l πq <µ q ν> + 2 τ π π <µ λ ω ν>λ 2 η ˆδ 2 π µν θ 2τ π π <µ λ σ ν>λ 2 λ πq q <µ ν> α + 2 λ ππ Π σ µν W. Israel, J.M. Stewart, Ann. Phys. 118 (1979) 341 W. Israel, J.M. Stewart, Ann. Phys. 118 (1979) 341 A. Muronga, PRC 76 (27) 1499 A. Muronga, PRC 76 (27) 1499 this work; B. Betz, D. Henkel, DHR, in preparation

Lecture at USTC, Hefei, China, October 6, 28 14 Results (V) Remarks: 1. Derivation is based on kinetic theory (Boltzmann equation) 2. R. Baier, P. Romatschke, D.T. Son, A.O. Starinets, M.A. Stephanov, arxiv:712.2451: second-order term λ 1 π <µ η 2 λ π ν>λ is contained in these equations: since π µν π µν NS = 2 η σ µν = λ 1 η 2 π <µ λ π ν>λ 2 λ 1 η π <µ λ σ ν>λ = λ 1 τ π η (second-order terms from collision integral may change this result!) 3. Coefficients ˆδ, ˆδ 1, ˆδ 2 are (complicated) functions of α, β 4. Viscosities and thermal conductivity ζ, η, κ, relaxation times τ Π, τ q, τ π, coefficients τ Πq, τ qπ, τ qπ, τ πq, l Πq, l qπ, l qπ, l πq, λ Πq, λ Ππ, λ qq, λ qπ, λ qπ, λ πq, λ ππ are (complicated) functions of α, β, divided by tensor coefficients of second moment of collision integral, C µν [f]: χ i (α, β)/ σ as cross section σ ( strong coupling limit!) = Π = q µ = π µν ideal fluid limit! 5. IS equations are formally independent of calculational frame (Eckart, Landau,...), but...

Lecture at USTC, Hefei, China, October 6, 28 15 Results (VI) 6. Values of coefficients are frame dependent! We have analyzed: (a) Eckart (N) or (net) charge frame: ν µ =, ǫ = ǫ, n = n ǫ, n : energy density and charge density in local thermodyn. equilibrium (b) Landau (E) or energy frame: q µ =, ǫ = ǫ, n = n Note: in IS equations q µ ǫ + p n νµ (c) Tsumura-Kunihiro-Ohnishi (TKO) frame: ν µ =, ǫ = ǫ 3 Π, n = n We have checked agreement with the results of IS for most coefficients computed by IS... 7. R.h.s.: all terms except NS terms are of second order, δ 2 = t < τ Π τ q τ π : dissipative terms relax towards their NS values, t > τ Π τ q τ π : last terms on r.h.s. and NS terms on l.h.s. largely cancel, second-order terms govern evolution!

Lecture at USTC, Hefei, China, October 6, 28 16 Conclusions and open problems 1. Derived Israel-Stewart (IS) equations from kinetic theory via Grad s 14-moment method = new second-order terms! 2. Results consistent with R. Baier, P. Romatschke, D.T. Son, A.O. Starinets, M.A. Stephanov, arxiv:712.2451 3. Coefficients of terms in IS equations are frame dependent = have (not yet completely) been computed in various frames (Eckart, Landau, TKO) 4. Generalization to a system of various particle species (done: quarks, antiquarks, gluons), various conserved charges 5. Numerical implementation

Lecture at USTC, Hefei, China, October 6, 28 17 Derivation: Grad s 14-moment method (I) 1. history: derivation of IS equations from kinetic theory H. Grad, Commun. Pure Appl. Math. 2 (1949) 381 J.M. Stewart, Proc. Roy. Soc. London Ser. A 357 (1977) 59 W. Israel, J.M. Stewart, Ann. Phys. 118 (1979) 341 DHR, PhD thesis, 1992 (unpublished) R. Baier, P. Romatschke, U.A. Wiedemann, PRC 73 (26) 6493 A. Muronga, PRC 76 (27) 1499, 1491 2. single-particle distribution function in local thermodynamical equilibrium: f = (e y + a) 1, y = α β K u a = ±1, for fermions/bosons, Boltzmann particles local: α α (X), β β (X), u µ u µ (X), X µ = (t, x) single-particle distribution function in local non-equilibrium: f = (e y + a) 1, y = α βk u K v + K µ K ν w µν v u =, w µ µ =

Lecture at USTC, Hefei, China, October 6, 28 18 Derivation: Grad s 14-moment method (II) 3. assume small deviations from equilibrium: y y = α α (β β ) K u K v + K µ K ν w µν δ 1 α α β β v µ w µν δ = expand f around f to linear order in y y : f f + f (1 a f )(y y ) + O(δ 2 ) 4. compute N µ and T µν as moments of f: N µ = d K K µ f = I µ + (α α ) J µ (β β ) J µν u ν J µν v ν + J µνλ w νλ T µν = d K K µ K ν f = I µν +(α α ) J µν (β β ) J µνλ u λ J µνλ v λ +J µνλρ w λρ where d K g d 3 k (2π) 3 E, E = k 2 + m 2, g no. of internal d.o.f. s, and I α 1 α n d K K α1 K α n f, J α 1 α n d K K α1 K α n f (1 af ) Note: I µ N µ net charge current in local thermodyn. equilibrium I µν T µν energy-momentum tensor in local thermodyn. equilibrium

Lecture at USTC, Hefei, China, October 6, 28 19 Derivation: Grad s 14-moment method (III) 5. tensor decomposition of I α 1 α n up to n = 3 and of J α 1 α n up to n = 5: I µ = I 1 u µ, I µν = I 2 u µ u ν + I 21 µν, I µνλ = I 3 u µ u ν u λ +3 I 31 u (µ νλ) J µ = J 1 u µ, J µν = J 2 u µ u ν + J 21 µν, J µνλ = J 3 u µ u ν u λ + 3 J 31 u (µ νλ) J µνλρ = J 4 u µ u ν u λ u ρ + 6 J 41 u (µ u ν λρ) + 3 J 42 µ(ν λρ) J µνλρσ = J 5 u µ u ν u λ u ρ u σ +1 J 51 u (µ u ν u λ ρσ) +15 J 52 u (µ νλ ρσ) Note: I 1 n, I 2 ǫ, I 21 p In general: I nq (2q+1)!! 1 d K E n 2q ( k 2 ) q f J nq 1 (2q+1)!! d K E n 2q ( k 2 ) q f (1 af ) 6. insert tensor decomposition for I α 1 α n, J α 1 α n, as well as w µν = w ( u µ u ν 1 3 µν) + 2 w(µ u ν) + w µν w µ u µ = w µν u ν = w µν u µ = ; w <µν> = w µν into expansion for N µ, T µν, cf. 4.

Lecture at USTC, Hefei, China, October 6, 28 2 Derivation: Grad s 14-moment method (IV) 7. choose a specific frame, i.e., impose matching conditions (cf. Results (V), 6.) = α α = w χ α (α, β), β β = w χ β (α, β), v µ = w µ χ v (α, β) 8. re-insert into tensor decomposition for N µ, T µν = w = Π χ Π (α, β), w µ = q µ χ q (α, β), w µν = π µν χ π (α, β) 9. compute third moment of f: S µνλ = d K K µ K ν K λ f = S 1 u µ u ν u λ + 3 S 2 u (µ νλ) + 3 ψ 1 q (µ u ν u λ) + 3 ψ 2 q (µ νλ) + 3 ψ 3 π (µν u λ) S 1 = I 3 + ψ 4 Π, S 2 = I 31 1 3 ψ 4 Π, ψ i = ψ i (α, β), i = 1,..., 4 1. compute second moment of collision integral C[f]: C νλ = d K K ν K λ C[f] = A Π ( u ν u λ 1 3 νλ) + B π νλ + 2 C u (ν q λ) A, B, C σ

Lecture at USTC, Hefei, China, October 6, 28 21 Derivation: Grad s 14-moment method (V) 11. compute second moment of Boltzmann equation, K f = C[f]: = µ S µνλ = C νλ or, tensor-decomposed: A Π ψ 4 Π = β J 41 θ + I 3 + ψ 4 Π + 5 3 ψ 4 Π θ + ψ 1 q q ψ 1 2 ψ 1 q u 2 ψ 3 π µν σ µν C q µ ψ 1 µν q ν = β J 41 u µ + µ J 31 1 3 ψ 4 µ Π 1 3 Π µ ψ 4 + 5 3 ψ 4 Π u µ + (ψ 1 2 ψ 2 ) q ν σ µν ψ 1 q ν ω µν + q µ [ ψ 1 + 1 3 (4 ψ 1 5 ψ 2 ) θ ] + ψ 3 µν λ π νλ + π µν ( ν ψ 3 ψ 3 u ν ) B π µν ψ 3 π <µν> = 2 I 31 σ µν 2 3 ψ 4 Π σ µν + 2 ψ 2 <µ q ν> + 2 q <µ ν> ψ 2 + 2 (ψ 1 ψ 2 ) q <µ u ν> + 2 ψ 3 π <µ λ σ ν>λ 2 ψ 3 π <µ λ ω ν>λ + π µν ( ψ 3 + 5 3 ψ 3 θ ) 12. substitute I 3, β J 41 u µ, µ I 31, ψ i, µ ψ i, i = 1,..., 4 with the help of energy conservation and acceleration equation = IS equations with explicit (frame-dependent) expressions for coefficients