Ideal relativistic fluid dynamics in heavy-ion collisions Strongly - - PowerPoint PPT Presentation

Strongly inspired by the lectures by J.-Y . Ollitrault, Eur. J. Phys. 29 (2008) 275 = arXiv:0708.2433 [nucl-th]

Ideal relativistic fluid dynamics in heavy-ion collisions

Length scales & dimensionless numbers

Three length scales: size over which macroscopic fields vary: ( ) mean free path density , mean cross-section : thermal wavelength ( for a massless particle) Rem.: L ℓmfp ℓmfp ≃ 1 nσ σ n λth ∼ 1 T ∼ 1/∂µ n ∼ T 3 ∼ 1 λ3

th

σ ≫ λ2

th

Knudsen number: : free-streaming limit : “hydrodynamical” limit assumed in the following! ⇔ : weak-coupling limit ⇔ : strong-coupling limit ( ill-defined...)

Length scales & dimensionless numbers

Kn ≡ ℓmfp L Kn ≫ 1 Kn ≪ 1 ℓmfp λth ≫ 1 ℓmfp λth ≪ 1 σ ≪ λ2

th

ℓmfp

σ ≫ λ2

th

Knudsen number: : free-streaming limit : “hydrodynamical” limit assumed in the following! ⇔ : weak-coupling limit ⇔ : strong-coupling limit ( ill-defined...)

Length scales & dimensionless numbers

Kn ≡ ℓmfp L Kn ≫ 1 Kn ≪ 1 ℓmfp λth ≫ 1 ℓmfp λth ≪ 1 σ ≪ λ2

th

ℓmfp

Knudsen number: Reynolds number: energy density, fluid velocity, shear viscosity : inviscid (“ideal”) flow : viscous flow (remark: should be replaced by the enthalpy density for a relativistic fluid) Mach number: : incompressible flow : compressible, supersonic flow

Length scales & dimensionless numbers

Kn ≡ ℓmfp L Re ≡ eLvfluid η Ma ≡ vfluid cs η ∼ eℓmfpcs Re ≪ 1 Re ≫ 1 e vfluid e e + P Ma ≪ 1 Ma 1

Knudsen number: Reynolds number: energy density, fluid velocity, shear viscosity : inviscid (“ideal”) flow : viscous flow (remark: should be replaced by the enthalpy density for a relativistic fluid) Mach number: : incompressible flow : compressible, supersonic flow

Length scales & dimensionless numbers

Kn ≡ ℓmfp L Re ≡ eLvfluid η Ma ≡ vfluid cs η ∼ eℓmfpcs Re ≪ 1 Re ≫ 1 e vfluid e e + P Ma ≪ 1 Ma 1

Length scales & dimensionless numbers

Knudsen number: Reynolds number: Mach number: Kn ≡ ℓmfp L Re ≡ eLvfluid η Ma ≡ vfluid cs Kn × Re ∼ Ma

Length scales & dimensionless numbers

Knudsen number: Reynolds number: Mach number: Kn ≡ ℓmfp L Re ≡ eLvfluid η Ma ≡ vfluid cs Kn × Re ∼ Ma In the case of a heavy-ion collision the fluid expands into the vacuum:

• bviously a compressible flow!

viscous effects measure the departure from (“ideal fluid dynamics”) Kn ≪ 1

dU = −P dV + T dS + µ dN Differential of internal energy U: P pressure, V volume, T temperature, S entropy, chemical potential; N is the number of particles in non-relativistic thermodynamics; this is not conserved in a relativistic system: should be replaced by a conserved quantity (for instance, baryon number). In a relativistic system, U includes the mass energy.

Thermodynamics

µ (1)

dU = −P dV + T dS + µ dN Differential of internal energy U: P pressure, V volume, T temperature, S entropy, chemical potential; N is the number of particles in non-relativistic thermodynamics; this is not conserved in a relativistic system: should be replaced by a conserved quantity (for instance, baryon number). In a relativistic system, U includes the mass energy.

Thermodynamics

µ U = −PV + TS + µN V dp = S dT + Ndµ Internal energy: Gibbs-Duhem relation: (1)

Thermodynamics

In fluid dynamics, the useful quantities are the densities: energy density , entropy density , baryon number density . e ≡ U/V s ≡ S/V n ≡ N/V e = −P + Ts + µn de = T ds + µ dn dP = s dT + n dµ (3) Gibbs-Duhem: (2) (4) Eq.(1) gives

Thermodynamics

In an inviscid flow, the entropy sV is conserved: Eq.(1) gives then ds s = −dV V = dn n de e + P = −dV V

Equation of state

Relation between P and e. For instance : gas of massless (relativistic!) particles. P = e 3

Equation of state

Relation between P and e. For instance : gas of massless (relativistic!) particles. P = e 3

• Phys. Rev. D 77 (2008) 045511

ε More realistic: lattice QCD equation of state (here energy density: ).

Relativistic fluid dynamics

THE assumption: local thermodynamic equilibrium thermodynamic quantities vary slowly with space and time Fluid rest frame: for a fluid element, the frame in which its momentum vanishes. This is the frame where thermodynamic quantities are defined. Local thermodynamic equilibrium in that frame, the properties

• f the fluid are isotropic.

Fluid velocity: velocity of the fluid rest frame in the lab. frame. 4-velocity , is a 4-vector. function of .

• v

u0 = 1 √ 1 − v2

• u =
• v

√ 1 − v2 uµ xµ

Relativistic fluid dynamics

THE assumption: local thermodynamic equilibrium thermodynamic quantities vary slowly with space and time Fluid rest frame: for a fluid element, the frame in which its momentum vanishes. This is the frame where thermodynamic quantities are defined. Local thermodynamic equilibrium in that frame, the properties

• f the fluid are isotropic.

Fluid velocity: velocity of the fluid rest frame in the lab. frame. 4-velocity , is a 4-vector. function of .

• v

u0 = 1 √ 1 − v2

• u =
• v

√ 1 − v2 uµ xµ

Relativistic fluid dynamics

THE assumption: local thermodynamic equilibrium thermodynamic quantities vary slowly with space and time Fluid rest frame: for a fluid element, the frame in which its momentum vanishes. This is the frame where thermodynamic quantities are defined. Local thermodynamic equilibrium in that frame, the properties

• f the fluid are isotropic.

Fluid velocity: velocity of the fluid rest frame in the lab. frame. 4-velocity , is a 4-vector. function of .

• v

u0 = 1 √ 1 − v2

• u =
• v

√ 1 − v2 uµ xµ

Relativistic fluid dynamics

Baryon number conservation

In non-relativistic fluid dynamics: In a relativistic system, the volume is contracted by a factor : the baryon density in the moving frame is . ∂ρ ∂t + ∇ · (ρ v) = 0 ∂µ(nuµ) = 0 (5) nu0 u0

Relativistic fluid dynamics

Baryon number conservation

In non-relativistic fluid dynamics: In a relativistic system, the volume is contracted by a factor : the baryon density in the moving frame is . ∂ρ ∂t + ∇ · (ρ v) = 0 ∂µ(nuµ) = 0 (5) In the rest frame of the fluid, the baryon flux vanishes, otherwise it would define a preferred direction and thus break the isotropy. (No longer true in viscous fluid dynamics: diffusion). n u nu0 u0

Relativistic fluid dynamics

Energy and momentum conservation

Energy-momentum tensor :

• : energy density;
• : density of the jth component of momentum ( j = 1,2,3);
• : energy flux along axis i;
• : flux along axis i of the jth component of momentum.

T µν T 00 T 0j T i0 T ij In the rest frame of the fluid, isotropy implies that and vanish and that is proportional to the identity matrix: T i0 T ij T 0j T(0) =     e P P P     T µν = (e + P)uµuν − Pgµν For an arbitrary fluid velocity (6)

Relativistic fluid dynamics

Energy and momentum conservation

T µν = (e + P)uµuν − Pgµν ∂µT µν = 0 (7) (6) ∂µ(nuµ) = 0 (5) 5 equations, 6 unknowns ( , P, , )... Closed systems of equations with the equation of state! n v e

Relativistic fluid dynamics

To first-order in the velocity: T =     e (e + P)vx (e + P)vy (e + P)vz (e + P)vx P (e + P)vy P (e + P)vz P     ∂e ∂t + ∇ · ((e + P) v) = 0 ∂ ∂t((e + P) v) + ∇P =

Energy and momentum conservation

(8) (9)

Small perturbations (sound waves): e(xµ) = e0 + δe(xµ) P(xµ) = P0 + δP(xµ)

Relativistic fluid dynamics

∂(δe) ∂t + (e0 + P0) ∇ · v = 0 Energy density decreases if the velocity field diverges (volume grows). Eq.(8) reads (10)

Small perturbations (sound waves): e(xµ) = e0 + δe(xµ) P(xµ) = P0 + δP(xµ)

Relativistic fluid dynamics

∂(δe) ∂t + (e0 + P0) ∇ · v = 0 Energy density decreases if the velocity field diverges (volume grows). Eq.(8) reads (10) (e0 + P0)∂ v ∂t + ∇δP = Eq.(9) becomes Inertia of the fluid times its acceleration is equal to the force. (11)

Small perturbations (sound waves): e(xµ) = e0 + δe(xµ) P(xµ) = P0 + δP(xµ)

Relativistic fluid dynamics

∂(δe) ∂t + (e0 + P0) ∇ · v = 0 Energy density decreases if the velocity field diverges (volume grows). Eq.(8) reads (10) (e0 + P0)∂ v ∂t + ∇δP = Eq.(9) becomes Inertia of the fluid times its acceleration is equal to the force. (11) ∂2(δe) ∂t2 − c2

s△(δe) = 0

c2

s ≡

∂P ∂e

• | s

n

Define Wave equation in 3+1 dimension, velocity of sound. cs Eqs.(10)-(11) give