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

Ideal relativistic fluid dynamics in heavy-ion collisions Strongly inspired by the lectures by J.-Y . Ollitrault, Eur. J. Phys. 29 (2008) 275 = arXiv:0708.2433 [nucl-th]

Length scales & dimensionless numbers Three length scales: size over which macroscopic fields vary: ( ) ∼ 1 / ∂ µ L mean free path ℓ mfp 1 density , mean cross-section : � σ � ℓ mfp ≃ n n � σ � ∼ 1 thermal wavelength ( for a massless particle) λ th T 1 Rem.: n ∼ T 3 ∼ λ 3 th

Length scales & dimensionless numbers Kn ≡ ℓ mfp Knudsen number: L : free-streaming limit Kn ≫ 1 : “hydrodynamical” limit Kn ≪ 1 assumed in the following! ℓ mfp ⇔ : weak-coupling limit � σ � ≪ λ 2 ≫ 1 th λ th ℓ mfp � σ � ≫ λ 2 ⇔ : strong-coupling limit ( ill-defined...) ≪ 1 ℓ mfp th λ th

Length scales & dimensionless numbers Kn ≡ ℓ mfp Knudsen number: L : free-streaming limit Kn ≫ 1 : “hydrodynamical” limit Kn ≪ 1 assumed in the following! ℓ mfp ⇔ : weak-coupling limit � σ � ≪ λ 2 ≫ 1 th λ th ℓ mfp � σ � ≫ λ 2 ⇔ : strong-coupling limit ( ill-defined...) ≪ 1 ℓ mfp th λ th

Length scales & dimensionless numbers Kn ≡ ℓ mfp Knudsen number: L Re ≡ eLv fluid Reynolds number: η energy density, fluid velocity, shear viscosity η ∼ e ℓ mfp c s e v fluid : inviscid (“ideal”) flow Re ≫ 1 : viscous flow Re ≪ 1 (remark: should be replaced by the enthalpy density for a e + P e relativistic fluid) Ma ≡ v fluid Mach number: c s : incompressible flow Ma ≪ 1 : compressible, supersonic flow Ma � 1

Length scales & dimensionless numbers Kn ≡ ℓ mfp Knudsen number: L Re ≡ eLv fluid Reynolds number: η energy density, fluid velocity, shear viscosity η ∼ e ℓ mfp c s e v fluid : inviscid (“ideal”) flow Re ≫ 1 : viscous flow Re ≪ 1 (remark: should be replaced by the enthalpy density for a e + P e relativistic fluid) Ma ≡ v fluid Mach number: c s : incompressible flow Ma ≪ 1 : compressible, supersonic flow Ma � 1

Length scales & dimensionless numbers Kn ≡ ℓ mfp Knudsen number: L Re ≡ eLv fluid Reynolds number: η Ma ≡ v fluid Mach number: c s Kn × Re ∼ Ma

Length scales & dimensionless numbers Kn ≡ ℓ mfp Knudsen number: L Re ≡ eLv fluid Reynolds number: η Ma ≡ v fluid Mach number: c s Kn × Re ∼ Ma In the case of a heavy-ion collision the fluid expands into the vacuum: obviously a compressible flow! viscous effects measure the departure from (“ideal fluid Kn ≪ 1 dynamics”)

Thermodynamics Differential of internal energy U : (1) d U = − P d V + T d S + µ d N P pressure, V volume, T temperature, S entropy, chemical potential; µ 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 Differential of internal energy U : (1) d U = − P d V + T d S + µ d N P pressure, V volume, T temperature, S entropy, chemical potential; µ 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. Internal energy: U = − PV + TS + µN Gibbs-Duhem relation: V d p = S d T + N d µ

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

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

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

Equation of state Relation between P and e . P = e For instance : gas of massless (relativistic!) particles. 3 More realistic: lattice QCD equation of state (here energy density: ). ε Phys. Rev. D 77 (2008) 045511

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 of the fluid are isotropic. Fluid velocity: velocity of the fluid rest frame in the lab. frame. � v � 1 u 0 = v 4-velocity , is a 4-vector. � u = √ √ v 2 v 2 1 − � 1 − � function of . u µ x µ

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 of the fluid are isotropic. Fluid velocity: velocity of the fluid rest frame in the lab. frame. � v � 1 u 0 = v 4-velocity , is a 4-vector. � u = √ √ v 2 v 2 1 − � 1 − � function of . u µ x µ

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 of the fluid are isotropic. Fluid velocity: velocity of the fluid rest frame in the lab. frame. � v � 1 u 0 = v 4-velocity , is a 4-vector. � u = √ √ v 2 v 2 1 − � 1 − � function of . u µ x µ

Relativistic fluid dynamics Baryon number conservation ∂ρ In non-relativistic fluid dynamics: ∂ t + � ∇ · ( ρ� v ) = 0 In a relativistic system, the volume is contracted by a factor : u 0 the baryon density in the moving frame is . nu 0 (5) ∂ µ ( nu µ ) = 0

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

Relativistic fluid dynamics Energy and momentum conservation Energy-momentum tensor : T µ ν - : energy density; T 00 - : density of the j th component of momentum ( j = 1,2,3); T 0 j T i 0 - : energy flux along axis i ; - : flux along axis i of the j th component of momentum. T ij In the rest frame of the fluid, isotropy implies that and vanish T i 0 T 0 j and that is proportional to the identity matrix: T ij   e 0 0 0 0 P 0 0   T (0) =   0 0 P 0   0 0 0 P T µ ν = ( e + P ) u µ u ν − Pg µ ν For an arbitrary fluid velocity (6)

Relativistic fluid dynamics Energy and momentum conservation T µ ν = ( e + P ) u µ u ν − Pg µ ν (6) ∂ µ T µ ν = 0 (7) (5) ∂ µ ( nu µ ) = 0 5 equations, 6 unknowns ( , P , , )... n � e v Closed systems of equations with the equation of state!

Relativistic fluid dynamics Energy and momentum conservation To first-order in the velocity:   ( e + P ) v x ( e + P ) v y ( e + P ) v z e ( e + P ) v x 0 0 P   T =   ( e + P ) v y 0 0 P   ( e + P ) v z 0 0 P ∂ e (8) ∂ t + � ∇ · (( e + P ) � v ) = 0 ∂ (9) v ) + � ∇ P = � ∂ t (( e + P ) � 0

Relativistic fluid dynamics � e ( x µ ) = e 0 + δ e ( x µ ) Small perturbations (sound waves): P ( x µ ) = P 0 + δ P ( x µ ) ∂ ( δ e ) Eq.(8) reads (10) + ( e 0 + P 0 ) � v = 0 ∇ · � ∂ t Energy density decreases if the velocity field diverges (volume grows).

Relativistic fluid dynamics � e ( x µ ) = e 0 + δ e ( x µ ) Small perturbations (sound waves): P ( x µ ) = P 0 + δ P ( x µ ) ∂ ( δ e ) Eq.(8) reads (10) + ( e 0 + P 0 ) � v = 0 ∇ · � ∂ t Energy density decreases if the velocity field diverges (volume grows). ( e 0 + P 0 ) ∂� v (11) Eq.(9) becomes ∂ t + � ∇ δ P = � 0 Inertia of the fluid times its acceleration is equal to the force.

Relativistic fluid dynamics � e ( x µ ) = e 0 + δ e ( x µ ) Small perturbations (sound waves): P ( x µ ) = P 0 + δ P ( x µ ) ∂ ( δ e ) Eq.(8) reads (10) + ( e 0 + P 0 ) � v = 0 ∇ · � ∂ t Energy density decreases if the velocity field diverges (volume grows). ( e 0 + P 0 ) ∂� v (11) Eq.(9) becomes ∂ t + � ∇ δ P = � 0 Inertia of the fluid times its acceleration is equal to the force. � ∂ P � Define c 2 s ≡ ∂ e | s n ∂ 2 ( δ e ) Eqs.(10)-(11) give − c 2 s △ ( δ e ) = 0 ∂ t 2 Wave equation in 3+1 dimension, velocity of sound. c s