Causality of fluid dynamics for high-energy nuclear collisions - PowerPoint PPT Presentation

Causality of fluid dynamics for high-energy nuclear collisions Eduardo Grossi ITP Heidelberg University with S. Floerchinger arXiv:1711.06687 Cold Quantum Coffee, Heidelberg, 21/11/2017 1

Relativistic Hydrodynamics The equation of relativistic hydrodynamics: At global and homogeneous equilibrium the energy momentum tensor and charge current are: The basic assumption of hydrodynamics is the local equilibrium condition 2

Ideal relativistic hydrodynamics D � + ( � + p ) � · u = 0 ✦ Energy conservation ✦ Charge conservation Dn + n ∂ · u = 0 ( � + p ) Du ν � � ν p = 0 ✦ Euler equation With the equation of state this set of equation can be solved in a closed form S µ = su µ � µ S µ = 0 Ts = � + p − µn 3

figure from [Rezzolla, Zanotti “Relativistic Hydrodynamics”] Relativistic Hydrodynamics The equation of relativistic hydrodynamics: four-velocity energy density ( � + p ) u µ � bulk viscous pressure stress p + π bulk tensor 4

Navier- Stokes equations The general form of energy momentum tensor: The new terms depend on the value of the transport coefficient ✦ Heat flux ✦ Shear stress tensor ✦ Bulk viscous pressure ✦ Diffusion current Mathematically correspond to a system of parabolic equation. 5

Instability of Navier-Stokes Equation [W. Hiscock et al. PRD (1985)] Consider a traverse perturbation around equilibrium with the fluid at rest. δ Q = δ Q 0 e ikx + Γ t For k=0 we will have a growing mode and a damping mode in the transverse direction Γ + � ( � c 2 + p ) c 2 � T If we neglect the heat conductivity we obtain the diffusive mode like in non relativistic theory Γ = − � k 2 � + p

Instability of Navier-Stokes Equation [W. Hiscock et al. PRD (1985)] Nevertheless if we boost the reference frame the growing mode at k=0 will appear again Γ + � ( � c 2 + p ) c 2 δ Q = δ Q 0 e ik ( − γ vt + γ x )+ Γ ( γ t − v γ x ) � v 2 � The characteristic time scale it is really short. In non relativistic limit the this time goes to zero. � v 2 � � = ( � c 2 + p ) c 4 → 0 The non relativistic limit is safe respect of this instability.

Second order in gradients [R. Baier et al. JHEP 0804 (2008)] [S. Bhattacharyya et al., JHEP 0802 (2008)] Shear Tensor: [P. Romatschke Class.Quant.Grav. 27 (2010)] Bulk pressure: 8

Israel Stewart theory The Navier-Stokes theory is unstable and a-causal. Adding a second order term the equation became casual and stable ✦ 2 order in gradient ✦ Promoting as a variabile The shear tensor becomes a dynamical variabile that relax at its Navier-Stokes value ✦ Relaxation type equation New transport coefficient, the relaxation time [Muller 1967, Israel, Stewart 1976] [Hiscock 1983] 9

Hydro-Equation ✦ Energy conservation and Euler equation ✦ Relaxation equation for shear tensor ✦ Relaxation equation for bulk pressure A more general theory will have more terms in the equation for the stress tensor. This equation can be derive from Kinetic theory in consistent way. [G. S. Denicol et al. Phys. Rev. D (2012)] 10

0+1 solution The simplest approximation: the Bjorken model J. Bjorken, (1983). ✦ boost invariance along the z direction ✦ rotational and translation invariance in the transverse plane. Effectively it corresponds to 1+0 problem ✦ Every quantity depend only on the proper time. Describe only the central rapidity region, no radial velocity. Figure from W. van der Schee, (2014).

1+1 solution 1+1 Hydro ✦ boost invariance along the z direction ✦ rotational invariance in the transverse plane. ✦ Temperature ✦ Four velocity ✦ Shear and bulk This simplified setup nevertheless can be used to modeling the hydro evolution of an ultra central collision, where the initial energy profile is almost rotational invariant. Picture stolen from C. Shen et al . (2016) 12

1+1D hydro equation The previous hydro equation can be recast in the following matrix form choosing as variables Radial rapidity Temperature Shear tensor Bulk pressure The matrices A and B depend non-linearly by the variables. The source term reflect the dissipative behavior of these equation 13

First order hyperbolic quasi-linear PDE ∂ 0 Φ i + M ij ∂ 1 Φ j + S i = 0 This type of equation are called hyperbolic when is possible to diagonalize the matrix M w ( m ) λ ( m ) Eigenvalue Left eigenvector j With this variables we can partially diagonalized the system of pde. d J ( m ) = w ( m ) Riemann Invariants: d Φ j j ∂ 0 J ( m ) + λ ( m ) ∂ 1 J ( m ) + w ( m ) S j = 0 j

Riemann invariant for the perfect fluid The perfect fluid equation in cartesian coordinates in 2 dimension can be written in a simple form � 1 + v � � J ± = 1 1 2 ln ± d T c 2 1 − v s T These variables are transport constant on the characteristic curves: d x d t = v ± c s 1 ± vc s

Characteristics speed To find the characteristic velocity of the system of PDE one has to solve the eigenvalue problem Speed of signal propagation of viscous 1+1 D hydro The theory is casual and stable if the signal velocity are less of the speed of light 16

Energy integrals Any Hyperbolic PDE is linear stable, the solution of linearized equation is bounded. P ( x 0 , x 1 ) [Courant and Hilbert, (1953)] ✦ If the initial data vanish,then 
 P l P r x 0 = t also the solution vanish. x 0 C l C r ✦ Γ This define the domani of 
 dependence of a given point. A l A r x 0 = 0 x 1

Starting condition � 1 − x � s ( τ 0 , r ) = s 0 n W N ( r ) + xn BC ( r ) 2 Number wounded Number binary nucleon collision ✦ Initial vanish 
 T [ GeV ] 0.4 four velocity u µ = ( − 1 , 0 , 0 , 0) 0.3 ✦ Initial stress tensor 
 0.2 equal to Navier- 
 Stokes value 0.1 ✦ No bulk r [ fm ] 5 10 15 20 25 30 Corresponding to a Au-Au-200 GeV collision

Transport properties Sound velocity Viscosity c s 2 η / s 1.2 0.30 1.0 0.25 0.8 0.20 0.6 0.15 0.4 0.2 0.10 T [ GeV ] 0.05 0.1 0.2 0.3 0.4 0.5 0.6 T [ GeV ] [ N. Christianse et al. PRL (2015)] 0.1 0.2 0.3 0.4 0.5 0.6 [ S. Borsanyi et al., Nature (2016) ] Relaxation Time The relaxation time τ shear 30 is taken proportional to 25 the viscosity 20 15 10 � � shear = 3 5 ( � + p ) T [ GeV ] 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Stress tensor initial condition For the initial condition for the stress tensor we select three type of initial condition ✦ Vanish initial condition π φ φ = π η η = 0 ✦ Navier-Stokes initial condition η = − 2 η 1 1 φ = η π φ π η 3 3 τ 0 τ 0 ✦ Modified Navier-Stokes initial condition η + 2 η 1 + 4 1 π η 3 τ shear π η = 0 η 3 2 τ 0 τ 0

Evolution of modified sound velocity Zero at 0.6 fm/c Navier-Stokes at 0.1 fm/c 1.2 1.2 1.1 1.1 1.0 1.0 c 0.9 c 0.9 � � 0.8 0.8 0.7 0.7 τ = 0.6 fm / c τ = 1.5 fm / c τ = 4 fm / c τ = 15 fm / c τ = 0.1 fm / c τ = 0.6 fm / c τ = 4 fm / c τ = 15 fm / c 0.6 0.6 0 5 10 15 0 5 10 15 r [ fm ] r [ fm ] Navier-Stokes at 0.6 fm/c Modified Navier-Stokes at 0.6 fm/c 1.2 1.2 1.1 1.1 1.0 1.0 c 0.9 c 0.9 � � 0.8 0.8 0.7 0.7 τ = 0.6 fm / c τ = 1.5 fm / c τ = 4 fm / c τ = 15 fm / c τ = 0.6 fm / c τ = 1.5 fm / c τ = 4 fm / c τ = 15 fm / c 0.6 0.6 0 5 10 15 0 5 10 15 r [ fm ] r [ fm ]

Estimation of the bound For Navier-Stokes initial condition is possible to estimate the causality bound If we define The constrain on the sound velocity leads to This number depends on the choice of transport coefficient

Domain of dependence λ (1) = v + ˜ Maximal c 14 velocity 1 + ˜ cv T = 0.145 GeV 12 Γ i λ (2) = v − ˜ Minmal c 10 x velocity 1 − ˜ τ [ fm / c ] cv 8 λ (3) = v Fluid velocity 6 Γ d 4 Γ d Domain of dependence λ ( 1 ) λ ( 3 ) λ ( 2 ) 2 Domain of influence Γ i 0 2 4 6 8 10 12 14 r [ fm ] The domain of dependence is contained in the past light cone

Conformal Transformation i + 1.0 r = const. τ = const 0.8 R = 15 α = 3 20 0.6 10 h ( x ) [ fm ] � + 0 σ 0.4 - 10 - 20 0.2 - 1.0 - 0.5 0.0 0.5 1.0 i 0 x 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ

Penrose Diagram of HIC i + i + λ ( 1 ) λ ( 2 ) 1.0 1.0 λ (1) = v + ˜ c 0.8 0.8 1 + ˜ cv T = 0.15 GeV T = 0.15 GeV 0.6 0.6 � + � + T = 0.20 GeV T = 0.20 GeV λ (2) = v − ˜ σ σ c 0.4 0.4 1 − ˜ cv 0.2 0.2 T = 0.25 GeV T = 0.25 GeV Σ 0 Σ 0 i 0 i 0 λ (3) = v 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ ρ i + λ ( 3 ) 1.0 0.8 T = 0.15 GeV 0.6 � + T = 0.20 GeV σ 0.4 0.2 T = 0.25 GeV Σ 0 i 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ

Pressure anisotropy Pressure anisotropy for different type of initial conditions satisfying the causality bound at r=0 1.0 1.0 0.5 0.5 P η / P r P η / P r 0.0 0.0 - 0.5 - 0.5 2 4 6 8 10 12 14 2 4 6 8 10 12 14 τ [ fm / c ] τ [ fm / c ] � � � shear = 30 � shear = 3 � + p � + p

Summary ✦ The dissipative hydro equation are hyperbolic partial differential equation. ✦ The causal structure is given by the characteristic velocity ✦ The characteristic velocity in general are state dependents ✦ It is possible to use the causality bound to constrain the allowed initial condition and the transport coefficient. 27

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