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

SLIDE 1

Causality of fluid dynamics for high-energy nuclear collisions

Eduardo Grossi ITP Heidelberg University

Cold Quantum Coffee, Heidelberg, 21/11/2017

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with S. Floerchinger arXiv:1711.06687

SLIDE 2

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

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SLIDE 3

Ideal relativistic hydrodynamics

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✦ Energy conservation ✦ Charge conservation ✦ Euler equation

With the equation of state this set of equation can be solved in a closed form D + ( + p) · u = 0 Dn + n∂ · u = 0 ( + p)Duν νp = 0 Ts = + p − µn Sµ = suµ µSµ = 0

SLIDE 4

Relativistic Hydrodynamics

The equation of relativistic hydrodynamics:

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p + πbulk

( + p)uµ energy density four-velocity stress tensor bulk viscous pressure

figure from [Rezzolla, Zanotti “Relativistic Hydrodynamics”]

SLIDE 5

Navier-Stokes equations

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

SLIDE 6

Instability of Navier-Stokes Equation

δQ = δQ0eikx+Γt Γ+ (c2 + p)c2 T For k=0 we will have a growing mode and a damping mode in the transverse direction Consider a traverse perturbation around equilibrium with the fluid at rest. If we neglect the heat conductivity we obtain the diffusive mode like in non relativistic theory Γ = − k2 + p

[W. Hiscock et al. PRD (1985)]

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δQ = δQ0eik(−γvt+γx)+Γ(γt−vγx) Γ+ (c2 + p)c2 v2

Instability of Navier-Stokes Equation

Nevertheless if we boost the reference frame the growing mode at k=0 will appear again The characteristic time scale it is really short. In non relativistic limit the this time goes to zero. The non relativistic limit is safe respect of this instability. = v2 (c2 + p)c4 → 0

[W. Hiscock et al. PRD (1985)]

SLIDE 8

Second order in gradients

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Shear Tensor: Bulk pressure:

[R. Baier et al. JHEP 0804 (2008)] [P. Romatschke Class.Quant.Grav. 27 (2010)] [S. Bhattacharyya et al., JHEP 0802 (2008)]

SLIDE 9

Israel Stewart theory

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The Navier-Stokes theory is unstable and a-causal. Adding a second order term the equation became casual and stable

[Muller 1967, Israel, Stewart 1976] [Hiscock 1983]

The shear tensor becomes a dynamical variabile that relax at its Navier-Stokes value ✦2 order in gradient ✦Promoting as a variabile ✦Relaxation type equation

New transport coefficient, the relaxation time

SLIDE 10

Hydro-Equation

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. ✦ Energy conservation and Euler equation ✦ Relaxation equation for shear tensor ✦ Relaxation equation for bulk pressure

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[G. S. Denicol et al. Phys. Rev. D (2012)]

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0+1 solution

The simplest approximation: the Bjorken model

✦ boost invariance along the z direction ✦ rotational and translation invariance in the transverse plane. ✦

Effectively it corresponds to 1+0 problem

Figure from W. van der Schee, (2014).

Every quantity depend

nly on the proper time.

Describe only the central rapidity region, no radial velocity.

J. Bjorken, (1983).

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1+1 solution

1+1 Hydro

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✦ boost invariance along the z direction ✦ rotational invariance in the transverse plane. ✦ Temperature ✦ Four velocity

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.

✦ Shear and bulk

Picture stolen from C. Shen et al. (2016)

SLIDE 13

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

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SLIDE 14

First order hyperbolic quasi-linear PDE

∂0Φi + Mij∂1Φj + Si = 0 This type of equation are called hyperbolic when is possible to diagonalize the matrix M w(m)

j

λ(m) dJ(m) = w(m)

j

dΦj ∂0J(m) + λ(m)∂1J(m) + w(m)

j

Sj = 0 Left eigenvector Eigenvalue With this variables we can partially diagonalized the system of pde. Riemann Invariants:

SLIDE 15

Riemann invariant for the perfect fluid

The perfect fluid equation in cartesian coordinates in 2 dimension can be written in a simple form J ± = 1 2 ln 1 + v 1 − v

±
dT

1 c2

sT

These variables are transport constant on the characteristic curves: dx dt = v ± cs 1 ± vcs

SLIDE 16

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

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SLIDE 17

Energy integrals

x1 x0

Cl Cr P(x0,x1) Pr Pl Ar Al Γ x0=t x0=0

Any Hyperbolic PDE is linear stable, the solution of linearized equation is bounded. ✦ If the initial data vanish,then   also the solution vanish. ✦ This define the domani of  dependence of a given point.

[Courant and Hilbert, (1953)]

SLIDE 18

Starting condition

Number wounded nucleon

5 10 15 20 25 30 r [fm] 0.1 0.2 0.3 0.4 T [GeV]

Number binary collision Corresponding to a Au-Au-200 GeV collision

✦ Initial vanish 

four velocity

✦ Initial stress tensor 

equal to Navier-  Stokes value uµ = (−1, 0, 0, 0)

✦ No bulk

s(τ0, r) = s0 1 − x 2 nW N(r) + xnBC(r)

SLIDE 19

Transport properties

0.0 0.1 0.2 0.3 0.4 0.5 0.6 T [GeV] 5 10 15 20 25 30 τshear

0.1 0.2 0.3 0.4 0.5 0.6 T [GeV] 0.05 0.10 0.15 0.20 0.25 0.30 cs2

0.1 0.2 0.3 0.4 0.5 0.6 T [GeV] 0.2 0.4 0.6 0.8 1.0 1.2 η/s

Viscosity Sound velocity Relaxation Time The relaxation time is taken proportional to the viscosity shear = 3

( + p)

[ N. Christianse et al. PRL (2015)] [ S. Borsanyi et al., Nature (2016) ]

SLIDE 20

Stress tensor initial condition

For the initial condition for the stress tensor we select three type of initial condition ✦Vanish initial condition ✦Navier-Stokes initial condition ✦Modified Navier-Stokes initial condition πφ

φ = πη η = 0

πη

η + 2η

3 1 τ0 + 4 3τshearπη

η

1 2τ0 = 0 πη

η = −2η

3 1 τ0 πφ

φ = η

3 1 τ0

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Evolution of modified sound velocity

5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 r [fm] c

τ=0.1 fm/c

τ=0.6 fm/c τ=4 fm/c τ=15 fm/c

5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 r [fm] c

τ=0.6 fm/c

τ=1.5 fm/c τ=4 fm/c τ=15 fm/c

5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 r [fm] c

τ=0.6 fm/c

τ=1.5 fm/c τ=4 fm/c τ=15 fm/c

5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 r [fm] c

τ=0.6 fm/c

τ=1.5 fm/c τ=4 fm/c τ=15 fm/c

Zero at 0.6 fm/c Navier-Stokes at 0.6 fm/c Navier-Stokes at 0.1 fm/c Modified Navier-Stokes at 0.6 fm/c

SLIDE 22

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

SLIDE 23

Domain of dependence

2 4 6 8 10 12 14 2 4 6 8 10 12 14 r [fm] τ [fm/c]

λ(1) λ(3) λ(2)

T=0.145 GeV

Γd Γi x

The domain of dependence is contained in the past light cone

Γd

Γi λ(1) = v + ˜ c 1 + ˜ cv λ(2) = v − ˜ c 1 − ˜ cv λ(3) = v Maximal velocity Minmal velocity Fluid velocity Domain of dependence Domain of influence

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Conformal Transformation

1.0
0.5

0.0 0.5 1.0

20
10

10 20 x h(x) [fm]

R=15 α=3

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 σ

+ i0 i+

r=const. τ=const

SLIDE 25

Penrose Diagram of HIC

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 σ

+ i0 i+ Σ0

T=0.15 GeV T=0.20 GeV T=0.25 GeV

λ(3)

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 σ

+ i0 i+ Σ0

T=0.15 GeV T=0.20 GeV T=0.25 GeV

λ(1)

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 σ

+ i0 i+ Σ0

T=0.15 GeV T=0.20 GeV T=0.25 GeV

λ(2)

λ(1) = v + ˜ c 1 + ˜ cv λ(2) = v − ˜ c 1 − ˜ cv λ(3) = v

SLIDE 26

Pressure anisotropy

2 4 6 8 10 12 14

0.5

0.0 0.5 1.0 τ [fm/c] Pη/Pr 2 4 6 8 10 12 14

0.5

0.0 0.5 1.0 τ [fm/c] Pη/Pr

shear = 3

+ p

shear = 30

+ p

Pressure anisotropy for different type of initial conditions satisfying the causality bound at r=0

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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.

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SLIDE 28

Causality of fluid dynamics for high-energy nuclear collisions

Cold Quantum Coffee, Heidelberg, 21/11/2017

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

Ideal relativistic hydrodynamics

With the equation of state this set of equation can be solved in a closed form D + ( + p) · u = 0 Dn + n∂ · u = 0 ( + p)Duν νp = 0 Ts = + p − µn Sµ = suµ µSµ = 0

Relativistic Hydrodynamics

The equation of relativistic hydrodynamics:

Navier-Stokes equations

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

Instability of Navier-Stokes Equation

Instability of Navier-Stokes Equation

Nevertheless if we boost the reference frame the growing mode at k=0 will appear again The characteristic time scale it is really short. In non relativistic limit the this time goes to zero. The non relativistic limit is safe respect of this instability. = v2 (c2 + p)c4 → 0

Second order in gradients

Shear Tensor: Bulk pressure:

Israel Stewart theory

The Navier-Stokes theory is unstable and a-causal. Adding a second order term the equation became casual and stable

The shear tensor becomes a dynamical variabile that relax at its Navier-Stokes value ✦2 order in gradient ✦Promoting as a variabile ✦Relaxation type equation

Hydro-Equation

0+1 solution

The simplest approximation: the Bjorken model

Effectively it corresponds to 1+0 problem

1+1 solution

1+1 Hydro

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

First order hyperbolic quasi-linear PDE

Riemann invariant for the perfect fluid

The perfect fluid equation in cartesian coordinates in 2 dimension can be written in a simple form J ± = 1 2 ln 1 + v 1 − v

These variables are transport constant on the characteristic curves: dx dt = v ± cs 1 ± vcs

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

Energy integrals

Any Hyperbolic PDE is linear stable, the solution of linearized equation is bounded. ✦ If the initial data vanish,then also the solution vanish. ✦ This define the domani of dependence of a given point.

Starting condition

Number wounded nucleon

Number binary collision Corresponding to a Au-Au-200 GeV collision

four velocity

equal to Navier- Stokes value uµ = (−1, 0, 0, 0)

Transport properties

Viscosity Sound velocity Relaxation Time The relaxation time is taken proportional to the viscosity shear = 3

Stress tensor initial condition

For the initial condition for the stress tensor we select three type of initial condition ✦Vanish initial condition ✦Navier-Stokes initial condition ✦Modified Navier-Stokes initial condition πφ

Evolution of modified sound velocity

Zero at 0.6 fm/c Navier-Stokes at 0.6 fm/c Navier-Stokes at 0.1 fm/c Modified Navier-Stokes at 0.6 fm/c

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

The domain of dependence is contained in the past light cone

Γd

Conformal Transformation

Penrose Diagram of HIC

Pressure anisotropy

Pressure anisotropy for different type of initial conditions satisfying the causality bound at r=0

Summary

Extra slide

Any Hyperbolic PDE is linear stable, the solution of linearized equation is bounded. ✦ If the initial data vanish,then   also the solution vanish. ✦ This define the domani of  dependence of a given point.

equal to Navier-  Stokes value uµ = (−1, 0, 0, 0)