Non-Newtonian nature of Causal Hydrodynamics

T. Koide

(Universidade Federal do Rio de Janeiro)

G.S. Denicol (UFRJ),T. Kodama (UFRJ),Ph. Mota (UFRJ)

Because of causality, the relativistic dissipative fluid will bea non-Newtonian fluid. Thus1) The GKN formula should be modified.2) 1/4can be a lower bound of the shear of Newtonian fluids.3) The fluid expands to vacuum by forming a stationary wave.4) The additional viscosity is still necessary stabilize solutions.

Infinite speed in diffusion process

ix

1

21

2

1i ix x x 1i it t t

Random walk

Infinite speed in diffusion process

1 1

1 1( , ) ( , ) ( , )

2 2n n nP x t t P x t P x t

2( , ) ( , )P x t D P x tt

2ik

2

, 0limx t

xD

t

xv

t

0, 0x t

Dispersion relation

Relativistic NR equation(1+1)

22

0 02 2

k ki k

w w

02 /crik w

At , the dispersion relation is reduced to the diffusion type.crik k

>0. stable

Linear perturbation analysis

Hiscock&Lindblom(‘85)

Introduction of memory effect

( ) / ( )R

tt s

R

DJ dse n s

Equation of continuityn J

t

J D n

2n D nt

Fick’s law

Acausal (Cattaneo)Breaking sum rules (Kadanoff&Martin, T.K.)

22

20R n n D n

t t

R

DV

Landau-Lifshitz theory

0T 0N

( ) ( )T P u u g P

N nu

Equation of Continuity

heat current

bulk viscosity shear viscosity

0u 0u

Introduction of Memory Effect

1( ) 0P u

T T

u

P u

PT

( , )dsG s u

( , )P dsG s u

( , )P dsG sT

T.K.,Denicol,Mota&Kodama(2007)Entropy four current

Landau-Lifshitz

Memory effect

Causal Hydrodynamics

Dispersion relation(1+1)

0 40 80 120 160m om e n tu m

0

0.2

0.4

0.6

0.8

1

Im w

0 40 80 120m om e n tu m

-120

-80

-40

0

40

80

120

Re

w

Stable in the sense of the linear analysis.

blue and red

Speed of light

Linear analysis of stability

LL equation CDhydro

Equilibrium

Lorentz boost

Scaling solution Stable Unstable

Stable Stable

StableUnstable

?

Hiscock& Lindblam (‘85), Kouno,Maruyama,Takagi&Saito (‘90)

The Eckert theory is more unstable than the LL theory.

Stability around scaling solution

10R

Ts

u

P u

PT

( , )dsG s u

( , )P dsG s u

( , )P dsG sT

Newtonian fluid

Memory effect

Causal HydrodynamicsLandau-Lifshitz

Anomalous viscosity

Newtonian

Dilatant

ThixotropicRheopectic

Pseudoplastic

Bingham flow

Velocity gradient

Vis

cosit

y

sludge, paint,blood, ketchup

latex, paper pulp, clay solns.

quicksand, candy compounds

tars, inks, glue

Navier-Stokes

QGP ?

Common sense?

• The transport coefficients are given by the Green-Kubo-Nakano formula.

• The lower bound of the shear viscosity of the CD hydro. is given by .

• The viscosity is small correction.

• We do not need any additional viscosity in the CD hydro.

1/(4 )

Not trivial !!

Linear response theory

system

Hamiltonian

H

( )H AF tExternal perturbation F

Green-Kubo-Nakano formula

0

( ) ( ) ( )eq

t d A i J t

( )

( )

J tD

F t

( ) ( ) ( )t

J t ds t s F s

Exact!

In Newtonian fluid, transport coefficients are defined by

0

( ) ( )J ds s F t

We ignore the memory effect

0

( )D ds s

GKN formula

( ) ( )t

J ds t s F s

0

( )D ds s

Newtonian

( ) ( )J t DF t

Non-Newtonian

( ) /( ) ( )R

tt s

R

DJ t dse F s

D=?

( ) /( ) ( )R

tt s

R

DJ t dse F s

( ) ( ) ( )t

J t ds t s F s

0

( ) (0) ( ) ( ) ( )t sJ t F t ds s F t

1( ) ( ) ( )t

R R

DJ t J t F t

( ) (0) ( ) ( ) ( )t

t tJ t F t ds t s F s

0

(0)( ) ( ) ( ) ( )t s

GKN

J t J t ds s F tD

( ) ( )GKNJ t D F t

Markov app.

1. In the GKN formula, we need In the generalized formula, we need and 2. characterizes the deviation from the GKN formula.3. When vanishes in the low momentum limit, the new formula reproduces the GKN formula.4. The result obtained in the new formula is consistent with sum rules.

Generalization of GKN formula

1

2

T.K.&Maruyama (2004), T.K.(2005), T.K.(2007), T.K.&Kodama (2008)

1 ( );J t J

2 ( );t J

GKN formula

New formula

12

2

Lower limit of shear viscosity?

3

0

1[ ( , ), (0,0)]

2lim i txy xydtd xe T x t T

1) GKN formula

2) Absorption cross section

38( ) [ ( , ), (0,0)]i t

abs xy xy

Gdtd xe T x t T

(0)

16abs

G

G: Newton’s constant

1

4s

(0)

4abss

G

Kovtun,Son&Starinets(‘05)

Lower limit of shear viscosity?

3

0

1[ ( , ), (0,0)]

2lim i txy xydtd xe T x t T

1) GKN formula

2) Absorption cross section

38( ) [ ( , ), (0,0)]i t

abs xy xy

Gdtd xe T x t T

(0)

16abs

G

G: Newton’s constant

1

4s

(0)

4abss

G

Only for Newtonian!!

Kovtun,Son&Starinets(‘05)

Common sense?

• The transport coefficients are given by the Green-Kubo-Nakano formula.

• The lower bound of the shear viscosity of the CD hydro. is given by .

• The viscosity is small correction.1/(4 )

In the following calculations, we consider only the 1+1 dimensional fluid.

Expansion of fluid

- 8 - 4 0 4 8x

0

2

4

6

8

en

trip

y d

en

sity

It seems to be a stationary wave.

Universal relation between pressure and viscosity

We assume that the fluid forms a stationary wave at the boundary to vacuum. Then at the boundary,

2 00 11 01 2(1 )sc T T vT v

11 2 00T v T

P

- 8 - 4 0 4 8

0

0 . 4

0 . 8

1 . 2

pressure

(-1) * viscosity

1. The causal fluid expands as a stationary wave.

2. Viscosity can be the same order as energy density and pressure.

Common sense?

• The transport coefficients are given by the Green-Kubo-Nakano formula.

• The lower bound of the shear viscosity of the CD hydro. is given by .

• The viscosity is small correction.

• We do not need any additional viscosity in the CD hydro.

1/(4 )

Numerical oscillation

-15 -10 -5 0 5 10x

0

100

200

300

400

en

tro

py

de

nsi

ty

The frequency is the size of the grid.

unstable

-20 -10 0 10 20x

0

200

400

600

800

en

tro

py

de

nsi

ty

Usual “artificial” viscosity

2 2/u x

2/u x

/ ( / )u x u x

Navier-Stokes type:

Burnett type:

Neumann-Richtmyer type:

Generalization to relativistic cases ? Causal?

Additional causal viscosityWe need to introduce the artificial viscosity consistent with causality.

total av

av av av av

du

d

av hP

h :the size of the grid (0.01 fm)

-15 -10 -5 0 5 10

x

0

100

200

300

400

en

tro

py

de

nsi

ty

-20 -10 0 10 20

x

0

200

400

600

800

en

tro

py

de

nsi

ty

Effect to entropy production

0 1 2 3 4 5t

4400

4410

4420

4430

4440

4450

4460

tota

l en

tro

py

The effect of the additional viscosity is

0.05%

This is the same order of the breaking of the energy conservation in our code.

Summary• The transport coefficients are given by the Green-Kubo-Nakano

formula.

• The lower bound of the shear viscosity of the CD hydro. is given by

.

• The viscosity is small correction.

• We do not need any additional viscosity in the CD hydro.

No. We need a new formula for non-Newtonian fluids

Not yet clear for causal dissipative hydro.

1/(4 )

P

We need.

No. We have a relationand forms a stationary wave.

We have to check

• Is the new formula really useful or not.

• What is the lower bound of the causal shear viscosity?

• CD hydro is a stable theory?

• Are numerical oscillations we encountered due to the problem of numerical calculation or the problem of the theory (turbulence)?

That is, the causal hydrodynamics will be one of non-Newtonian fluids!!

Newtonian Non-Newtonian

Stability and Causality(1+1)

3 2 2 2

0

10

R R R

ik i k

w

0

1

3R R

ik

w

0

Re 1limk

R

vk w

Linear perturbation analysis of the causal hydro.

Sound velocity is faster than ideal.

1Im 0

3 R

Stable, Independent of momentum

0w P

P

T.K. Denicol,Kodama&Mota (‘07)

It is difficult to cut high momentum modes to avoid unstable modes.

u uu

t x

( )ik x utu e 2 ( )i k x utu

iket

crik k 2 crik k

We need to find a new theory to satisfy causality.

Model equation of hydro

assumption

Non-linear effect

-15 -10 -5 0 5 10x

0

100

200

300

400

en

tro

py

de

nsi

ty

-15 -10 -5 0 5 10x

0

100

200

300

400

en

tro

py

de

nsi

ty

Additional viscosity