Dissipative relativistic hydrodynamics

P. VánDepartment of Theoretical Physics

Research Institute of Particle and Nuclear Physics, Budapest, Hungary

– Motivation– Problems with second order theories – Thermodynamics, fluids and stability

– Generic stability of relativistic dissipative fluids– Temperature of moving bodies– Summary

Internal energy:

22 q e

Nonrelativistic Relativistic

Local equilibrium Fourier+Navier-Stokes Eckart (1940),(1st order) Tsumura-Kunihiro (2008)

Beyond local equilibrium Cattaneo-Vernotte, Israel-Stewart (1969-72),(2nd order) gen. Navier-Stokes Pavón, Müller-Ruggieri,

Geroch, Öttinger, Carter, conformal, etc.

Eckart:

Extended (Israel–Stewart – Pavón–Jou–Casas-Vázquez):

T

qunesNTS

aaaaba ),(),(

babaa

abc

bcbb

aaba

qqqT

uT

qqTT

nesNTS

10

2120

1

222),(),(

Dissipative relativistic fluids

(+ order estimates)

Remarks on causality and stability:

Symmetric hyperbolic equations ~ causality

– The extended theories are not proved to be symmetric hyperbolic.

– In Israel-Stewart theory the symmetric hyperbolicity conditions of the perturbation equations follow from the stability conditions.

– Parabolic theories cannot be excluded – speed of the validity range can be small. Moreover, they can be extended later.

Stability of the homogeneous equilibrium (generic stability) is required.

– Fourier-Navier-Stokes limit. Relaxation to the (unstable) first order theory? (Geroch 1995, Lindblom 1995)

02)(

,0

~~4

2

~~2~~~~

Tc

v

c

vTv

TT

tttxxxxt

xxt

.2

,j

iijk

kij

ii

vv

Tq

Isotropic linear constitutive relations,<> is symmetric, traceless part

Equilibrium:

.),(.,),(.,),( consttxvconsttxconsttxn ii

ii

Linearization, …, Routh-Hurwitz criteria:

00)(

,0

,0,0,0

TT

TTpnpp

T

nnn

,0

,0

,0

iijj

jj

ii

jiiji

ii

i

ii

Pvkk

vPqv

vnn

Hydrodynamic stability )( 22 sDetT

Thermodynamic stability(concave entropy)

Fourier-Navier-Stokes

0)(11

jiijij

ii vnTsP

TTq

p

Remarks on stability and Second Law:

Non-equilibrium thermodynamics:

basic variables Second Lawevolution equations (basic balances)

Stability of homogeneous equilibrium

Entropy ~ Lyapunov function

Homogeneous systems (equilibrium thermodynamics):dynamic reinterpretation – ordinary differential equations

clear, mathematically strictSee e.g. Matolcsi, T.: Ordinary thermodynamics, Academic Publishers, 2005

partial differential equations – Lyapunov theorem is more technical

Continuum systems (irreversible thermodynamics):

Linear stability (of homogeneous equilibrium)

Stability conditions of the Israel-Stewart theory (Hiscock-Lindblom 1985)

,0)(

1

2221

enn

s

TnTn

eT

pe

T

p

e

pe

,0...)/()/(

12

nTp

ns

p

ns

e

pe

,02

,0,02

21

172805

,01

3

22

2

21

0

20

16

ne

T

Tn

,0

2

22121

1124

pe

,03

211)(

6

2

203

K

e

ppe

n

s .0

3

21

2

1

0

0

ne

pK

aa

a jTT

qJ

0),( aa

aa

aa JusnesS

aa

aa

bb

abba

aa

aa

aa

abb

a

junnN

PuququeeTu

0

Special relativistic fluids (Eckart):

0)()(1

2 aa

a

ababab

aa

s uTTT

qupP

TTj

Eckart term

eue aa

ba

aba

baa

aa

a uPuPujuq 0,0 .

,aaa

ababbabaab

jnuN

PuququeuT

General representations by local rest frame quantities.

energy-momentum density

particle density vector

011

TqvpP

T ii

jiijij

qa – momentum density or energy flux??

Second Law (Liu procedure) – first order weakly nonlocal:

Entropy inequality with the conditions of energy-momentum and particle number balances as constraints:

2222 ~)(ˆ),( qq

esesqes

e

sq

q

se

a

aa

)),,(,(),,( nueqesnues aaa 1)

2)

Ván: JMMS, 2008, 3/6, 1161, (arXiv:07121437)

3) aa

a jTT

qJ

0 aa

abba

aa NTS

Consequences:),,( nue aState space:

0),( aa

aa

aa JusnsS

Modified relativistic irreversible thermodynamics:

aa

aa

bb

abba

aa

aa

aa

abb

a

junnN

PuququeeTu

0

cac

baba

a uTTue 22 qInternal energy:

01

2

e

qTuTT

T

qupP

TTj a

aa

a

ababab

aa

s

Eckart term

aa

a jTT

qJ

Ván and Bíró EPJ, (2008), 155, 201. (arXiv:0704.2039v2)

Dissipative hydrodynamics

< > symmetric traceless spacelike part.2

,

,

,

,0)()(

,0)(

,0

ab

ab

cc

aa

aa

caca

a

ccaca

acbb

cac

ab

bbb

aacbb

ac

abab

aaa

aa

aab

ba

aa

aa

aa

u

upP

T

e

qTuTTq

quququpeT

uuqqupeeTu

junnN

linear stability of homogeneous equilibriumConditions: thermodynamic stability, nothing more.

(Ván: arXiv:0811.0257)

nqesnqqesdnTdsdqe

qde a

aa

a

a

,,ˆ,2

Thermostatics:

Temperatures and other intensives are doubled:

eeTTe

ss,

1ˆ;

1

Different roles:

Equations of state: Θ, MConstitutive functions: T, μ

e

qTuTTq

a

ccaca

inertial observer

moving body

About the temperature of moving bodies:

inertial observer

moving body

About the temperature of moving bodies:

K

K0

v

body

001

00 ,, ppTTeess

21

1

v

translational work

Einstein-Planck: entropy is vector, energy + work is scalar

pdVTdSddE Gv

About the temperature of moving bodies:

K

K0

v

pdVTdSdE

body

Ott - hydro: entropy is vector, energy-pressure are from a tensor

002

02

0 ,, TTppeess

21

1

v

Mdndsdqq

deedqqede

qedd aa

aa

22

000 ,, ss Landsberg

01

02, TTee

eT

Einstein-Planck

0TT Ottnon-dissipative

aa energy(-momentum) vectorba

ba Tu

EddEE

EMdNdS aa

aaaaaaaaa uudEE

E

E

EdE

E

EdE

E

ESSd 211

2

2

1

12

22

21

11

121 )(

Thermal interaction requires uniform velocities.

Equilibration: Two bodies A and B have relative speed v. What must be the relation between their temperatures TA and TB, measured in their rest frames, if they are to be in thermal equilibrium?

Simple transformation properties?

21

21

.

.

constN

constEE aa

Integration, homogeneity:

.,,22 NnVSsVEEEVV aa

aa G

Quasi-hyperbolic extension – relaxation of viscosity:

nDesnesns bcbc ,,),( 22

22

022 q

Relaxation:

.02

11

,013

,0111

abbaab

bb

ab

abab

ab

ue

ue

qT

uTT

qe

Simpler than Israel-Stewart: there are no β derivatives.

.1,1 20

Bíró, Molnár and Ván: PRC, (2008), 78, 014909 (arXiv:0805.1061)

1) Generalized Bjorken flow - the role of q:

tetrad : ; axial symmetry

Only for the q=0 solution remains the v=0 Bjorken-flow stationary.

aaa reex 20 aaaa eeee 3210 ,,,

).(),( 1010aaaaaa eveqqveeu

2) Temperatures:

-qgp eos- τ0 = 0.6fm/c,-e0=ε0 =30GeV/fm3

- η/s=0.4,- π0=0.

3) Reheating:

Eckart: R-1<1 (p<4π) stability

η0 Eckart IS HO0.3 6·10−4 5.6·10−7 2.67·10−4

0.08 3·10−6 2.89·10−9 1.75·10−4

40

4

0

00 3

4

beEckart

RHICLHC

Summary

– Extended theories are not ultimate.

– energy ≠ internal energy→ generic stability without extra conditions

– hyperbolic(-like) extensions, generalized Bjorken solutions, reheating conditions, etc…

– different temperatures in Fourier-law (equilibration) and

in EOS out of local equilibrium → temperature of moving bodies - interpretation

Thank you for your attention!

K1

K

v1

K2

v2

0

.

.''

21

21

dSdS

constN

constEE aa

1

1

2

2

1

1

12

'

'11

E

Q

E

Q

E

E

TT

2212

11

212

1

2

2

121

,)1(1

0,11

0'

,11

0

EQT

v

TEQ

QTT

Q

vE

Q

TTQ

lightlike

Einstein-Planck

Ott

K0

K

v

BodyVelocity distributions:

u

Averages? (Cubero et. al. PRL 2007, 99 170601)

Heavy-ion experiments, cosmology.

021ˆ)( T

c

uvT

Tfuf

– basic state (fields):– constitutive state:– constitutive functions:

Thermodynamics – local rest frame

.0

0

a

aa

aa

a

abb

bb

aaab

bbb

aabb

aaabb

JussS

PuqquqquueueuueT

),( aue

),,,( aaba JsPq),,,( a

baa ueue

0 abba

aa TS aaa lu 4-vector (temperature ?)

Aqles

aPlqJb

b

aabb

aa

Solution of Liu equations ( are local):aa A,

Liu procedure for relativistic fluids

0)()(

)()()(

)(

22

222

11111

ababbabcc

b

baac

cabaababba

aa

baaabba

qluqeluqls

alPqAPu

alPquqlse

Dissipation inequality

2222 ~)(ˆ),( qq

esesqese

sq

q

se a

aa

....2

222

eeeq

q

??),( aues

)),(,(),( aaa ueqesues 1)

2)

Energy-momentum – momentum density and energy flux

ijj

iab

aba

aa

abbababaab

Pq

qeT

PuquPuqquueuT

ˆ

0,0,ˆ

.0)ˆ(

,0

,0

acbb

cac

ab

bbb

aacbb

ac

abab

aaa

aa

aab

ba

aabb

PquququeT

uPuqqueeTu

T

Landau choice: aaq 0

aaa

aaaean

aaean

aa

aaaaean

aa

aa

aa

aa

jquen

quTT

eTT

nTT

jeT

nT

u

queepnp

qupee

junn

.0

,0

,0

,0

,0)(

,0

222

LinearizationAAA 0

yy

xy

y

y

v

q

u

ik

ike

TT

ikpe

100~010

001

0)(

R

x

x

x

nn

en

en

j

q

u

e

n

ik

Tik

Tik

e

TTTikTik

ikpepikpik

ikpeik

ikikn

~1

0000

01

00

001

0)(

00)(0

000

Q

)exp(0 ikxtAA exponential plane-waves

/4/3~

Routh-Hurwitz:

0

0,0

Ten

T

Tne

TTne

T

thermodynamic stability

TT

TTk

pTpTnT

pT

ppeTk

TT

TTpek

TTTkpnpepTk

TTTepTk

peT

QDet

enne

enneenne

enne

nene

ne

2

24

2

2222

2222

23

)(

)(

))((

))((

)(

)(

Causality hyperbolic or parabolic?

Well posedness Speed of signal propagation

Hydrodynamic range of validity:ξ – mean free pathτ – collision time

TT

TT tx

,

More complicated equations, more spacetime dimensions, ….

Water at room temperature:

t

x

et

AtxT

4

2

2),(

Vcv max

s

m

cv

V

14max

3 2 1 0 1 2 3x

0 .5

1 .0

1 .5

2 .0

3 2 1 0 1 2 3x

0 .5

1 .0

1 .5

2 .0

1) Hyperbolicity does not result in automatic causality, because the propagation speed of small perturbations can be large.

hyperbolic causal

2) Parabolic equations and first order theories are not automatically excluded. The validity range of the theory could prevent large speeds if the perturbations were relaxingfast.

parabolic+stable causal 3) Instability in first order theories is not acceptable.

Second order dissipative theories are corrections to first order stable theories.

Remarks on hyperbolicity

Causality hyperbolic or parabolic?

Well posedness Speed of signal propagation

0),,,,(2 TTTtxFTCTBTA txttxtxx

Second order linear partial differential equation:

02(*) 22 ttxx CBA

Corresponding equation of characteristics:

i) Hyperbolic equation: two distinct families of real characteristicsParabolic equation: one distinct families of real characteristicsElliptic equation: no real characteristics

Well posedness: existence, unicity, continuous dependence on initial data.

A characteristic Cauchy problem of (1) is well posed. (initial data on the characteristic surface: ))()0,( xfxT

iii) The outer real characteristics that pass through a given point give its domain of influence .

),( 00 tx0

02)(

,0

~~4

2

~~2~~~~

Tc

v

c

vTv

TT

tttxxxxt

xxt

(1)

ii) (*) is transformation invariant ),(~~),,(~~ txtttxxx

t t

x x 0 vtx

E.g.

)()0,(

,0

xxT

TT xxt

t

x

et

AtxT

4

2

2),(

Infinite speed of signal propagation?physics - mathematics

Hydrodynamic range of validity:ξ – mean free pathτ – collision time

TT

TT tx

,

More complicated equations, more spacetime dimensions, ….

Water at room temperature:

Fermi gas of light quarks at :

t

x

et

AtxT

4

2

2),(

Vcv max

s

m

cv

V

14max

cTs

mmT

c

vmc

cv

V

V

V

31max 10

Non-relativistic fluid mechanics local equilibrium, Fourier-Navier-Stokes

.0

,0

,0

iijj

jj

ii

jiiji

ii

i

ii

Pvkk

vPqv

vnn

n particle number densityvi relative (3-)velocitye internal energy densityqi internal energy (heat) fluxPij pressureki momentum density

Thermodynamics

.2

,,2mnv

enTspdnTdsd

...1

),(T

qvsn

TTJvsns

i

ii

ii

ii

i

0)(11

jiijij

ii vnTsP

TTq

p

inertial observer

moving body

Sardegna

About the temperature of moving bodies: