Hydrodynamics, stability and temperature

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)(

11

enn

s

np

nep

pe

T

p

e

pe

,0...)/()/(

12

nTp

ns

p

ns

e

pe

,02

,0,02

21

172805

,0)/(

1

3

22

2

21

0

20

16

n

ns

T

Tn

,0

2

222121

1124

pe

,03

211)(

6

2

203

K

e

ppe

n

s .0

3

21

/2

1

0

0

nsn

T

T

nK

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??

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)

inertial observer

moving body

About the temperature of moving bodies:

inertial observer

moving body

About the temperature of moving bodies:

K

K0

v

body

01

00 ,, TTssTdsd

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

0 ,, TTeessTdsde

21

1

v

Landsberg: 0TT

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

Mdndsdqq

deedqqede

qedd aa

aa

22

000 ,, ss Landsberg

01

02, TTee

eT

Einstein-Planck

0TT Ottnon-dissipative

aa energy(-momentum) vectorba

ba Tu

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

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

Thank you for your attention!