Relativistic thermodynamics: discrete, continuum and kinetic aspects

Peter VánKFKI, RMKI, Dep. Theoretical Physics

– Temperature of moving bodies – the story

– Relativistic equilibrium – kinetic theory

– Stability and causality – hydrodynamics

– Temperature of moving bodies – the conclusion

with Tamás Biró, Etele Molnár

Planck and Einstein

body

vobserver

K0

K

pdVTdSdE Relativistic thermodynamics?

About the temperature of moving bodies (part 1)

• Planck-Einstein (1907): cooler

• Ott (1963) [Blanusa (1947)] : hotter

• Landsberg (1966-67): equal

• Costa-Matsas-Landsberg (1995): direction dependent (Doppler)

0TT

)cos1(

0

v

TT

v

body

observer

K0

K

0T

T

0TT

21

1

v

0

0

0

/

SS

VV

pp

– Lorentz

– Planck: adiabatic change

observers are equivalent

2

0

1

0 SS

21 SS

?pdVTdSdE

What are the transformations? – hidden covariance

0

00

01

1

vE

EE

v

v

G

E

vtx

cvxt

cvvtx

vxt

x

t

v

v

x

t 2

2

/

)/(1

1

)(

)(

1

1

'

'

Vector of energy-momentum

v

body

observer

K0

K

21

1,1

vc

1

0S 2

0S

1S 2S

observer

body

0

0

0

0

0

/

vdEdG

dEdE

dSdS

dVdV

pp

vdGpdVTdSdE

translational work –

heat = momentum

00000

000

0

0

20000

dVpdSTdE

dVpTdS

dE

dEvdV

pTdSdE

vdGpdVTdSdE

vobserver

K0

K

reciprocal temperature

- vector?0T

T

21

1

v

Rest frame arguments: Ott (1963)

v

body

reservoir

KK0

dQ

Planck-Einstein

v

body

reservoir

K

K0

dQ

Ott

vdGpdVTdSdE

0

0

0

0

2

/

dEdE

dSdS

dVdV

pp

00000

00

2

00 /

dVpdSTdE

dVpTdSdE

pdVTdSdE

v

body

observer

KK0

0TT

No translational work

Blanusa (1947)

Einstein (1952) (letter to Laue)

temperature – vector?

Outcome

T

pu

epxf

),(0 0T

T Einstein-Planck

(Ott?)

Relativistic statistical physics and kinetic theory:

Jüttner distribution (1911):

Historical discussion (~1963-70, Møller, von Treder, Israel,

ter Haar, Callen, …, renewed Dunkel-Talkner-Hänggi 2007, …):

new (?) arguments/ no (re)solution.

→ Doppler transformation

e.g. solar system, microwave background

→ Velocity is thermodynamic variable?

Landsberg

van Kampen

Questions

• What is moving (flowing)?

– barion, electric, etc. charge (Eckart)

– energy (Landau-Lifshitz)

• What is a thermodynamic body?

– volume

– expansion (Hubble)

• What is the covariant form of an e.o.s.?

– S(E,V,N,…)

• Interaction: how is the temperature transforming?

→ kinetic theory and/or hydrodynamics

Boltzmann equation)( fCfp

Kinetic theory → thermodynamics

'' 11 ffff

pxx

epxf)()(

0 ),(

(local) equilibrium distribution

)1(ln

0

3

ffpp

pdS

Thermodynamic equilibrium = no dissipation:

2mpp

Boltzmann gas

01ln4

1|0

3

0

3

0

3

,,,0

3

klijji

ji

lk

ji

lk

l

l

k

k

j

j

lkji i

i Wffff

ff

ff

ff

p

pd

p

pd

p

pd

p

pd

T

pu

epxf

),(0T

u

T

,

Thermodynamic relations - normalization

Jüttner distribution?

pxx

epxf)()(

0 ),(

00 fpN

00 fppT

ppfpffpN 0000

000 TNN

000 )1(: TNS

000 TNS

Legendre transformation

0000

TNS

0

TNS

covariant Gibbs relation?

(Israel,1963)

Lagrange multipliers – non-equilibrium

Constrained inequality – A. Cimmelli

qeuE

PuqEuT

jnuN

JsuS

Rest frame is required:

.0,0

;0,0

;,1

uPPuqu

juJu

uuuu

Remark:

0,

wu

T

wu

0)()(

)(

000

pEwunTsT

wu

upwEwunsTT

wu

TNS

a

aa

a

a duqwpedqwde

dupwdEwudnTds

))((

)(

pEwunTs )(

Thermodynamics:

qeuE

Summary of kinetic equilibrium:

- Gibbs relation:

- Spacelike parts in equilibrium:

nwnuN 0

pe

wwpwuuwpeueuT

)(0

dednTds

a

aa

a

a duqwpedqwdednTds ))((

• What is dissipative?

– dissipative and non-dissipative parts

• Free choice of flow frames?

– QGP - effective hydrodynamics.

• Kinetic theory → hydrodynamics

– local equilibrium in the moment series expansion

• What is the role and manifestation of local

thermodynamic equilibrium? – generic stability and causality

Questions

Nonrelativistic Relativistic

Local equilibrium Fourier+Navier-Stokes Eckart (1940),

(1st order) Tsumura-Kunihiro

Beyond local equilibrium Cattaneo-Vernotte, Israel-Stewart (1969-72),

(2nd order) gen. Navier-Stokes Pavón, Müller-Ruggieri-Liu,

Geroch, Öttinger, Carter,

conformal, Rishke-Betz,

etc…Eckart:

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

T

jqunesNTS

),(),(

qqjqT

uT

qqTT

nesNTS

10

2120

1

222),(),(

(+ order estimates)

Thermodynamics → hydrodynamics (which one?)

Causality and stability:

Symmetric hyperbolic equations ~ causality

– The extended theories are not proved to be symmetric hyperbolic (exception:

Müller-Ruggeri-Liu).

– In Israel-Stewart theory the symmetric hyperbolicity conditions of the perturbation

equations follow from the stability conditions.

– Generic stable parabolic theories can be extended later.

– Stability of the homogeneous equilibrium (generic stability) is related to

thermodynamics.

Thermodynamics → generic stability → causality

02)(

,0

~~4

2

~~2

~~~~

Tc

v

c

vTv

TT

tttxxxxt

xxt

Conditions of generic stability in extended thermodynamics:Israel–Stewart - conditional suppression (Hiscock and Lindblom, 1985):

qqjqT

uT

qqTT

nesNTS

10

2120

1

222),(),(

,0

)(

11

en

n

s

n

pn

e

ppe

T

p

e

pe,0...

)/()/(

12

nTp

ns

p

ns

e

pe

,02

,0,02

2

1172805

,0)/(

1

3

22

2

2

1

0

2

016

nns

T

Tn

,0

2

222

121

1124

pe

,03

211)(

6

2

20

3

K

e

ppe

n

s .0

3

21

/2

1

0

0

nsn

T

T

nK

j

TT

qJ

0),(

JusnesS

junnN

PuququeeTu

0

Special relativistic fluids (Eckart):

0)()(1

2

uTT

T

qupP

TTj

Eckart term

., jnuNPuququeuT

011

TqvpP

Ti

i

ji

ijij

qa – momentum density or energy flux?

ijj

i

Pq

qeT

dednTds

Bouras, I. et. al., PRC PRC 82, 024910, 2010 (arxiv:1006.0387v2)

Heat flow problem –

kinetic theory versus Israel-Stewart hydro in Riemann shocks:

0),,(

JusnqesS

Improved Eckart theory:

junnN

PuququeeTu

0

hwwpeq )(Special choice:

0)(

1

2

1

uquqqupepe

TT

T

q

uqqhpPTT

j

j

TT

qJ

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

duqwpedqwdedupwdEwudnTds ))(()(

Summary of hydrodynamics:

Modified Gibbs relation restores generic stability.

In the acceleration independent case:

linear stability of homogeneous equilibrium

Conditions: thermodynamic stability, nothing more. Ván P.: J. Stat. Mech. (2009) P02054

Israel-Stewart like relaxational (quasi-causal) extensionBiró T.S. et. al.: PRC (2008) 78, 014909

integrating multiplier

Hydrodynamics → thermodynamics

}3,2,1,0{,

,,,1

GuEE

dVqGeVedVEdVuV

u

QdAquVupGuE

HH H

H

H(2)

H(1)

Volume integrals: work, heat, internal energy

Change of heat and entropy:

pdVAA

uAdG

AA

AdE

AA

uAdS

VESSGuEE

ASVupEQ

),(,

pdVdEgTdS

temperature1

ug

integrating multiplier

• there are four different velocities

• only one of them can be eliminated

• the motion of the body and the energy-momentum currents are slower than light

dEwuTdS )( 1)(

uwu

Interaction

v2

observer

w2

v1

w1

w spacelike, but |w|<1 -- velocity of the heat current

About the temperature of moving bodies (part 2)

v2

w2

v1

w1

2

222

1

111

2

222

1

111

T

)wv(

T

)wv(

T

)wv1(

T

)wv1(

2

2

2

1

2

1

22

22

11

11

T

w1

T

w1

wv1

wv

wv1

wv

2

22

1

11 )()()(

T

wu

T

wudEwuTdS

1+1 dimensions:

),(),,( wvwwvu

2

2

2

1

1

1

vw

v

T

T

Four velocities: v1, v2, w1, w2

Transformation of temperatures

2

2

2

1

2

1

22

22

11

1111

,11 T

w

T

w

wv

wv

wv

wv

v

w2w1Relative velocity

(Lorentz transformation) 21

12

1 vv

vvv

general Doppler-like form!

2

21

1 vw

wvw

0

2

0 1

1

vw

v

T

T

Special:

w0 = 0 T = T0 / γ Planck-Einstein

w = 0 T = γ T0 Ott

w0 = 1, v > 0 T = T0 • red Doppler

w0 = 1, v < 0 T = T0 • blue Doppler

w0 + w = 0 T= T0 Landsberg

v

w0w

K K0thermometer

T T0

Biró T.S. and Ván P.: EPL, 89 (2010) 30001

V=0.6, c=1

Summary

Generalized Gibbs relation:

– consistent kinetic equilibrium

– improves hydrodynamics

– explains temperature of moving bodies

Outlook: what is the local lest frame/flow?(Eckart or Landau-Lifshitz)!?

is dissipation frame independent?

non-relativistic (Brenner)

dupwdEwudnTds )(

Thermodynamics

Hydrodynamics Kinetic

theory

homogeneity equilibrium

local

equilibrium

Thank you for the attention!

Biró-Ván, EPL, 89, 30001, 2010

u1

u2

w1

w2

g1

g2

Blue shifted doppler

u1

u2

w1 w2

g1

g2

Planck-Einstein

u1

u2

w1

w2

g1 g2

Landsberg

u2

w1

w2

g1 u1g2

Ott-Blanusa

u1

u2

w1

w2

g1

g2

Red shifted doppler

wug

Energy-momentum density:

T

pu

T

pwu

peeepxf

ˆ

ˆˆ)(

0 ),(

21

ˆw

wuu

,1

ˆ,1

ˆ22 w

T

w

TT

nwnuunfpN ˆˆ00

TeT

mKTmwnn

ˆˆ41ˆ

2

22

00ˆˆˆˆˆ fpppuueT

T

mK

T

mK

nmTne

ˆ

ˆˆˆˆ3ˆ

2

1

pe

qqpquuqueuT

0

wpeqw

wpeepp )(,

1

ˆˆ,ˆ

2

2

Heat flux:0

w

n

peqI

.2

,

j

i

ijk

k

ij

ii

vv

Tq

Isotropic linear constitutive relations,

<> is symmetric, traceless part

Equilibrium:

.),(.,),(.,),( consttxvconsttxconsttxn i

i

ii

Linearization, …, Routh-Hurwitz criteria:

00)(

,0

,0,0,0

TT

TTpnpp

T

nnn

,0

,0

,0

iij

j

j

j

ii

ji

iji

i

i

i

i

i

Pvkk

vPqv

vnn

Hydrodynamic stability)( 22 sDetT

Thermodynamic stability

(concave entropy)

Fourier-Navier-Stokes

0)(11

ji

ijij

i

i vnTsPTT

q

p

Remarks on stability and Second Law:

Non-equilibrium thermodynamics:

basic variables Second Law

evolution equations (basic balances)Stability of homogeneous equilibrium

Entropy ~ Lyapunov function

Homogeneous systems (equilibrium thermodynamics):

dynamic reinterpretation – ordinary differential equations

clear, mathematically strict

See 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)

u1

u2

w1

w2

g1

g2

Blue shifted doppler

u1

u2

w1 w2

g1

g2

Planck-Einstein

u1

u2

w1

w2

g1 g2

Landsberg

u2

w1

w2

g1 u1g2

Ott-Blanusa

u1

u2

w1

w2

g1

g2

Red shifted doppler

Thermodynamics

Hydrodynamics Kinetic

theory

homogeneity equilibrium

moment series

general balances

concepts

concepts dissipation