Temperature of moving bodies – thermodynamic, hydrodynamic 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– Outlook

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

0

0

/

vdEdG

dEdE

dSdS

dVdV

pp

vdGpdVTdSdE

translational work – heat = momentum

00000

000

0

020

000

dVpdSTdE

dVpTdS

dE

dEvdV

pTdSdE

vdGpdVTdSdE

vobserver

K0

K

reciprocal temperature - vector?

0TT

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

02

/

dEdE

dSdS

dVdV

pp

00000

002

00 /

dVpdSTdE

dVpTdSdE

pdVTdSdE

v

body

observer

KK0

0TT

No translational work

Blanusa (1947)Einstein (1952) (letter to Laue)

temperature – vector?

Outcome

T

ku

ekxf

),(0 0T

T Einstein-Planck (Ott?)

Relativistic statistical physics and kinetic theory:

Jüttner distribution (1911):

Historical discussion (~1963-70, Moller, 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)( fCfk

Kinetic theory → thermodynamics

'' 11 ffff

kxxekxf )()(

0 ),( (local) equilibrium distribution

)1(ln

0

3

ffkk

kdS

Thermodynamic equilibrium = no dissipation:

2mkk

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

k

kd

k

kd

k

kd

k

kd

T

ku

ekxf

),(0T

u

T

,

Thermodynamic relations - normalization

Jüttner distribution?

kxxekxf )()(

0 ),( 00 fkN

00 fkkT

kkfkffkN 0000

000 TNN

000 )1(: TNS

000 TNS

Legendre transformation

0000

TNS

0

TNS

covariant Gibbs relation(Israel,1963)

Lagrange multipliers – non-equilibrium

qeuE

PuqEuT

jnuN

JsuS

Rest frame quantities:

.0,0

;0,0

;,1

uPPuqu

juJu

uuuu

0)(

00

Puqj

uEnsEns

TNS

Remark:

0,0,0 wT

0)(

PuqjuEnsEns

0 Ens

T

wu

T

g

E

s

Tn

s

Ens

),(

0wu

0)(

PuqjuEns

nTEnTsp ideal gas

rest frame/uniform intensives

duqewdqwdeqeudwudEdnTds )()()(

Velocity dependence?

deviation from Jüttner

A)

B)

Energy-momentum density:

T

ku

T

kwuk eeekxf ˆ

ˆˆ)(

0 ),(

21

ˆw

wuu

,1

ˆ,1

ˆ22 w

T

w

TT

nwnuunfkN ˆˆ00

TeT

mKTmwnn

ˆˆ41ˆ 2

22

00ˆˆˆˆˆ fkkpuueT

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

Summary of kinetic equilibrium:

- Gibbs relation (of Israel):

- Equilibrium spacelike parts:

nwnuN 0

pe

wwpwuuwpeueuT

)(0

dEwudEgdnTds )(

• 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

→ talk of Etele Molnár

• 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?)

Israel–Stewart - conditional suppression (Hiscock and Lindblom, 1985):

qqjqT

uT

qqTT

nesNTS

10

2120

1

222),(),(

,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

Remarks on 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

aa

a jTT

qJ

0),(

JusnesS

junnN

PuququeeTu

0

Special relativistic fluids (Eckart):

0)()(1

2

uTT

T

qupP

TTj

Eckart term

., jnuNPuququeuT

011

TqvpP

T ii

jiijij

qa – momentum density or energy flux?

ijj

i

Pq

qeT

Bouras, I. et. al., PRC 2010 under publication (arxiv:1006.0387v2)

Heat flow problem – kinetic theory versus Israel-Stewart hydro in Riemann shocks:

0),( aa

aa

aa JusnsS

Improved Eckart theory:

junnN

PuququeeTu

0

EEe 22 qInternal energy:

01

2

e

qTuTT

T

qupP

TTj b

Eckart term

aa

a jTT

qJ

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

duqewdqwdedEdnTds )(

Dissipative hydrodynamics

< > symmetric traceless spacelike part

.2

,

,

,

,0)()(

,0)(

,0

u

upP

T

e

qTuTTq

quququpeT

uuqqupeeTu

junnN

vv

vvv

vvv

vv

vvv

v

linear stability of homogeneous equilibriumConditions: 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

pdVdEgTdS

ugwug 1

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

2

1

1

T

g

T

gdEgTdS

1+1 dimension:

),(),,( wvwwvu

2

2

2

1

1

1

vw

v

T

T

Four velocities: v1, v2, w1, w2

Transformation of temperatures

2

22

1

21

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

KEY: no freedom in flow frames (Eckart or Landau-Lifshitz)!?evolving frame?is dissipation frame independent?QGP - effective hydro

Outlook:Outlook:

Dissipation beyond a single viscosity?Dissipation beyond a single viscosity?Causal and generic stable hydro from improved moment series expansion.

dEgdnTds

Thank you for the attention!

u1

u 2

w 1

w 2

g 1

g2

Blue shifted doppler

u 1

u 2

w 1 w 2

g1

g2

Planck-Einstein

u1

u 2

w 1

w 2

g1 g2

Landsberg

u 2

w 1

w 2

g1 u1g2

Ott-Blanusa

u1

u 2

w 1

w 2

g 1

g2

Red shifted doppler

.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)

Thermodynamics

Hydrodynamics Kinetictheory

homogeneity equilibrium

moment series

general balances

concepts

concepts homogeneity

Summary

– S = S(E,V,N)– Work with momentum exchange– Relative velocity v is zero– Cooler, hotter, equal or Doppler?

Ván: J. Stat. Mech. P02054, 2009.Bíró-Molnár-Ván: PRC 78, 014909, 2008.

Bíró-Ván: EPL, 89, 30001, 2010, (arXiv:0905.1650)Wolfram Demonstration Project, Transformation of …

Internal energy:

E Ea

– S = S(Ea, V, N)– energy-momentum exchange– T and v do not equilibrate– γwT and w v are equilibrating

– T: Doppler of w with the speed v

Heavy ion physics: dissipative relativistic fluids