Relativistic thermodynamics: discrete, continuum and kinetic aspects Peter Ván KFKI, 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
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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,