relativistic irreversible thermodynamics
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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,
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
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