hydrodynamics, stability and temperature p. ván department of theoretical physics research...
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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!