dissipative relativistic hydrodynamics
DESCRIPTION
Internal energy:. Dissipative relativistic hydrodynamics. P. Ván Department of Theoretical Physics Research Institute of Particle and Nuclear Physics, Budapest, Hungary. Motivation Problems with s econd order theories Thermodynamics, fluids and stability - PowerPoint PPT PresentationTRANSCRIPT
Dissipative relativistic hydrodynamics
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)(
1
2221
enn
s
TnTn
eT
pe
T
p
e
pe
,0...)/()/(
12
nTp
ns
p
ns
e
pe
,02
,0,02
21
172805
,01
3
22
2
21
0
20
16
ne
T
Tn
,0
2
22121
1124
pe
,03
211)(
6
2
203
K
e
ppe
n
s .0
3
21
2
1
0
0
ne
pK
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??
Second Law (Liu procedure) – first order weakly nonlocal:
Entropy inequality with the conditions of energy-momentum and particle number balances as constraints:
2222 ~)(ˆ),( qq
esesqes
e
sq
q
se
a
aa
)),,(,(),,( nueqesnues aaa 1)
2)
Ván: JMMS, 2008, 3/6, 1161, (arXiv:07121437)
3) aa
a jTT
qJ
0 aa
abba
aa NTS
Consequences:),,( nue aState space:
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)
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
K
K0
v
body
001
00 ,, ppTTeess
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
02
0 ,, TTppeess
21
1
v
Mdndsdqq
deedqqede
qedd aa
aa
22
000 ,, ss Landsberg
01
02, TTee
eT
Einstein-Planck
0TT Ottnon-dissipative
aa energy(-momentum) vectorba
ba Tu
EddEE
EMdNdS aa
aaaaaaaaa uudEE
E
E
EdE
E
EdE
E
ESSd 211
2
2
1
12
22
21
11
121 )(
Thermal interaction requires uniform velocities.
Equilibration: Two bodies A and B have relative speed v. What must be the relation between their temperatures TA and TB, measured in their rest frames, if they are to be in thermal equilibrium?
Simple transformation properties?
21
21
.
.
constN
constEE aa
Integration, homogeneity:
.,,22 NnVSsVEEEVV aa
aa G
Quasi-hyperbolic extension – relaxation of viscosity:
nDesnesns bcbc ,,),( 22
22
022 q
Relaxation:
.02
11
,013
,0111
abbaab
bb
ab
abab
ab
ue
ue
qT
uTT
qe
Simpler than Israel-Stewart: there are no β derivatives.
.1,1 20
Bíró, Molnár and Ván: PRC, (2008), 78, 014909 (arXiv:0805.1061)
1) Generalized Bjorken flow - the role of q:
tetrad : ; axial symmetry
Only for the q=0 solution remains the v=0 Bjorken-flow stationary.
aaa reex 20 aaaa eeee 3210 ,,,
).(),( 1010aaaaaa eveqqveeu
2) Temperatures:
-qgp eos- τ0 = 0.6fm/c,-e0=ε0 =30GeV/fm3
- η/s=0.4,- π0=0.
3) Reheating:
Eckart: R-1<1 (p<4π) stability
η0 Eckart IS HO0.3 6·10−4 5.6·10−7 2.67·10−4
0.08 3·10−6 2.89·10−9 1.75·10−4
40
4
0
00 3
4
beEckart
RHICLHC
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
K1
K
v1
K2
v2
0
.
.''
21
21
dSdS
constN
constEE aa
1
1
2
2
1
1
12
'
'11
E
Q
E
Q
E
E
TT
2212
11
212
1
2
2
121
,)1(1
0,11
0'
,11
0
EQT
v
TEQ
QTT
Q
vE
Q
TTQ
lightlike
Einstein-Planck
Ott
K0
K
v
BodyVelocity distributions:
u
Averages? (Cubero et. al. PRL 2007, 99 170601)
Heavy-ion experiments, cosmology.
021ˆ)( T
c
uvT
Tfuf
– basic state (fields):– constitutive state:– constitutive functions:
Thermodynamics – local rest frame
.0
0
a
aa
aa
a
abb
bb
aaab
bbb
aabb
aaabb
JussS
PuqquqquueueuueT
),( aue
),,,( aaba JsPq),,,( a
baa ueue
0 abba
aa TS aaa lu 4-vector (temperature ?)
Aqles
aPlqJb
b
aabb
aa
Solution of Liu equations ( are local):aa A,
Liu procedure for relativistic fluids
0)()(
)()()(
)(
22
222
11111
ababbabcc
b
baac
cabaababba
aa
baaabba
qluqeluqls
alPqAPu
alPquqlse
Dissipation inequality
2222 ~)(ˆ),( qq
esesqese
sq
q
se a
aa
....2
222
eeeq
q
??),( aues
)),(,(),( aaa ueqesues 1)
2)
Energy-momentum – momentum density and energy flux
ijj
iab
aba
aa
abbababaab
Pq
qeT
PuquPuqquueuT
ˆ
0,0,ˆ
.0)ˆ(
,0
,0
acbb
cac
ab
bbb
aacbb
ac
abab
aaa
aa
aab
ba
aabb
PquququeT
uPuqqueeTu
T
Landau choice: aaq 0
aaa
aaaean
aaean
aa
aaaaean
aa
aa
aa
aa
jquen
quTT
eTT
nTT
jeT
nT
u
queepnp
qupee
junn
.0
,0
,0
,0
,0)(
,0
222
LinearizationAAA 0
yy
xy
y
y
v
q
u
ik
ike
TT
ikpe
100~010
001
0)(
R
x
x
x
nn
en
en
j
q
u
e
n
ik
Tik
Tik
e
TTTikTik
ikpepikpik
ikpeik
ikikn
~1
0000
01
00
001
0)(
00)(0
000
Q
)exp(0 ikxtAA exponential plane-waves
/4/3~
Routh-Hurwitz:
0
0,0
Ten
T
Tne
TTne
T
thermodynamic stability
TT
TTk
pTpTnT
pT
ppeTk
TT
TTpek
TTTkpnpepTk
TTTepTk
peT
QDet
enne
enneenne
enne
nene
ne
2
24
2
2222
2222
23
)(
)(
))((
))((
)(
)(
Causality hyperbolic or parabolic?
Well posedness Speed of signal propagation
Hydrodynamic range of validity:ξ – mean free pathτ – collision time
TT
TT tx
,
More complicated equations, more spacetime dimensions, ….
Water at room temperature:
t
x
et
AtxT
4
2
2),(
Vcv max
s
m
cv
V
14max
3 2 1 0 1 2 3x
0 .5
1 .0
1 .5
2 .0
3 2 1 0 1 2 3x
0 .5
1 .0
1 .5
2 .0
1) Hyperbolicity does not result in automatic causality, because the propagation speed of small perturbations can be large.
hyperbolic causal
2) Parabolic equations and first order theories are not automatically excluded. The validity range of the theory could prevent large speeds if the perturbations were relaxingfast.
parabolic+stable causal 3) Instability in first order theories is not acceptable.
Second order dissipative theories are corrections to first order stable theories.
Remarks on hyperbolicity
Causality hyperbolic or parabolic?
Well posedness Speed of signal propagation
0),,,,(2 TTTtxFTCTBTA txttxtxx
Second order linear partial differential equation:
02(*) 22 ttxx CBA
Corresponding equation of characteristics:
i) Hyperbolic equation: two distinct families of real characteristicsParabolic equation: one distinct families of real characteristicsElliptic equation: no real characteristics
Well posedness: existence, unicity, continuous dependence on initial data.
A characteristic Cauchy problem of (1) is well posed. (initial data on the characteristic surface: ))()0,( xfxT
iii) The outer real characteristics that pass through a given point give its domain of influence .
),( 00 tx0
02)(
,0
~~4
2
~~2~~~~
Tc
v
c
vTv
TT
tttxxxxt
xxt
(1)
ii) (*) is transformation invariant ),(~~),,(~~ txtttxxx
t t
x x 0 vtx
E.g.
)()0,(
,0
xxT
TT xxt
t
x
et
AtxT
4
2
2),(
Infinite speed of signal propagation?physics - mathematics
Hydrodynamic range of validity:ξ – mean free pathτ – collision time
TT
TT tx
,
More complicated equations, more spacetime dimensions, ….
Water at room temperature:
Fermi gas of light quarks at :
t
x
et
AtxT
4
2
2),(
Vcv max
s
m
cv
V
14max
cTs
mmT
c
vmc
cv
V
V
V
31max 10
Non-relativistic fluid mechanics local equilibrium, Fourier-Navier-Stokes
.0
,0
,0
iijj
jj
ii
jiiji
ii
i
ii
Pvkk
vPqv
vnn
n particle number densityvi relative (3-)velocitye internal energy densityqi internal energy (heat) fluxPij pressureki momentum density
Thermodynamics
.2
,,2mnv
enTspdnTdsd
...1
),(T
qvsn
TTJvsns
i
ii
ii
ii
i
0)(11
jiijij
ii vnTsP
TTq
p