Download Relativistic irreversible thermodynamics

Dissipative relativistic
hydrodynamics
P. Ván
Department of Theoretical Physics
Research Institute of Particle and Nuclear Physics,
Budapest, Hungary
– Motivation
– Problems with second order theories
– Thermodynamics, fluids and stability
Internal energy:
  e2  q2
– Generic stability of relativistic dissipative fluids
– Temperature of moving bodies
– Summary
Dissipative relativistic fluids
Nonrelativistic
Relativistic
Local equilibrium
(1st order)
Fourier+Navier-Stokes
Eckart (1940),
Tsumura-Kunihiro (2008)
Beyond local equilibrium
(2nd order)
Cattaneo-Vernotte,
gen. Navier-Stokes
Israel-Stewart (1969-72),
Pavón, Müller-Ruggieri,
Geroch, Öttinger, Carter,
conformal, etc.
Eckart:
a
q
S a (T ab , N a )  s(e, n)u a 
T
Extended (Israel–Stewart – Pavón–Jou–Casas-Vázquez):





S a (T ab , N a )   s(e, n)  0  2  1 qb q b  2  bc bc u a 
2T
2T
2T


1
 q a   0 q a  1 ab qb
(+ order estimates)
T


Remarks on causality and stability:
Symmetric hyperbolic equations ~ causality
 tT   xxT  0,


v
v2
 ( ~t  v ~x )T     ~x~x  2 2  ~x ~t  4  ~t ~t T  0
c
c


– 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)
Fourier-Navier-Stokes
n  n i v i  0,
   i v i   i q i  P ij  i v j  0,
k i  k i  j v j   j P ij  0i ,
qii
q    T ,
i
i
 ij   k v k  ij  2  i v j .


1 1 ij

P  (Ts  n   ) ij  i v j  0
T T
p
Isotropic linear constitutive relations,
<> is symmetric, traceless part
Equilibrium:
n( xi , t )  const.,
 ( xi , t )  const.,
v i ( xi , t )  const.
Linearization, …, Routh-Hurwitz criteria:
  0,   0,   0,
Thermodynamic stability
(concave entropy)
  T  0,
(  p )  p  n n p  0    T n
Hydrodynamic stability

T
  nT 
 T 2 Det ( 2 s)

T
0
Remarks on stability and Second Law:
Non-equilibrium thermodynamics:
basic variables
evolution equations (basic balances)
Second Law
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
Continuum systems (irreversible thermodynamics):
partial differential equations – Lyapunov theorem is more technical
Linear stability (of homogeneous equilibrium)
Stability conditions of the Israel-Stewart theory
(Hiscock-Lindblom 1985)
1 e
1 
e  p p
2 
T

s
n
(e  p ) 2
1
e
p
e  p  ( s / n) p  ( s / n)
5   0  0,
T
 
 n 2T 2
e n
n T

nT
8   2  0,
2  2  1  1
4  e  p 
 0,
2
21 2  1

 p
3  (e  p)1 
 e

 0,
e
 ...  0,
12
7  1 
 0,
2 2
 02 212
1 T
6  1 

 2
 0,
 0 3 2 n T e n

2
K2
 1



 0,

s 
 0 3 2 6
n 
K  1
 0 21 p


 0.
 0 3 2 e n
Special relativistic fluids (Eckart):
T ab  eu au b  q au b  q bu a  P ab ,
energy-momentum density
N a  nu a  j a .
particle density vector
q a ua  j a ua  0, P baua  P abua  0b
qa – momentum density
or energy flux??
General representations by local rest frame quantities.
 a S a  s(e, n)  s a u a   a J a  0
a
q
 a
a
J 
 j
T T
e  u a  a e

u a  bT ab  e  e  au a   a q a  ua q a  u1a  b Pijab  0
 b N  n  n  au   a j
b
a
a

1
 P  p  i v j  q  i  0
T
T
ij
i

a
1
q
 s  j a  a  ( P ab  p ab ) bua  2 ( aT  Tua )  0
T T
T
Eckart term
Second Law (Liu procedure) – first order weakly nonlocal:
Entropy inequality with the conditions of energy-momentum and particle number
balances as constraints:
 a S a   a  bT ab   a N a  0
Consequences:
1)
2)
State space: (e, u a , n)
s(e, u a , n)  s(e, q a (e, u a ), n)
s
s
e a  qa

q
e

s(e, q a )  sˆ(e 2  q 2 )  ~
s e2  q2
3)
a
q
 a
a
J 
 j
T T

Ván: JMMS, 2008, 3/6, 1161, (arXiv:07121437)
Modified relativistic irreversible thermodynamics:
Internal energy:
  e2  q2   a a  ubT baTacu c
 a S a  s( , n)  s a u a   a J a  0
u a  bT ab  e  e  au a   a q a  ua q a  ua  b P ab  0
a
q
 a
a
J 
 j
T T
 b N b  n  n  au a   a j a
a
qa 

1
q

a
ab
ab
 s  j  a  P  p  bua  2   aT  Tua  T   0
T T
T 
e
Eckart term
Ván and Bíró EPJ, (2008), 155, 201. (arXiv:0704.2039v2)
Dissipative hydrodynamics
a N a
 n  n a u a   a j a  0,
u a  bT ab  e  (e  p ) a u a   a q a  q a ua   ab bu a  0,
ac  bT cb  (e  p )u a  q a  bu b  q b  bu a  ac (q c   b  cb )  0 a ,
qa

a
 aa
a



q
ac
,
    cT  Tuc  T
e 

   c
ac

,
T
 Paa  p   c u c ,
 ab  2   bu a  .
< > symmetric traceless spacelike part
 linear stability of homogeneous equilibrium
Conditions: thermodynamic stability, nothing more.
(Ván: arXiv:0811.0257)
Thermostatics:



qa
de  dqa  Tds  dn  s e 2  q a qa , n  sˆ e, q a , n
e
Temperatures and other intensives are doubled:
s 1
sˆ 1
 ;

  e T
 eT   , e  
Different roles:
Equations of state:
Θ, M
Constitutive functions: T, μ
a



q
a
ac
q     cT  Tuc  T 
e 


About the temperature of moving bodies:
moving body
inertial observer
About the temperature of moving bodies:
moving body
inertial observer
About the temperature of moving bodies:
body
v
K0

K
dE  vdG  TdS  pdV
translational work
Einstein-Planck: entropy is vector, energy + work is scalar
s  s0 , e  e0
 T   1T0 ,
p  p0
1
1 v2
body
v
K0
K
dE  TdS  pdV

Ott - hydro: entropy is vector, energy-pressure are from a tensor
s  s0 , e   2 e0 ,
p   2 p0
 T  T0
1
1 v2
d  d e 2  q 2 
ede  q a dqa
   a a

e
qa a 
  de  dq   ds  Mdn



 a  ubT ba energy(-momentum) vector
s  s0 ,    0 ,    0

T  , e   2 e0
e
non-dissipative
 T   1T0
 T  T0
Landsberg
Einstein-Planck
Ott
Simple transformation properties?
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?
Integration, homogeneity:
V  V  a a  Ea E a  E 2  G 2 , sV  S , nV  N.
E1  E2  const .
a
Ea
dS  MdN  dE a  d E
E
a
N  const .
1   2  
 E1a
E1a
E2 a
E2 a
a
a

d ( S1  S 2 ) 
dE1 
dE2  

1 E1
 2 E2
  E1  E2
 a
dE1  u1a  u2a


Thermal interaction requires uniform velocities.
Quasi-hyperbolic extension – relaxation of viscosity:
 

s( , n)  s e2  q 2  0 2   2 bc bc , n  s e2  D2 , n

 0  1,  2  1.
Relaxation:
1 ab
1
1 a
 qb   aT  abub 
q  0,
e
T
T
3
1
b
   b u    0,
e

1
1
 ab    a ub    ab  0.
e
2
Simpler than Israel-Stewart: there are no β derivatives.
Bíró, Molnár and Ván: PRC, (2008), 78, 014909 (arXiv:0805.1061)
1) Generalized Bjorken flow - the role of q:
tetrad :
e0a , e1a , e2a , e3a; axial symmetry
x a  e0a  re2a
u   (e  ve ), q  q(ve  e ).
a
a
0
a
1
a
a
0
a
1
Only for the q=0 solution remains the v=0 Bjorken-flow stationary.
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
4
 4 
e0   0 
 b 04
 3 0  Eckart
η0 Eckart IS
HO
0.3 6·10−4 5.6·10−7 2.67·10−4
0.08 3·10−6 2.89·10−9 1.75·10−4
RHIC
LHC
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
Thank you for your attention!
v2
v1
K1
K2
K
1
1 E '1

T2 T1 E1
E '1  E '2  const.
a
a
N  const.
Q2 Q '1

E2
E1
dS1  dS 2  0
Q1  0

1
1

,
T2
T1
Q2
 v
E2
Einstein-Planck
Q '1  0

1
1

,
T2
T1
Q2  0
Ott
Q1  E1

1
 (1  v )

,
T2
T1
Q2  E2
lightlike
Body
Velocity distributions:
u
v
K
K0
 
 uv 
ˆ
f (u )  f    T   1  2 T0
T 
 c 
Averages? (Cubero et. al. PRL 2007, 99 170601)
Heavy-ion experiments, cosmology.
Liu procedure for relativistic fluids
 bT ab  eu a  eu a  bu b  eu a  u a  b q b  q b  bu a  q a  q a  bu b   b P ab  0
 a S a  s  s au a   a J a  0.
Thermodynamics – local rest frame
– basic state (fields):
– constitutive state:
– constitutive functions:
 a S a   a  bT ab  0
(e, u aa )
(e, u ,  a e,  bu a )
(q a , P ab , s, J a )
 a  ua  la
Solution of Liu equations (  a , Aa are local):
J a  q a  lb P ab  a a
s  e  lb q b  A
4-vector (temperature ?)
Dissipation inequality

s(e, u a ) ??

 a e (1s    lb 1q b )u a  q a 1  P ba 1la  1a a 

 a ub  P ab  Aab  q a ( 2 ) b  P ca ( 2lc ) a  ( 2 a a ) b 
( s)
2
1)
2)
b




 lc ( 2 q c )b u a  l b e  q b u a  l b q a  0
s(e, u a )  s(e, q a (e, u a ))

s
s
e a  qa
 s(e, q a )  sˆ(e 2  q 2 )  ~
s e2  q2
q
e
2
q
2
2
e  q  e   ....
2e

Energy-momentum – momentum density and energy flux
T ab  eu a u b  u a q b  qˆ a u b  P ab , ua q a  0, ua P ab  0 
T ab
e
  j
 qˆ
qi 

P ij 
 bT ab  0a ,
u a  bT ab  e  e a u a   a q a  q a ua  P ab bua  0,
a  T cb  eu a  q a  u b  q b  u a  a (qˆ c   P cb )  0a.
c
b
b
Landau choice:
q a  0a
b
c
b
Linearization
A  A0  A
 0,
 a j a
n a u a 
n 
 0,
( e  p )  a u a   a  q a
q a 
 e p a e  eua 
 a    0,
 a u a 
   0,
e 
 n p a n 


 a n   e  a e 
T
T



 nT a n  2  eT a e  2 Tua 
2
T
T
T
n
e
u a
 0,
j a
 n
 0.
q a
q a j a

A  A0 exp( t  ikx)


0
 ik p
 n

ik T
Q n

 ik 
 nT

0


exponential plane-waves
0
ikn
0
ik

ik (e  p)
ik
0
(e  p ) 

0
ik e p
ik eT
T

T
0
ik n
0
ik
1 T
  0
 e
1
0

0

ik 0   u y 
 (e  p )



T
y
 T  1  
0 0   q 
R
e
  xy 
0
1 0    
 ik
 ik~
   yy 
0
0
1

v


0
0 

0  n 
ik  e 


 u x 
0 

x
 q 
0  j x 

   

1 
~
 
~ 

3 / 4  / 
 Det (Q) 
 3 T 2 (e  p ) 


 2 k 2 (T 2  ( p  e))   eT  T 2  n  
T




 k 2  T 2 ( e p(e  p )  n n p)  k 2    eT  T 2  n  
T



 

k 2 (e  p)  eT n   nT e   
T
T 





k 4  T 2  (e  p)  e p n   n p e   n eT n p   nT e p  
T
T




 k 2   eT n

T
  nT e
 

T  
Routh-Hurwitz:
T
 
 0,
0
e
n T
T   T  

0
e n T n e T
thermodynamic stability
Causality hyperbolic or parabolic?
 Well posedness
 Speed of signal propagation
Hydrodynamic range of validity:
ξ – mean free path
τ – collision time
x2
A 4 t
T ( x, t ) 
e
2 t
 xT 
vmax
T
T
,  tT 




 
 cV 
2.0
Water at room temperature:
vmax 
1.5

m
 14
cV 
s
1.0
0.5
More complicated equations,
more spacetime dimensions, ….
x
3
2
1
0
1
2
3
Remarks on hyperbolicity
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 relaxing
fast.
parabolic+stable  causal
3) Instability in first order theories is not acceptable.
 Second order dissipative theories are corrections to first
order stable theories.
Causality hyperbolic or parabolic?
 Well posedness
 Speed of signal propagation
Second order linear partial differential equation:
A xxT  2B xtT  C ttT  F ( x, t , T ,  xT ,  tT )  0
Corresponding equation of characteristics:
A x   2B x  t   Ct   0
2
(*)
i)
Hyperbolic equation:
Parabolic equation:
Elliptic equation:
2
two distinct families of real characteristics
one distinct families of real characteristics
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: T ( x,0)  f ( x) )
ii)
(*) is transformation invariant
~ ~
~
x ~
x ( x, t ), t  t ( x, t )
(1)  tT   xxT  0,


v
v2

 ( ~t  v ~x )T    ~x~x  2 2  ~x ~t  4  ~t ~t T  0
c
c


iii) The outer real characteristics that pass through a given point ( x0 , t0 ) give its
domain of influence  0 .
x
x
t
E.g.
 tT   xxT  0,
T ( x,0)   ( x)
x  vt  0
t
T ( x, t ) 
A
e
2 t

x2
4 t
Infinite speed of signal propagation?
physics - mathematics
Hydrodynamic range of validity:
ξ – mean free path
τ – collision time
 xT 
T
T
,  tT 


vmax 



 cV 
Water at room temperature:
vmax 

m
 14
cV 
s
Fermi gas of light quarks
at Tc :
vmax 
mcV v



cV 
cV 
x2
A 4 t
T ( x, t ) 
e
2 t
More complicated equations, more spacetime dimensions, ….
mT
m
 10 31

s
Non-relativistic fluid mechanics
local equilibrium, Fourier-Navier-Stokes
n  n i v  0,
i
   i v i   i q i  P ij  i v j  0,
k i  k i  j v j   j P ij  0i.
Thermodynamics
n
vi
e
qi
Pij
ki
particle number density
relative (3-)velocity
internal energy density
internal energy (heat) flux
pressure
momentum density
mnv2
p  Ts  n   ,   e 
.
2
d  Tds  dn,
1

qi
i
s( , n)  s i v   i J    n  s i v   i  ... 
T
T
T
i
i


1 1 ij
q  i  P  (Ts  n   ) ij  i v j  0
T T
p
i
About the temperature of moving bodies:
moving body
Sardegna
inertial observer