Temperature in relativistic dissipative fluids

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Temperature in relativistic dissipative fluids

• 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
• Generic stability of relativistic dissipative
  fluids
• Temperature of moving bodies
• Summary

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Dissipative relativistic fluids
Nonrelativistic Relativistic Local
equilibrium FourierNavier-Stokes Eckart
(1940), (1st order) Tsumura-Kunihiro
(2008) Beyond local equilibrium Cattaneo-Vernotte
, Israel-Stewart (1969-72), (2nd order) gen.
Navier-Stokes Pavón-extended, etc Müller-R
uggieri, Geroch, Öttinger, Carter, etc.
Eckart Extended (IsraelStewart
PavónJouCasas-Vázquez)
( order estimates)
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Remarks 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. They can be extended.
Fourier-Navier-Stokes limit. Relaxation to the
(unstable) first order theory? (Geroch 1995,
Lindblom 1995) Generic stability is important.
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Fourier-Navier-Stokes
Isotropic linear constitutive relations, ltgt is
symmetric, traceless part
Equilibrium
Linearization, , Routh-Hurwitz criteria
Thermodynamic stability (concave entropy)
Hydrodynamic stability
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Stability and Second Law
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
partial differential equations Lyapunov
theorem is more technical
Linear stability (of homogeneous equilibrium)
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Stability conditions of the Israel-Stewart theory
(Hiscock-Lindblom 1985)
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Special relativistic fluids (Eckart)
energy-momentum density particle density vector
qa momentum density or energy flux??
General representations by local rest frame
quantities.
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Second Law (Liu procedure) first order weakly
nonlocal Entropy inequality with the
conditions of energy-momentum and particle number
balances as constraints
Consequences
1)
2)
3)
Ván JMMS, 2008, 3/6, 1161, (arXiv07121437)
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Modified relativistic irreversible thermodynamics
Internal energy
Eckart term
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1) Thermostatics
Temperatures and other intensives are doubled
Different roles Equations of state
T, M Constitutive functions T, µ
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2) Quasy-hyperbolic extension relaxation of
viscosity
Relaxation
There are no ß derivatives.
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Dissipative hydrodynamics
lt gt symmetric traceless spacelike part

• linear stability of homogeneous equilibrium
• CONDITION thermodynamic stability

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About the temperature of moving bodies
moving body
inertial observer
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About the temperature of moving bodies
moving body
inertial observer
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About the temperature of moving bodies
translational work
Einstein-Planck entropy is vector, energy work
is scalar
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body
v
K0
K
Ott - hydro entropy is vector, energy-pressure
are from a tensor
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Ea energy-momentum vector
Integration, homogeneity
Landsberg
Einstein-Planck
Ott
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Hovewer, Gibbs relation gives more than the
transformation properties!
The real question is 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?
Thermal interaction requires uniform velocities.

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Summary Extended theories are not
ultimate. energy ? internal energy ?
generic stability without extra conditions
hyperbolic(-like) extensions, solutions
/Bíró, Molnár and Ván PRC, (2008), 78, 014909
(arXiv0805.1061)/ different temperatures in
Fourier-law (equilibration) and in state
functions out of local equilibrium ?
interpretation general arguments (no
Boltzmann) ? universality
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Thank you for your attention!
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Body
Velocity distributions
u
v
K
K0
Averages? (Cubero et. al. PRL 2007, 99
170601) Heavy-ion experiments, cosmology.
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Liu procedure for relativistic fluids
Thermodynamics local rest frame

• basic state (fields)
• constitutive state
• constitutive functions

4-vector (temperature ?)
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Dissipation inequality
1)
2)
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Energy-momentum momentum density and energy flux
Landau choice
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Linearization
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exponential plane-waves
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Routh-Hurwitz
thermodynamic stability
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Causality hyperbolic or parabolic?
? Well posedness ? Speed of signal propagation
Hydrodynamic range of validity ? mean free
path t collision time
Water at room temperature
More complicated equations, more spacetime
dimensions, .
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Remarks on hyperbolicity
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Causality hyperbolic or parabolic?
? Well posedness ? Speed of signal propagation
Second order linear partial differential equation
Corresponding equation of characteristics
i) Hyperbolic equation two distinct families of
real characteristics Parabolic equation one
distinct families of real characteristics Ellipti
c equation no real characteristics
Well posedness existence, unicity, continuous
dependence on initial data.
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ii) () is transformation invariant
x
x
t
t
E.g.
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Infinite speed of signal propagation? physics -
mathematics
Hydrodynamic range of validity ? mean free
path t collision time
Water at room temperature Fermi gas of light
quarks at
More complicated equations, more spacetime
dimensions, .
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Non-relativistic fluid mechanics local
equilibrium, Fourier-Navier-Stokes
n particle number density vi relative
(3-)velocity e internal energy density qi
internal energy (heat) flux Pij pressure ki moment
um density
Thermodynamics
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3) Generalized Bjorken flow - the role of
q tetrad axial symmetry Only
for the q0 solution remains the v0 Bjorken-flow
stationary.
4) Temperatures

• qgp eos
• t0 0.6fm/c,
• e0e0 30GeV/fm3
• ?/s0.4,
• p00.

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5) Reheating Eckart R-1lt1 (plt4p) stability
?0 Eckart IS HO 0.3 610-4
5.610-7 2.6710-4 0.08 310-6 2.8910-9
1.7510-4
LHC
RHIC
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About the temperature of moving bodies
moving body
Sardegna
inertial observer