Stable first order special relativistic dissipative hydrodynamics

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Stable first order special relativistic
dissipative hydrodynamics

• P. Ván and T. S. Biró
• University of Bergen and RMKI, Budapest

• Introduction
• Origin of the instabilities
• irreversible thermodynamics of Eckart
• Nonrelativistic theory
• Separation of dissipative and nondissipative
  parts
• Conclusions

NCIRH07, Frankfurt
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Introduction
Nonrelativistic Relativistic Local
equilibrium (1st) Fourier, Navier-Stokes Eckart B
eyond local equilibrium Cattaneo-Vernotte,
Israel-Stewart, (2nd) gen. Navier-Stokes
Müller-Ruggieri Öttinger, Carter, etc..
Dissipative nondissipative Causality
parabolic or hyperbolic Stability homogeneous
equilibrium (linear and nonlinear) ?
Israel-Stewart stability condition
?
?Relaxation to the first order theory (- E.
Molnár).
Remark Carter (Olson and Hiscock, 1990)
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Eckart term
Ideal hydro hits the target. ? feelings on
dissipation
water
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Energy-momentum
Landau
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Thermodynamics
Statics
Dynamics
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Isotropic current-force functions
Stability? For water blows up in
Let us eliminate q Landau convention? (not
stable - P. Romatshke)
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Nonrelativistic experience a four vector
formalism
Energy units of mass
mass velocity (momentum ?) internal
energy velocity-momentum (relativistic?)
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Nonrelativistic spacetime there is time
(absolute)
spacelike, timelike, vectors and
covectors, substantial time derivative
energy-momentum tensor
?
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mass-momentum vector
total energy-momentum tensor
separation of dissipative and nondissiaptive
parts
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Relativistic theory
flow energy
Separation condition
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Meaning?
(a) energies total internal flow (mass?) (b)
velocity momentum (heat) flow energy heat
flux
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Thermodynamics
normal with internal energy e, or
Statics
q dependence
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Balance of entropy
Stable!
Net balances
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Summary momentum density but heat flow
energy internal energy flow energy
ADDS entropy flux and
can ben justified (Liu procedure) linear
stability of homogeneous equilibrium Therm
odynamics ? stability of matter
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Thank you for your attention!
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Energy-momentum - general
Landau choice
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Complete system of equations
lt gt - symmetric traceless spacelike part
Equilibrium
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Linearization
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exponential plane-waves
Stability condition for R
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Routh-Hurwitz
thermodynamic stability
hydrodynamic stability