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Hydrodynamics, stability and temperature

Internal energy:. Hydrodynamics, stability and temperature. 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

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Hydrodynamics, stability and temperature

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  1. Internal energy: Hydrodynamics, stability and temperature 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

  2. Dissipative relativistic fluids 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): (+ order estimates)

  3. Remarks on causality and stability: Symmetric hyperbolic equations ~ causality –The extended theoriesarenot 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)

  4. p Fourier-Navier-Stokes Isotropic linear constitutive relations, <> is symmetric, traceless part Equilibrium: Linearization, …, Routh-Hurwitz criteria: Thermodynamic stability (concave entropy) Hydrodynamic stability

  5. Remarks on stability and Second Law: Non-equilibrium thermodynamics: basic variables Second Law evolution equations (basic balances) 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)

  6. Stability conditions of the Israel-Stewart theory (Hiscock-Lindblom 1985)

  7. Eckart term Special relativistic fluids (Eckart): energy-momentum density particle density vector qa – momentum density or energy flux?? General representations by local rest frame quantities.

  8. Eckart term Modified relativistic irreversible thermodynamics: Internal energy: Ván and Bíró EPJ, (2008), 155, 201. (arXiv:0704.2039v2)

  9. Dissipative hydrodynamics < > symmetric traceless spacelike part • linear stability of homogeneous equilibrium Conditions: thermodynamic stability, nothing more. (Ván: arXiv:0811.0257)

  10. About the temperature of moving bodies: moving body inertial observer

  11. About the temperature of moving bodies: moving body inertial observer

  12. body v K0 K About the temperature of moving bodies: translational work Einstein-Planck: entropy is vector, energy + work is scalar

  13. body v K0 K Ott - hydro: entropy is vector, energy-pressure are from a tensor Landsberg:

  14. Thermostatics: Temperatures and other intensives are doubled: Different roles: Equations of state: Θ, M Constitutive functions: T, μ

  15. energy(-momentum) vector Landsberg Einstein-Planck non-dissipative Ott

  16. 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)

  17. Thank you for your attention!

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