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Thermodynamics of relativistic fluids

Internal energy:. Thermodynamics of relativistic fluids. P. Ván Department of Theoretical Physics Research Institute of Particle and Nuclear Physics, Budapest, Hungary. Introduction Thermodynamics and stability N on-relativistic fluids

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Thermodynamics of relativistic fluids

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  1. Internal energy: Thermodynamics of relativistic fluids P. Ván Department of Theoretical Physics Research Institute of Particle and Nuclear Physics, Budapest, Hungary • Introduction • Thermodynamics and stability • Non-relativistic fluids • Stability paradoxof dissipative relativistic fluids • About the temperature of moving bodies • Conclusions common work with T. S. Bíró and E. Molnár

  2. Non-equilibrium thermodynamics: basic variables Second Law evolution equations (basic balances) Stability of homogeneous equilibrium Introduction – role of the Second Law: Entropy ~ Lyapunov function Homogeneous systems (equilibrium thermodynamics): dynamic reinterpretation – ordinary differential equations clear, mathematically strict - Finite time thermodynamics …. - Matolcsi T.: Ordinary thermodynamics (Academic Publishers, 2005)

  3. Continuum systems partial differential equations – Lyapunov theorem is more technical Linear stability (of homogeneous equilibrium) Example: 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 pi momentum density Thermodynamics p

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

  5. Dissipative relativistic fluids Nonrelativistic Relativistic Local equilibrium Fourier+Navier-Stokes Eckart (1940) (1st order) Beyond local equilibrium Cattaneo-Vernotte, Israel-Stewart (1969-72), (2nd order) gen. Navier-Stokes Müller-Ruggieri, Öttinger, Carter, etc. Eckart: Israel-Stewart:

  6. Eckart term Special relativistic fluids (Eckart): energy-momentum density particle density vector General representations by local rest frame quantities.

  7. Stabilityof homogeneousequilibrium Eckart theory: instable – due to heat conduction water Israel-Stewart theory: stability is conditional: complicated conditions  relaxation to the first order theory?(Geroch 1995, Lindblom 1995)

  8. Second Law as a constrained inequality (Liu procedure): 1) 2) Ván: under publication in JMMS, (arXiv:07121437)

  9. Modified relativistic irreversible thermodynamics: Internal energy: Eckart term

  10. Dissipative hydrodynamics < > symmetric traceless spacelike part • linear stability of homogeneous equilibrium CONDITION: thermodynamic stablity

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

  12. About the temperature of moving bodies: moving body CRETE inertial observer

  13. Body v K0 K About the temperature of moving bodies: translational work Einstein-Planck: entropy is invariant, energy is vector

  14. Body v K0 K Ott - hydro: entropy is vector, energy-presssure are from a tensor Our:

  15. Summary – energy≠ internal energy → generic stability without extra conditions - relativistic thermodynamics: there is no local equilibrium - different temperatures in Fourier-law (equilibration) and in state functions out of local equilibrium. - causality /Ván and Bíró, EPJ, (2007), 155, p201-212, (arXiv:0704.2039v2)/ - hyperbolic(-like) extensions, solutions /Bíró, Molnár and Ván: under publication in PRC, (arXiv:0805.1061)/

  16. Thank you for your attention!

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