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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 paradox of dissipative relativistic fluids – About the temperature of moving bodies – Conclusions common work with T. S. Bíró and E. Molnár Internal energy:

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Introduction – role of the Second Law: Non-equilibrium thermodynamics: basic variablesSecond Law evolution equations (basic balances) Stability of homogeneous equilibrium 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)

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partial differential equations – Lyapunov theorem is more technical Continuum systems Linear stability (of homogeneous equilibrium) Example: non-relativistic fluid mechanics local equilibrium, Fourier-Navier-Stokes nparticle number density v i relative (3-)velocity einternal energy density q i internal energy (heat) flux P ij pressure p i momentum density Thermodynamics p

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

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

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

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instable – due to heat conduction Stability of homogeneous equilibrium water Israel-Stewart theory: stability is conditional: complicated conditions relaxation to the first order theory? (Geroch 1995, Lindblom 1995) Eckart theory:

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

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Modified relativistic irreversible thermodynamics: Eckart term Internal energy:

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Dissipative hydrodynamics symmetric traceless spacelike part linear stability of homogeneous equilibrium CONDITION: thermodynamic stablity

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

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

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K K0K0 v Body translational work Einstein-Planck: entropy is invariant, energy is vector About the temperature of moving bodies:

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K K0K0 v Body Ott - hydro: entropy is vector, energy-presssure are from a tensor Our:

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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) /

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