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Thermodynamics of relativistic fluids P. Ván Department of Theoretical Physics Research Institute of Particle and Nuclear Physics, Budapest, Hungary –

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Presentation on theme: "Thermodynamics of relativistic fluids P. Ván Department of Theoretical Physics Research Institute of Particle and Nuclear Physics, Budapest, Hungary –"— Presentation transcript:

1 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:

2 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)

3 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

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

5 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

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

7 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:

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

9 Modified relativistic irreversible thermodynamics: Eckart term Internal energy:

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

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

14 K K0K0 v Body 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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