temperature of moving bodies – thermodynamic, hydrodynamic and kinetic aspects peter ván kfki,...

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  • Slide 1
  • Temperature of moving bodies thermodynamic, hydrodynamic and kinetic aspects Peter Vn KFKI, RMKI, Dep. Theoretical Physics Temperature of moving bodies the story Relativistic equilibrium kinetic theory Stability and causality hydrodynamics Temperature of moving bodies the conclusion Outlook with Tams Bir, Etele Molnr
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  • Planck and Einstein body v observer K0K0 K Relativistic thermodynamics? About the temperature of moving bodies (part 1)
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  • Planck-Einstein (1907): cooler Ott (1963) [Blanusa (1947)] : hotter Landsberg (1966-67): equal Costa-Matsas-Landsberg (1995): direction dependent (Doppler) v body observer K0K0 K
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  • translational work heat = momentum v observer K0K0 K reciprocal temperature - vector?
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  • Rest frame arguments: Ott (1963) v body reservoir K K0K0 dQ Planck-Einstein v body reservoir K K0K0 dQ Ott
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  • v body observer K K0K0 No translational work Blanusa (1947) Einstein (1952) (letter to Laue) temperature vector?
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  • Outcome Einstein-Planck (Ott?) Relativistic statistical physics and kinetic theory: Jttner distribution (1911): Historical discussion (~1963-70, Moller, von Treder, Israel, ter Haar, Callen, , renewed Dunkel-Talkner-Hnggi 2007 ): new arguments/ no (re)solution. Doppler transformation e.g. solar system, microwave background Velocity is thermodynamic variable? Landsberg van Kampen
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  • Questions What is moving (flowing)? barion, electric, etc. charge (Eckart) energy (Landau-Lifshitz) What is a thermodynamic body? volume expansion (Hubble) What is the covariant form of an e.o.s.? S(E,V,N,) Interaction: how is the temperature transforming kinetic theory and/or hydrodynamics
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  • Boltzmann equation Kinetic theory thermodynamics (local) equilibrium distribution Thermodynamic equilibrium = no dissipation: Boltzmann gas
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  • Thermodynamic relations - normalization Jttner distribution? Legendre transformation
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  • covariant Gibbs relation (Israel,1963) Lagrange multipliers non-equilibrium Rest frame quantities: Remark:
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  • ideal gas rest frame/uniform intensives Velocity dependence? deviation from Jttner A) B)
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  • Energy-momentum density: Heat flux:
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  • Summary of kinetic equilibrium: - Gibbs relation (of Israel): - Equilibrium spacelike parts:
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  • What is dissipative? dissipative and non-dissipative parts Free choice of flow frames? QGP - effective hydrodynamics. Kinetic theory hydrodynamics local equilibrium in the moment series expansion talk of Etele Molnr What is the role and manifestation of local thermodynamic equilibrium? generic stability and causality Questions
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  • NonrelativisticRelativistic Local equilibrium Fourier+Navier-StokesEckart (1940), (1st order)Tsumura-Kunihiro Beyond local equilibriumCattaneo-Vernotte, Israel-Stewart (1969-72), (2 nd order) gen. Navier-Stokes Pavn, Mller-Ruggieri-Liu, Geroch, ttinger, Carter, conformal, Rishke-Betz, etc Eckart: Extended (Israel Stewart PavnJou Casas-Vzquez): (+ order estimates) Thermodynamics hydrodynamics (which one?)
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  • Israel Stewart - conditional suppression (Hiscock and Lindblom, 1985):
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  • Remarks on causality and stability: Symmetric hyperbolic equations ~ causality The extended theories are not proved to be symmetric hyperbolic (exception: Mller-Ruggeri-Liu). In Israel-Stewart theory the symmetric hyperbolicity conditions of the perturbation equations follow from the stability conditions. Generic stable parabolic theories can be extended later. Stability of the homogeneous equilibrium (generic stability) is related to thermodynamics. Thermodynamics generic stability causality
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  • Special relativistic fluids (Eckart): Eckart term q a momentum density or energy flux?
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  • Bouras, I. et. al., PRC 2010 under publication (arxiv:1006.0387v2) Heat flow problem kinetic theory versus Israel-Stewart hydro in Riemann shocks:
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  • Improved Eckart theory: Internal energy: Eckart term Vn and Br EPJ, (2008), 155, 201. (arXiv:0704.2039v2)
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  • Dissipative hydrodynamics symmetric traceless spacelike part linear stability of homogeneous equilibrium Conditions: thermodynamic stability, nothing more. Vn P.: J. Stat. Mech. (2009) P02054 Israel-Stewart like relaxational (quasi-causal) extension Bir T.S. et. al.: PRC (2008) 78, 014909
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  • integrating multiplier Hydrodynamics thermodynamics H( 2 ) H( 1 ) Volume integrals: work, heat, internal energy Change of heat and entropy: temperature integrating multiplier
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  • there are four different velocities only one of them can be eliminated the motion of the body and the energy-momentum currents are slower than light Interaction v2v2 observer w2w2 v1v1 w1w1 w spacelike, but |w|
  • Special: w 0 = 0T = T 0 / Planck-Einstein w = 0T = T 0 Ott w 0 = 1, v > 0T = T 0 red Doppler w 0 = 1, v < 0T = T 0 blue Doppler w 0 + w = 0T= T 0 Landsberg v w0w0 w KK0K0 thermometer T T0T0 Bir T.S. and Vn P.: EPL, 89 (2010) 30001
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  • V=0.6, c=1
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  • Summary Generalized Gibbs relation: consistent kinetic equilibrium improves hydrodynamics explains temperature of moving bodies KEY: no freedom in flow frames (Eckart or Landau-Lifshitz)!? evolving frame? is dissipation frame independent? QGP - effective hydroOutlook: Dissipation beyond a single viscosity? Causal and generic stable hydro from improved moment series expansion.
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  • Thank you for the attention!
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  • Blue shifted dopplerPlanck-EinsteinLandsbergOtt-BlanusaRed shifted doppler
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  • Isotropic linear constitutive relations, is symmetric, traceless part Equilibrium: Linearization, , Routh-Hurwitz criteria: Hydrodynamic stability Thermodynamic stability (concave entropy) Fourier-Navier-Stokes p
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  • Remarks on stability and 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 See e.g. Matolcsi, T.: Ordinary thermodynamics, Academic Publishers, 2005 partial differential equations Lyapunov theorem is more technical Continuum systems (irreversible thermodynamics): Linear stability (of homogeneous equilibrium)
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  • Thermodynamics Hydrodynamics Kinetic theory homogeneity equilibrium moment series general balances concepts homogeneity
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  • Summary S = S(E,V,N) Work with momentum exchange Relative velocity v is zero Cooler, hotter, equal or Doppler? Vn: J. Stat. Mech. P02054, 2009. Br-Molnr-Vn: PRC 78, 014909, 2008. Br-Vn: EPL, 89, 30001, 2010, (arXiv:0905.1650) Wolfram Demonstration Project, Transformation of Internal energy: E E a S = S(E a, V, N) energy-momentum exchange T and v do not equilibrate w T and w v are equilibrating T: Doppler of w with the speed v Heavy ion physics: dissipative relativistic fluids


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