Temperature of moving bodies – thermodynamic, hydrodynamic and kinetic aspects Peter Ván 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 Tamás Biró, Etele Molnár

Planck and Einstein body v observer K0K0 K Relativistic thermodynamics? About the temperature of moving bodies (part 1)

Planck-Einstein (1907): cooler Ott (1963) [Blanusa (1947)] : hotter Landsberg ( ): equal Costa-Matsas-Landsberg (1995): direction dependent (Doppler) v body observer K0K0 K

translational work – heat = momentum v observer K0K0 K reciprocal temperature - vector?

Rest frame arguments: Ott (1963) v body reservoir K K0K0 dQ Planck-Einstein v body reservoir K K0K0 dQ Ott

v body observer K K0K0 No translational work Blanusa (1947) Einstein (1952) (letter to Laue) temperature – vector?

Outcome Einstein-Planck (Ott?) Relativistic statistical physics and kinetic theory: Jüttner distribution (1911): Historical discussion (~ , Moller, von Treder, Israel, ter Haar, Callen, …, renewed Dunkel-Talkner-Hänggi 2007 ): new arguments/ no (re)solution. → Doppler transformation e.g. solar system, microwave background → Velocity is thermodynamic variable? Landsberg van Kampen

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

Boltzmann equation Kinetic theory → thermodynamics (local) equilibrium distribution Thermodynamic equilibrium = no dissipation: Boltzmann gas

Thermodynamic relations - normalization Jüttner distribution? Legendre transformation

covariant Gibbs relation (Israel,1963) Lagrange multipliers – non-equilibrium Rest frame quantities: Remark:

ideal gas rest frame/uniform intensives Velocity dependence? deviation from Jüttner A) B)

Energy-momentum density: Heat flux:

Summary of kinetic equilibrium: - Gibbs relation (of Israel): - Equilibrium spacelike parts:

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 Molnár What is the role and manifestation of local thermodynamic equilibrium? – generic stability and causality Questions

NonrelativisticRelativistic Local equilibrium Fourier+Navier-StokesEckart (1940), (1st order)Tsumura-Kunihiro Beyond local equilibriumCattaneo-Vernotte, Israel-Stewart ( ), (2 nd order) gen. Navier-Stokes Pavón, Müller-Ruggieri-Liu, Geroch, Öttinger, Carter, conformal, Rishke-Betz, etc… Eckart: Extended (Israel – Stewart – Pavón–Jou – Casas-Vázquez): (+ order estimates) Thermodynamics → hydrodynamics (which one?)

Israel – Stewart - conditional suppression (Hiscock and Lindblom, 1985):

Remarks on causality and stability: Symmetric hyperbolic equations ~ causality – The extended theories are not proved to be symmetric hyperbolic (exception: Müller-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

Special relativistic fluids (Eckart): Eckart term q a – momentum density or energy flux?

Bouras, I. et. al., PRC 2010 under publication (arxiv: v2) Heat flow problem – kinetic theory versus Israel-Stewart hydro in Riemann shocks:

Improved Eckart theory: Internal energy: Eckart term Ván and Bíró EPJ, (2008), 155, 201. (arXiv: v2)

Dissipative hydrodynamics symmetric traceless spacelike part  linear stability of homogeneous equilibrium Conditions: thermodynamic stability, nothing more. Ván P.: J. Stat. Mech. (2009) P02054  Israel-Stewart like relaxational (quasi-causal) extension Biró T.S. et. al.: PRC (2008) 78,

integrating multiplier Hydrodynamics → thermodynamics H(  2 ) H(  1 ) Volume integrals: work, heat, internal energy Change of heat and entropy: temperature integrating multiplier

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|<1 -- velocity of the heat current About the temperature of moving bodies (part 2)

v2v2 w2w2 v1v1 w1w1 1+1 dimension:

Four velocities: v 1, v 2, w 1, w 2 Transformation of temperatures v w2w2 w1w1 Relative velocity (Lorentz transformation) general Doppler-like form!

Special: w 0 = 0T = T 0 / γ Planck-Einstein w = 0T = γ 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 Biró T.S. and Ván P.: EPL, 89 (2010) 30001

V=0.6, c=1

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.

Thank you for the attention!

Blue shifted dopplerPlanck-EinsteinLandsbergOtt-BlanusaRed shifted doppler

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

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)

Thermodynamics Hydrodynamics Kinetic theory homogeneity equilibrium moment series general balances concepts homogeneity

Summary –S = S(E,V,N) –Work with momentum exchange –Relative velocity v is zero –Cooler, hotter, equal or Doppler? Ván: J. Stat. Mech. P02054, Bíró-Molnár-Ván: PRC 78, , Bíró-Ván: EPL, 89, 30001, 2010, (arXiv: ) 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 – γ w T and w  v are equilibrating – T: Doppler of w with the speed v Heavy ion physics: dissipative relativistic fluids