Temperature of moving bodies thermodynamic hydrodynamic and kinetic

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

About the temperature of moving bodies (part 1) Planck and Einstein body v observer K 0 K Relativistic thermodynamics?

body v observer K 0 K • Planck-Einstein (1907): cooler • Ott (1963) [Blanusa (1947)] : hotter • Landsberg (1966 -67): equal • Costa-Matsas-Landsberg (1995): direction dependent (Doppler)

translational work – heat = momentum v observer K 0 K reciprocal temperature - vector?

Rest frame arguments: Ott (1963) d. Q body v reservoir K K 0 Planck-Einstein d. Q body v K 0 K reservoir Ott

body v observer No translational work K K 0 temperature – vector? Blanusa (1947) Einstein (1952) (letter to Laue)

Outcome Historical discussion (~1963 -70, Moller, von Treder, Israel, ter Haar, Callen, …, renewed Dunkel-Talkner-Hänggi 2007): new arguments/ no (re)solution. → Doppler transformation Landsberg e. g. solar system, microwave background → Velocity is thermodynamic variable? van Kampen Relativistic statistical physics and kinetic theory: Jüttner distribution (1911): Einstein-Planck (Ott? )

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

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

Thermodynamic relations - normalization Jüttner distribution? Legendre transformation

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

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

Energy-momentum density: Heat flux:

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

Questions • 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

Thermodynamics → hydrodynamics (which one? ) Nonrelativistic Relativistic Local equilibrium (1 st order) Fourier+Navier-Stokes Eckart (1940), Tsumura-Kunihiro Beyond local equilibrium (2 nd order) Cattaneo-Vernotte, gen. Navier-Stokes Israel-Stewart (1969 -72), Pavón, Müller-Ruggieri-Liu, Geroch, Öttinger, Carter, conformal, Rishke-Betz, etc… Eckart: Extended (Israel–Stewart – Pavón–Jou–Casas-Vázquez): (+ order estimates)

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): qa – momentum density or energy flux? Eckart term

Heat flow problem – kinetic theory versus Israel-Stewart hydro in Riemann shocks: Bouras, I. et. al. , PRC 2010 under publication (arxiv: 1006. 0387 v 2)

Improved Eckart theory: Internal energy: Eckart term Ván and Bíró EPJ, (2008), 155, 201. (ar. Xiv: 0704. 2039 v 2)

Dissipative hydrodynamics < > symmetric traceless spacelike part Þ linear stability of homogeneous equilibrium Conditions: thermodynamic stability, nothing more. Ván P. : J. Stat. Mech. (2009) P 02054 Þ Israel-Stewart like relaxational (quasi-causal) extension Biró T. S. et. al. : PRC (2008) 78, 014909

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

About the temperature of moving bodies (part 2) w spacelike, but |w|<1 -- velocity of the heat current Interaction w 1 v 1 w 2 v 2 observer • 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

w 1 v 1 1+1 dimension: w 2 v 2

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

T T 0 w w 0 v K thermometer K 0 Special: w 0 = 0 T = T 0 / γ Planck-Einstein w=0 T = γ T 0 Ott w 0 = 1, v > 0 T = T 0 • red Doppler w 0 = 1, v < 0 T = T 0 • blue Doppler T= T 0 Landsberg w 0 + w = 0 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 hydro Outlook: Dissipation beyond a single viscosity? Causal and generic stable hydro from improved moment series expansion.

Thank you for the attention!

Red Blue Ott-Blanusa Planck-Einstein Landsberg shifted doppler

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

Remarks on stability and Second Law: Non-equilibrium thermodynamics: basic variables evolution equations (basic balances) Second Law 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 Continuum systems (irreversible thermodynamics): partial differential equations – Lyapunov theorem is more technical Linear stability (of homogeneous equilibrium)

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

Summary Internal energy: E – – Ea S = S(E, V, N) Work with momentum exchange Relative velocity v is zero Cooler, hotter, equal or Doppler? – – – S = S(Ea, V, N) energy-momentum exchange T and v do not equilibrate γw. T and w v are equilibrating T: Doppler of w with the speed v Bíró-Ván: EPL, 89, 30001, 2010, (ar. Xiv: 0905. 1650) Wolfram Demonstration Project, Transformation of … Heavy ion physics: dissipative relativistic fluids Ván: J. Stat. Mech. P 02054, 2009. Bíró-Molnár-Ván: PRC 78, 014909, 2008.