Dissipative relativistic hydrodynamics

1. Internal energy: Dissipative relativistic hydrodynamics P. Ván Department of Theoretical Physics Research Institute of Particle and Nuclear Physics, Budapest, Hungary • Motivation • Problems with second order theories • Thermodynamics, fluids and stability • Generic stability of relativistic dissipative fluids • Temperature of moving bodies • Summary

2. Dissipative relativistic fluids Nonrelativistic Relativistic Local equilibrium Fourier+Navier-Stokes Eckart (1940), (1st order) Tsumura-Kunihiro (2008) Beyond local equilibrium Cattaneo-Vernotte, Israel-Stewart (1969-72), (2nd order) gen. Navier-Stokes Pavón, Müller-Ruggieri, Geroch, Öttinger, Carter, conformal, etc. Eckart: Extended (Israel–Stewart – Pavón–Jou–Casas-Vázquez): (+ order estimates)

3. Remarks on causality and stability: Symmetric hyperbolic equations ~ causality –The extended theoriesarenot proved to be symmetric hyperbolic. – In Israel-Stewart theory the symmetric hyperbolicity conditions of the perturbation equations follow from the stability conditions. – Parabolic theories cannot be excluded – speed of the validity range can be small. Moreover, they can be extended later. Stability of the homogeneous equilibrium (generic stability) is required. – Fourier-Navier-Stokes limit. Relaxation to the (unstable) first order theory? (Geroch 1995, Lindblom 1995)

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

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

6. Stability conditions of the Israel-Stewart theory (Hiscock-Lindblom 1985)

7. Eckart term Special relativistic fluids (Eckart): energy-momentum density particle density vector qa – momentum density or energy flux?? General representations by local rest frame quantities.

8. State space: Second Law (Liu procedure) – first order weakly nonlocal: Entropy inequality with the conditions of energy-momentum and particle number balances as constraints: Consequences: 1) 2) 3) Ván: JMMS, 2008, 3/6, 1161, (arXiv:07121437)

9. Eckart term Modified relativistic irreversible thermodynamics: Internal energy: Ván and Bíró EPJ, (2008), 155, 201. (arXiv:0704.2039v2)

10. Dissipative hydrodynamics < > symmetric traceless spacelike part • linear stability of homogeneous equilibrium Conditions: thermodynamic stability, nothing more. (Ván: arXiv:0811.0257)

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

12. About the temperature of moving bodies: moving body inertial observer

13. About the temperature of moving bodies: moving body inertial observer

14. body v K0 K About the temperature of moving bodies: translational work Einstein-Planck: entropy is vector, energy + work is scalar

15. body v K0 K Ott - hydro: entropy is vector, energy-pressure are from a tensor

16. energy(-momentum) vector Landsberg Einstein-Planck non-dissipative Ott

17. Simple transformation properties? Equilibration: Two bodies A and B have relative speed v. What must be the relation between their temperatures TA and TB, measured in their rest frames, if they are to be in thermal equilibrium? Integration, homogeneity: Thermal interaction requires uniform velocities.

18. Quasi-hyperbolic extension – relaxation of viscosity: Relaxation: Simpler than Israel-Stewart: there are no βderivatives. Bíró, Molnár and Ván: PRC, (2008), 78, 014909 (arXiv:0805.1061)

19. 1) Generalized Bjorken flow - the role of q: tetrad : ; axial symmetry Only for the q=0 solution remains the v=0 Bjorken-flow stationary. 2) Temperatures: • qgp eos • τ0 = 0.6fm/c, • e0=ε0 =30GeV/fm3 • η/s=0.4, • π0=0.

20. 3) Reheating: Eckart: R-1<1 (p<4π) stability η0 Eckart IS HO 0.3 6·10−4 5.6·10−7 2.67·10−4 0.08 3·10−6 2.89·10−9 1.75·10−4 LHC RHIC

21. Summary – Extended theories are not ultimate. – energy≠ internal energy → generic stability without extra conditions – hyperbolic(-like) extensions, generalized Bjorken solutions, reheating conditions, etc… – different temperatures in Fourier-law (equilibration) and in EOS out of local equilibrium →temperature of moving bodies - interpretation

22. Thank you for your attention!

23. v2 v1 K1 K2 K Einstein-Planck Ott lightlike

24. Body Velocity distributions: u v K K0 Averages? (Cubero et. al. PRL 2007, 99 170601) Heavy-ion experiments, cosmology.

25. Solution of Liu equations ( are local): Liu procedure for relativistic fluids Thermodynamics – local rest frame • basic state (fields): • constitutive state: • constitutive functions: 4-vector (temperature ?)

26. Dissipation inequality 1) 2)

27. Energy-momentum – momentum density and energy flux Landau choice:

28. Linearization

29. exponential plane-waves

30. Routh-Hurwitz: thermodynamic stability

31. Causality hyperbolic or parabolic? Well posedness Speed of signal propagation Hydrodynamic range of validity: ξ – mean free path τ – collision time Water at room temperature: More complicated equations, more spacetime dimensions, ….

32. 1) Hyperbolicity does not result in automatic causality, because the propagation speed of small perturbations can be large. hyperbolic  causal 2) Parabolic equations and first order theories are not automatically excluded. The validity range of the theory could prevent large speeds if the perturbations were relaxing fast. parabolic+stable causal 3) Instability in first order theories is not acceptable. Second order dissipative theories are corrections to first order stable theories. Remarks on hyperbolicity

33. A characteristic Cauchy problem of (1) is well posed. (initial data on the characteristic surface: ) Causality hyperbolic or parabolic? Well posedness Speed of signal propagation Second order linear partial differential equation: Corresponding equation of characteristics: i) Hyperbolic equation: two distinct families of real characteristics Parabolic equation: one distinct families of real characteristics Elliptic equation: no real characteristics Well posedness: existence, unicity, continuous dependence on initial data.

34. iii) The outer real characteristics that pass through a given point give its domain of influence . (1) ii) (*) is transformation invariant x x t t E.g.

35. Infinite speed of signal propagation? physics - mathematics Hydrodynamic range of validity: ξ – mean free path τ – collision time Water at room temperature: Fermi gas of light quarks at : More complicated equations, more spacetime dimensions, ….

36. p Non-relativistic fluid mechanics local equilibrium, Fourier-Navier-Stokes n particle number density vi relative (3-)velocity e internal energy density qi internal energy (heat) flux Pij pressure ki momentum density Thermodynamics

37. About the temperature of moving bodies: moving body Sardegna inertial observer