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Explore dissipative relativistic fluids, stability theories, thermodynamic equilibrium, and extended theories in this study on fluid dynamics. Analyze temperature in moving bodies and the implications of non-equilibrium thermodynamics. Investigate the properties of internal energy and linear stability to deepen your understanding of hydrodynamics.
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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
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)
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)
p Fourier-Navier-Stokes 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 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)
Stability conditions of the Israel-Stewart theory (Hiscock-Lindblom 1985)
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.
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)
Eckart term Modified relativistic irreversible thermodynamics: Internal energy: Ván and Bíró EPJ, (2008), 155, 201. (arXiv:0704.2039v2)
Dissipative hydrodynamics < > symmetric traceless spacelike part • linear stability of homogeneous equilibrium Conditions: thermodynamic stability, nothing more. (Ván: arXiv:0811.0257)
Thermostatics: Temperatures and other intensives are doubled: Different roles: Equations of state: Θ, M Constitutive functions: T, μ
About the temperature of moving bodies: moving body inertial observer
About the temperature of moving bodies: moving body inertial observer
body v K0 K About the temperature of moving bodies: translational work Einstein-Planck: entropy is vector, energy + work is scalar
body v K0 K Ott - hydro: entropy is vector, energy-pressure are from a tensor
energy(-momentum) vector Landsberg Einstein-Planck non-dissipative Ott
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.
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)
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.
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
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
v2 v1 K1 K2 K Einstein-Planck Ott lightlike
Body Velocity distributions: u v K K0 Averages? (Cubero et. al. PRL 2007, 99 170601) Heavy-ion experiments, cosmology.
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 ?)
Dissipation inequality 1) 2)
Energy-momentum – momentum density and energy flux Landau choice:
Routh-Hurwitz: thermodynamic stability
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, ….
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
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.
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.
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, ….
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
About the temperature of moving bodies: moving body Sardegna inertial observer