Relativistic hydrodynamics stability and causality P Vn and
Relativistic hydrodynamics – stability and causality P. Ván and T. S. Bíró University of Bergen and RMKI, Budapest – Introduction – Stability – Eckart problem – Causality – parabolic equations – Separation of dissipative and nondissipative parts – Conclusions Zimányi 75 Memorial Workshop’ 07, Budapest
Introduction: Nonrelativistic Relativistic Local equilibrium (1 st) Fourier, Navier-Stokes Eckart Beyond local equilibrium (2 nd) Cattaneo-Vernotte, gen. Navier-Stokes Israel-Stewart, Müller-Ruggieri Öttinger, Carter, etc. . Conceptual issues plaguing relativistic hydrodynamics: Causality – first order is bad – noncausal second order is good - causal Stability – first order is bad – instable second order is good - stable Dissipative – nondissipative: symmetric part? ?
Stability of what and in what sense? homogeneous equilibrium (thermodynamics = theory of stability of …) linear and nonlinear – necessary condition Eckart theory: instable – due to heat conduction water Israel-Stewart theory: strange condition relaxation to the first order theory (Geroch 1995, Lindblom 1995)
Structure of dissipative hydrodynamic theories: Irreversible thermodynamics (heuristic method): Eckart term
Complete Eckart system < > - symmetric traceless spacelike part Equilibrium:
exponential plane-waves Stability condition for transverse modes: $root with a positive real part instability coupling of shear viscosity and heat conduction Landau frame?
Causality hyperbolic or parabolic? Well posedness Speed of signal propagation Second order linear partial differential equation: Corresponding equation of characteristics: i) Hyperbolic equation: Parabolic equation: Elliptic equation: two distinct families of real characteristics one distinct families of real characteristics no real characteristics Well posedness: existence, unicity, continuous dependence on initial data. A characteristic Cauchy problem of (1) is well posed. (initial data on the characteristic surface: )
ii) (*) is transformation invariant (1) iii) The outer real characteristics that pass through a given point domain of influence. x x t E. g. give its t
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, ….
Second order theory? stability – problematic Landau choice? As long as it is acceptable from a physical point of view. causality – no problem validity – second order effects can be important First order theory? stability – very problematic Landau choice? As long as it is acceptable from a physical point of view. Numerical problems!! causality – small problem – estimation? validity – second order effects can be important Origin of stability problem: wrong separation of dissipative and non dissipative terms and effects the choice of velocity field is not free (e. g. entropy production)
Separation of dissipation physics of flow energy Separation condition:
Meaning? (a) energies: total= internal+ flow (mass? ) (b) velocity – momentum (heat) flow energy – heat flux
Thermodynamics: normal with internal energy e, or: Statics: q dependence:
Summary momentum density = but ≡ heat flow – energy = internal energy + flow energy – ADDS: – entropy flux and can ben justified (thermodynamic theory construction – Liu procedure) – linear stability of homogeneous equilibrium Thermodynamics stability of matter
Thank you for your attention!
Balance of entropy: Stable! Net balances:
Energy-momentum - general Landau choice:
Linearization
Routh-Hurwitz: thermodynamic stability hydrodynamic stability
Eckart term: Ideal hydro hits the target. feelings on dissipation… water
Thermodynamics: Statics: Dynamics:
Isotropic current-force functions: Stability? For water blows up in Let us eliminate q: Landau convention? (not stable - P. Romatshke)
Nonrelativistic experience – a four vector formalism Energy units of mass: mass velocity (momentum ? ) internal energy velocity-momentum (relativistic? )
Nonrelativistic spacetime: there is time (absolute) spacelike, timelike, vectors and covectors, substantial time derivative ? energy-momentum tensor
mass-momentum vector 1442 4 43 total energy-momentum tensor separation of dissipative and nondissiaptive parts
- Slides: 28