Non-Newtonian nature of Causal Hydrodynamics

1. Non-Newtonian nature of Causal Hydrodynamics T. Koide (Universidade Federal do Rio de Janeiro) G.S. Denicol (UFRJ),T. Kodama (UFRJ),Ph. Mota (UFRJ) • Because of causality, the relativistic dissipative fluid will be • a non-Newtonian fluid. Thus • The GKN formula should be modified. • 1/4p can be a lower bound of the shear of Newtonian fluids. • The fluid expands to vacuum by forming a stationary wave. • The additional viscosity is still necessary stabilize solutions.

2. Infinite speed in diffusion process Random walk

3. Infinite speed in diffusion process Dispersion relation

4. Relativistic NR equation(1+1) Hiscock&Lindblom(‘85) Linear perturbation analysis >0. stable At , the dispersion relation is reduced to the diffusion type.

5. Introduction of memory effect Equation of continuity Fick’s law Acausal (Cattaneo) Breaking sum rules (Kadanoff&Martin, T.K.)

6. Landau-Lifshitz theory Equation of Continuity shear viscosity bulk viscosity heat current

7. Introduction of Memory Effect T.K.,Denicol,Mota&Kodama(2007) Entropy four current Memory effect Landau-Lifshitz Causal Hydrodynamics

8. Dispersion relation(1+1) Speed of light blue and red Stable in the sense of the linear analysis.

9. LL equation CDhydro Equilibrium Stable Stable Lorentz boost Unstable Stable ? Scaling solution Stable Unstable Linear analysis of stability Hiscock& Lindblam (‘85), Kouno,Maruyama,Takagi&Saito (‘90) The Eckert theory is more unstable than the LL theory.

10. Stability around scaling solution

11. Memory effect Causal Hydrodynamics Landau-Lifshitz Newtonian fluid

12. Anomalous viscosity Pseudoplastic latex, paper pulp, clay solns. Newtonian Navier-Stokes Dilatant Bingham flow quicksand, candy compounds sludge, paint, blood, ketchup Viscosity Thixotropic Rheopectic tars, inks, glue QGP ? Velocity gradient

13. Common sense? • The transport coefficients are given by the Green-Kubo-Nakano formula. • The lower bound of the shear viscosity of the CD hydro. is given by . • The viscosity is small correction. • We do not need any additional viscosity in the CD hydro. Not trivial !!

14. Linear response theory Hamiltonian system External perturbation

15. Green-Kubo-Nakano formula Exact! In Newtonian fluid, transport coefficients are defined by We ignore the memory effect GKN formula

16. Newtonian Non-Newtonian D=?

17. Markov app.

18. Generalization of GKN formula T.K.&Maruyama (2004), T.K.(2005), T.K.(2007), T.K.&Kodama (2008) GKN formula New formula • In the GKN formula, we need • In the generalized formula, we need and • 2. characterizes the deviation from the GKN formula. • When vanishes in the low momentum limit, • the new formula reproduces the GKN formula. • 4. The result obtained in the new formula is consistent with • sum rules.

19. Lower limit of shear viscosity? Kovtun,Son&Starinets(‘05) 1) GKN formula 2) Absorption cross section G: Newton’s constant

20. Lower limit of shear viscosity? Kovtun,Son&Starinets(‘05) Only for Newtonian!! 1) GKN formula 2) Absorption cross section G: Newton’s constant

21. Common sense? • The transport coefficients are given by the Green-Kubo-Nakano formula. • The lower bound of the shear viscosity of the CD hydro. is given by . • The viscosity is small correction.

22. In the following calculations, we consider only the 1+1 dimensional fluid.

23. Expansion of fluid It seems to be a stationary wave.

24. Universal relation between pressure and viscosity We assume that the fluid forms a stationary wave at the boundary to vacuum. Then at the boundary,

25. pressure (-1) * viscosity • The causal fluid expands as a stationary wave. • Viscosity can be the same order as energy density and pressure.

26. Common sense? • The transport coefficients are given by the Green-Kubo-Nakano formula. • The lower bound of the shear viscosity of the CD hydro. is given by . • The viscosity is small correction. • We do not need any additional viscosity in the CD hydro.

27. Numerical oscillation The frequency is the size of the grid. unstable

28. Usual “artificial” viscosity Navier-Stokes type: Burnett type: Neumann-Richtmyer type: Causal? Generalization to relativistic cases ?

29. Additional causal viscosity We need to introduce the artificial viscosity consistent with causality. :the size of the grid (0.01 fm)

32. Effect to entropy production The effect of the additional viscosity is This is the same order of the breaking of the energy conservation in our code.

33. Summary • The transport coefficients are given by the Green-Kubo-Nakano formula. • The lower bound of the shear viscosity of the CD hydro. is given by . • The viscosity is small correction. • We do not need any additional viscosity in the CD hydro. No. We need a new formula for non-Newtonian fluids Not yet clear for causal dissipative hydro. No. We have a relation and forms a stationary wave. We need.

34. We have to check • Is the new formula really useful or not. • What is the lower bound of the causal shear viscosity? • CD hydro is a stable theory? • Are numerical oscillations we encountered due to the problem of numerical calculation or the problem of the theory (turbulence)?

35. That is, the causal hydrodynamics will be one of non-Newtonian fluids!! Newtonian Non-Newtonian

36. Stability and Causality(1+1) T.K. Denicol,Kodama&Mota (‘07) Linear perturbation analysis of the causal hydro. Sound velocity is faster than ideal. Stable, Independent of momentum

37. It is difficult to cut high momentum modes to avoid unstable modes. Model equation of hydro Non-linear effect assumption We need to find a new theory to satisfy causality.

38. Additional viscosity