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 1)The GKN formula should be modified. 2)1/4  can be a lower bound of the shear of Newtonian fluids. 3)The fluid expands to vacuum by forming a stationary wave. 4)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) At, the dispersion relation is reduced to the diffusion type. >0. stable Linear perturbation analysis Hiscock&Lindblom(‘85)

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 heat current bulk viscosity shear viscosity

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

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

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

10. Stability around scaling solution

11. Newtonian fluid Memory effect Causal Hydrodynamics Landau-Lifshitz

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

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 system Hamiltonian External perturbation

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

16. NewtonianNon-Newtonian D=?

17. Markov app.

18. 1. In the GKN formula, we need In the generalized formula, we need and 2. characterizes the deviation from the GKN formula. 3.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. Generalization of GKN formula T.K.&Maruyama (2004), T.K.(2005), T.K.(2007), T.K.&Kodama (2008) GKN formula New formula

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

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

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 1.The causal fluid expands as a stationary wave. 2.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: Generalization to relativistic cases ? Causal?

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. We need. No. We have a relation and forms a stationary wave.

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

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

38. Additional viscosity