1. Viscous hydrodynamics and Transport Models Azwinndini Muronga 1,2 1 Centre for Theoretical Physics and Astrophysics Department of Physics, University of Cape Town, South Africa 2 UCT-CERN Research Centre Department of Physics, University of Cape Town, South Africa Workshop on “Viscous Hydrodynamics and Transport Models in heavy Ion Collisions” May 2, 2008 : BNL, Long Island, NY, USA

2. Dissipative Relativistic Fluid Dynamics Summary and Conclusions  It concerns conservation of net charges, energy-momentum and balance of fluxes. The primary state quantities such as number current, energy-momentum- tensor and entropy current differ from the ideal fluid by additional dissipative fluxes  It concerns non-linear, coupled partial differential equations  Formulation is relativistic and this add another complexity.  The system of equations is still closed by the equation of state.  In addition the balance of fluxes is controlled by the transport coefficients. Together with the equation of state they determine the relaxation times/lengths  Analytic solutions are rare.  Numerical solution poses a challenge.  Initial conditions are more interesting.  DFD open s a window that one can use to connect the macroscopic and microscopic dynamics of a system under consideration: in our case the system is the hot and dense matter produced in relativistic nuclear collisions.  The statement: transport coefficients are as important as the equation of state can no longer be overemphasized. Refers to: A. Muronga (2007) I&II

3. Non-equilibrium fluid dynamics from kinetic theory The equations for the first three moments of distribution function where

4. Non-equilibrium fluid dynamics from kinetic theory

5. Thermodynamic integrals for relaxation/coupling coefficients See A. Muronga (2007) II

7. 14-Fields theory of non-equilibrium fluid dynamics The conservation of net charge and of energy- momentum and the balance of fluxes 2 nd order entropy 4-current

8. 2 nd order relaxation/coupling coefficients

9. Entropy production and transport coefficients

10. Relaxation equations for dissipative fluxes Relaxation equations for the dissipative fluxes Transport and relaxation times/lengths

11. Make the equations tractable Macroscopic dynamics where See A. Muronga (2007) I

12. Make equations attractable Microscopic dynamics where

13. Physical problems: People’s ideas Simple scaling solution: A. Muronga (2001/2002/2004)

14. Ideal fluid vs non-ideal fluid Energy equation EoS and Transport coefficients Temperature evolution

15. Time evolution of thermodynamic quantities A. Muronga (2002/2004)

16. Transport coefficients and relaxation times A. Muronga (2004)

17. Physical problems Boost invariance + symmetric transverse: A. Muronga + D. H. Rischke (2004)

18. Physical problems: People’s realizations (2+1) viscous hydro: Formulations U. Heinz, H. Song and A.K. Chaudhuri (2006); A. Muronga (2007) Applications A.K. Chaudhuri (2007) P. Romatchke and U. Romatschke (2007); H. Song and U. Heinz (2008); K. Dusling and D. Teaney (2008); P. Huovinen and D. Molnar (2008); See the talks by P. Huovinen, P. Romatschke, H. Song and K. Dusling.

19. Physical problems: People’s realizations (2+1) viscous hydro: Formulations U. Heinz, H. Song and A.K. Chaudhuri (2006); A. Muronga (2007) Applications A.K. Chaudhuri (2007) P. Romatchke and U. Romatschke (2007); H. Song and U. Heinz (2008); K. Dusling and D. Teaney (2008); P. Huovinen and D. Molnar (2008); See the talks by P. Huovinen, P. Romatschke, H. Song and K. Dusling.

20. Viscous hydro vs transport models Bin Zhang et. al.

21. Viscous hydro vs transport models

22. P. Huovinen and D. Molnar

23. Viscous hydro vs transport models Slide from J.Y. Ollitrault’s talk

24. Viscous hydro vs transport models Extracting transport coefficients from transport

25. Viscous hydro vs transport models Extracting transport coefficients from transport A. El, A. Muronga (2007)