1. Relativistic Theory of Hydrodynamic Fluctuations Joe Kapusta University of Minnesota Nuclear Physics Seminar October 21, 2011 Collaborators: Berndt Muller and Misha Stephanov

2. WMAP picture WMAP 7 years Fluctuations in temperature of cosmic microwave background radiation

3. WMAP7 Most cosmological information comes from fluctuations. Can something similar be done with heavy ion collisions?

5. Elliptic Flow Nonzero impact parameter b > 0: –Spatial anisotropy in the initial state –Momentum anisotropy in the final state coordinate space  momentum space Fourier analysis => v 2 = elliptic flow x y z

7. Correlations as observed by ALICE The Ridge!Jet fragmentation?

8. ALICE Decomposition into 5 Fourier modes

9. Sources of Fluctuations in High Energy Nuclear Collisions Initial state fluctuations Hydrodynamic fluctuations due to finite particle number Energy and momentum deposition by jets traversing the medium Freeze-out fluctuations

10. Hydrodynamic Fluctuations Hydrodynamic fluctuations (noise) have been applied to a wide variety of physical, chemical, and biological systems. There are fluctuations in high energy heavy ion collisions due to the finite size and finite particle content of the system.

11. Molecular Dynamics Lubrication Equation Stochastic Lubrication Equation

12. Relativistic Dissipative Fluid Dynamics In the Landau-Lifshitz approach u is the velocity of energy transport.

13. Extend Landau’s theory of hydrodynamic fluctuations to the relativistic regime Stochastic source Similar expressions arise in the Eckart approach.

14. Procedure Solve equations of motion for arbitrary source function Perform averaging to obtain correlations/fluctuations Stochastic fluctuations need not be perturbative

15. Example: Boost Invariant Bjorken Model Fluctuation Solution response function noise calculable and similarly for fluctuations in the local flow velocity…

16. In the small viscosity limit This leads to delta functions and their derivatives. Cause: Space-time delta functions (white noise) in the original correlation functions. Cure: Use finite range correlations (colored noise).

17. Regular part of response function

18. Singular part of the response function smeared with Gaussians

19. Fluctuations in the local temperature and flow velocity fields give rise to a nontrivial 2-particle correlation function when the fluid elements freeze-out to free-streaming hadrons.

20. Magnitude and shape of correlation function depend on shear viscosity

21. 3D viscous - smooth initial conditions 3D viscous - lumpy initial conditions McGill group

22. Will hydrodynamic fluctuations have an impact on our ability to discern a critical point in the phase diagram (if one exists)?

23. quarks & gluons baryons & mesons What are the relevant degrees of freedom? What happens to confinement?

24. Expansion away from equilibrium states using Landau theory 0 along coexistence curve The relative probability to be at a density other than the equilibrium one is

25. Volume = 400 fm 3

26. =(n-n c )/n c

27. Look in a window of longitudinal and transverse rapidities to correlate position and momentum space. (CMB: spherical symmetry / High Energy Nuclear Collisions: cylindrical symmetry)

28. Many Accelerators Relativistic Heavy Ion Collider (RHIC) low energy runs - BNL Facility for Antiproton and Ion Research (FAIR) - Germany SPS Heavy Ion and Neutrino Experiment (SHINE) - CERN Nuclotron-based Ion Collider Facility (NICA) - Dubna

29. Summary Fluctuations are interesting and provide essential information on transport coefficients. We are learning and are having fun. There is plenty of work for both theorists and experimentalists! Supported by the Office Science, U.S. Department of Energy.