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Far-from-equilibrium isotropisation , quasi-normal modes and radial flow

Work with Michał Heller, David Mateos, Michał Spalinski, Diego Trancanelli and Miquel Triana References: 1202.0981 (PRL 108) and 1210.xxxx. Far-from-equilibrium isotropisation , quasi-normal modes and radial flow. Wilke van der Schee Supervisors: Gleb Arutyunov, Thomas Peitzmann,

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Far-from-equilibrium isotropisation , quasi-normal modes and radial flow

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  1. Work with Michał Heller, David Mateos, Michał Spalinski, Diego Trancanelli and Miquel Triana References: 1202.0981 (PRL 108) and 1210.xxxx Far-from-equilibrium isotropisation, quasi-normal modes and radial flow Wilke van der Schee Supervisors: Gleb Arutyunov, Thomas Peitzmann, Koenraad Schalm and Raimond Snellings Comparing numerical evolution with linearisation Workshop Holographic Thermalization, Leiden October 11, 2012

  2. Outline • Simple set-up for anisotropy • Quasi-normal modes and linearised evolution • Radial flow (new results, pictures only)

  3. Holographic context • Simplest set-up: • Pure gravity in AdS5 • Background field theory is flat • Translational- and SO(2)-invariant field theory • We keep anisotropy: • Caveat: energy density is constant so final state is known P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma (2008)

  4. The geometry • Symmetry allows metric to be: • A, B, Sare functions of r and t • B measures anisotropy • Einstein’s equations simplify • Null coordinates • Attractive nature of horizon • Key differences with Chesler, Yaffe (2008) are • Flat boundary • Initial non-vacuum state

  5. The close-limit approximation • Early work of BH mergers in flat space • Suggests perturbations of an horizon are always small •  Linearise evolution around final state (planar-AdS-Schw): • Evolution determined by single LDE: R. H. Price and J. Pullin, Colliding black holes: The Close limit (1994)

  6. Quasi-normal mode expansion • Expansion: • Solution possible for discrete • Imaginary part always positive G.T. Horowitz and V.E. Hubeny, Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium(1999) J. Friess, S. Gubser, G. Michalogiorgakis, and S. Pufu, Expanding plasmas and quasinormal modes of anti-de Sitter black holes (2006)

  7. First results (Full/Linearized/QNM)

  8. Bouncing off the boundary

  9. IR, normal, UV

  10. Statistics of 2000 profiles

  11. Recent additions • Same linearised calculations with a boost-invariant direction • Subtlety: final state is not known initially • Add-on: non-homogeneous and includes hydrodynamics • Works well  • Second and third order corrections • The expansion seems to converge • Works quite well 

  12. Radial flow • Calculation incorporating longitudinal and radial expansion • Numerical scheme very similar to colliding shock-waves: • Assume boost-invariance on collision axis • Assume rotational symmetry (central collision) •  2+1D nested Einstein equations in AdS P.M. Chesler and L.G. Yaffe, Holography and colliding gravitational shock waves in asymptotically AdS5 spacetime (2010)

  13. Radial flow – initial conditions • Two scales: T and size nucleus • Energy density is from Glauber model (~Gaussian) • No momentum flow (start at t ~ 0.05fm/c) • Scale solution such that • Metric functions ~ vacuum AdS (not a solution with energy!) H. Niemi, G.S. Denicol, P. Huovinen, E. Molnár and D.H. Rischke, Influence of the shear viscosity of the quark-gluon plasma on elliptic flow (2011)

  14. Radial flow – results

  15. Radial flow - acceleration • Velocity increases rapidly: • Acceleration is roughly with R size nucleus • Small nucleus reaches maximum quickly

  16. Radial flow – energy profile • Energy spreads out:

  17. Radial flow - hydrodynamics • Thermalisation is quick, but viscosity contributes

  18. Radial flow - discussion • Radial velocity at thermalisation was basically unknown • Initial condition is slightly ad-hoc, need more physics? • We get reasonable pressures • Velocity increases consistently in other runs • Results are intuitive • Input welcome 

  19. Conclusion • Studied (fast!) isotropisation for over 2000 states • UV anisotropy can be large, but thermalises fast (though no bound) • Linearised approximation works unexpectedly well • Works even better for realistic and UV profiles • Numerical scheme provides excellent basis • Radial flow, fluctuations, elliptic flow • What happens universally? What is the initial state?

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