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

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, S are 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
Opposed to Chesler Yaffe approach

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
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)
Note the QNM-expansion, only 10 complex numbers! Note: initial agreement very unexpected.

8. Bouncing off the boundary

9. IR, normal, UV

10. Statistics of 2000 profiles
NOTE: thermalization defined as hydro works
Thermalization time is (usually!) very fast:
Linear approximation always accurate within “20%”

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
Pressure in transverse plane is not the same
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?