1. Inhomogeneous thermalization and strongly coupled thermal flow via holography
Daniel Fernández
Max Planck Institute for Physics in Munich
Iberian Strings Madrid IFT

2. Time-dependent Holography
Quark-gluon plasma thermalization [Chesler, Yaffe, Heller, Romatschke, Mateos, vd Schee, Bantilan]
Turbulence in Gravity [Lehner, Green, Yang, Zimmerman, Chesler, Adams, Liu]
Driven superconductors [Rangamani, Rozali, Wong]
Quantum quenches [Balasubramanian, Buchel, Myers, van Niekerk, Das]
Revivals [Mas, Lopez, Serantes, da Silva] 1 2
Collisions modeled as shockwaves
Thermal flow modeled as dynamical horizon
1/14

3. Part 1: Heavy Ion Collisions

4. Hydrodynamics approach
Conservation Law:
Slow evolution of expectation values.
Assumption: local quasi-equilibrium.
Gradient expansion:
Include viscosity term:
1st and higher order gradient terms
In HICs, the kind of physics that dominates is viscous fluid dynamics.
Connection to holography:
Condition to work: small gradients.
But good description of HIC data!
[Luzum, Romatschke ‘09]
2/14

5. Evolution of a Heavy Ion Collision
(Picture not to scale)
Out of equilibrium
Non-hydro degrees of freedom are dominant.
Gravitational model: Shock wave collisions.
Hydrodynamization
Transition to the hydrodynamics approximation.
Thermalization
Isotropy of the diagonal components of Tmn in local rest frame.
Hadronization
Kinetic theory is applicable.
3/14

6. Gravitational formulation
Ansatz:
Boundary EOM:
Generalize
Simplification from 3+1 to 2+1 by:
Alternative: Use polar coordinates.
Assuming boost invariance and breaking rotational symmetry,
4/14

7. Initial Condition 5/14 Gaussian shocks
Two separated shocks with finite thickness and energy density, moving toward each other:
Exact analytic solution, in Fefferman-Graham coordinates.
is the bdry. energy density, arbitrary if
Choice: t r
null
geodesics AH EH
Characteristic Formulation
[Winicour ’01, Chesler, Yaffe ‘08]
Ingoing Eddington-Finkelstein coordinates.
Foliation of null hypersurfaces.
IR cutoff → require fixed-r Apparent Horizon condition.
(Generalize for perturbations)
5/14

8. Stress-Energy Tensor results
Energy density:
The inhomogeneities are evolved on top of the dynamical background:
6/14

9. Comparing with hydrodynamics
Compare with hydro results
Mid-rapidity
Dramatic rise in the pressures.
Very anisotropic and out-of-eq.
Hydro holds almost from the beginning.
[Chesler, Yaffe ‘10]
Isotropization time and
hydrodynamization time are completely different.
[Fernández ’15]
7/14

10. Outlook: Chiral Magnetic Effect
Strong magnetic fields are produced by the charged “spectators” ( )
Anomalous electric currents → Observable effects in hadron production.
In a Heavy Ion Collision:
In relativistic chiral plasma, the anomaly is present:
→ Anomalous electric current:
Holomodel ingredients
Massive → Emulate dynamical gluons for
Scalar field → Recover gauge invariance.
CS term → Non-dynamical part of anomaly.
[Rebhan, Schmitt, Stricker '09]
[Gürsoy, Kharzeev, Rajagopal ’14]
[Jiménez-Alba, Landsteiner, Melgar ‘15]
Stückelberg-Chern-Simons theory:
…evolution of axial charge?
8/14

11. Part 2: Thermal Flow

12. Universal regime of thermal transport
[Bernard, Doyon ’12]
Two exact copies initially at equilibrium,
independently thermalized.
Thermal quench in 1+1
Conservation eqs & tracelessness:
NOPE
Expectation for CFT:
shock waves emanating from interface,
converge to non-equilibrium Steady State.
9/14

13. Thermal transport in d>1
[Bhaseen, Doyon, Lucas, Schalm ’13]
Generalization to any d
Assume ctant. homogeneous heat flow as well:
Effective dimension reduction to 1+1.
Linear response regime:
→ Hydro eqs. explicitly solvable.
Long time limit
Final steady state characterized by
Generic d prediction
where
10/14

14. Hydrodynamics approach
Heat transport dominated by diffusion instead of flow?
Shockwave velocities: I
III II
Hydrodynamical evolution of 3 regions
Match solutions ↔ Asymptotics of the central region
with
Two configurations:
- Thermodynamic branch - “second branch”
[Bhaseen, Doyon, Lucas, Schalm ’13]
[Chang, Karch, Yarom ’13]
11/14

15. Holographic approach 12/14 [Amado, Yarom ’15]
Energy transport: Lorentz-boosted thermal distribution
→ Expectation: Boosted black brane
Gravitational problem:
Full out-of-equilibrium evolution of Einstein’s equations. Ansatz:
Energy density
Energy flux
12/14

16. Information Flow
How does information get exchanged between
the systems which are isolated at t=0?
Entanglement entropy
→ Holographic measure:
Regularization: Substract result at T=0: y x z
13/14

17. Entanglement EntropiesAc B A
Mutual Information
What do we learn about A by looking at B?
Shockwaves transport information.
M.I. grows as shocks pass through the region.
[Erdmenger, Fernández, Flory, Megías, Straub ’15]
14/14

18. Thank you
for your attention