Numerical techniques

• What can we learn about hydrodynamics using gauge/gravity duality?... • What can we learn about hydrodynamics using gauge/gravity duality?. • What can we learn about hydrodynamics usin

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Koushik Balasubramanian

YITP, Stony Brook University

New Frontiers in Dynamical Gravity, 2014

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Saturday, March 29, 14

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• What can we learn about hydrodynamics

using gauge/gravity duality?

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• What can we learn about gravity?

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• What happens far from equilibrium?

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• What happens far from equilibrium?

• When is hydro not a good description?

(Breakdown of gradient expansion)

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Thanks to Computers

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Why numerics?

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Why numerics?

• I can’t think of any other way

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Why numerics?

• Numerical techniques are well-developed

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Why numerics?

• We can face nonlinear PDEs with courage

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Why numerics?

• Computers can stay awake longer than

humans

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Why numerics?

• Computers can stay awake longer than

humans

• We can produce some nice screen-savers

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• I’ll start by showing some screen-savers

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• Counterflow

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• Counterflow

• Shockwave

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• Counterflow

• Shockwave

• Lattice induced momentum-relaxation.

• linear regime-hydro & gravity

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• Counterflow

• Shockwave

• Lattice induced momentum-relaxation.

• linear regime-hydro & gravity

• nonlinear regime-hydro & gravity

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Counterflow

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Vorticity Profile as a function of time

Counterflow

• 2+1 D Second order hydrodynamics (Israel-Stewart type equations)

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• Flat metric.

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Counterflow

• Flat metric.

• Transport properties - ABJM Plasma

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• Flat metric.

• periodic bc

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Counterflow

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• Flat metric.

• periodic bc

• Python/F2py (and Matlab)

Carrasco, Lehner, Myers (2012); Adams, Chesler, Liu (2013)

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Counterflow

• Flat metric.

• periodic bc

• Kolmogrov scaling; fractal-like structure of horizon

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• Flat metric.

• periodic bc

• Forced/Driven Turbulence?

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Counterflow

• Flat metric.

• periodic bc

• Forced/Driven Turbulence?

driving happens on short length scales?

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Moving Ball

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Moving Ball

Temperature profile as a function of time

• Ideal hydro description is not good (steepening of waves)

• How do we determine shock width and shock standoff

distance?

gtt

• Metric source

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• Metric source

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Shock Tube

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Effects of Hall Viscosity

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Losing Forward

Momentum Holographically

based on KB, Christopher P Herzog arXiv:1312.4953

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Momentum Relaxation

• In most realistic systems, translation invariance

is broken by the presence of impurities

• In the absence of impurities the DC

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Linear Response Theory

• Memory Function Formalism (c.f Foster’s book)

• Momentum relaxation time

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Our Setup

• It is possible to break translation invariance by

introducing scalar perturbations, spatially dependent chemical potential (ionic lattice) or metric

perturbations

• In our setup we break translation invariance by

introducing metric perturbations.

• Relaxation time scale can be computed using the

gtt = (1 + e m/t cos(kx)), O(x) ⌘ Ttt

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⌧ =

2 2⌘k23✏0T0

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Relaxation Time

• At late time, the flow relaxes to the following

steady state solution:

• We can obtain an expression for the relaxation

rate at late times using linear perturbation theory around this steady state solution

T = pTg0

tt

, u = 0

p

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Relaxation Time

• For large k, we can use gauge/gravity correspondence

to obtain relaxation time scale

2.3k

8 7 6 5 4 3 2 1 0

equations for small .

wavenumber behavior (simple WKB approximation

is not good enough).

are measured in units where

T = 4⇡3

The markers show values obtained by solving the full nonlinear equations.

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Numerical Scheme

• Pseudospectral methods for discretizing spatial

derivatives.

• Runge-Kutta and Adams-Bashforth for time stepping.

• We have used the null characteristic formulation for

solving Einstein’s equations.

• In gravity, we need to solve 2 boundary evolution

equations, 2 bulk graviton evolution equations and one evolution equation at the apparent horizon.

• Number of propagating degrees of freedom is the

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Numerical Scheme

• Bondi-Sachs coordinates

• Einstein’s equations have a nested structure

• Gauge Choice: The location of apparent

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0 1 2 3 4 5 6 7 8

k2 2t 8⇡Ti

8 7 6 5 4 3 2 1 0

k2 2t 8⇡Ti

6 5 4 3 2 1 0

Numerical Simulations

= 0.2, v = 0.2

• Gravity and hydro agree initially Gradient corrections

become important at late times.

• Reference line shows the linear response theory result

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0.006 0.004 0.002 0.000 0.002

Ttx

0.003 0.002 0.001 0.000 0.001

Ttt

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

k2 2t 8⇡T

0.003 0.002 0.001 0.000

Txx

Difference in Stress Tensor

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0 1 2 3 4 5

k2 2t 8⇡Ti

5 4 3 2 1 0

Gravity vs Hydro

k = 20⇡

50 , = 0.2, v = 0.2

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

k2f ( )t 8⇡T0

1.5 1.0 0.5 0.0

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0.0 0.2 0.4 0.6 0.8 1.0

k2f ( )t 8⇡T0

1.0 0.8 0.6 0.4 0.2 0.0 0.2

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40 20 0 20 40 0.6

0.4 0.2 0.0 0.2 0.4 0.6 0.8

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0 20 40 60 80 100

k2f ( )t 8⇡T0

10 8 6 4 2 0

result!!!

for large lattice strength at

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• Linear response theory seems to work for

small values of lattice strength

• For large lattice strengths, we can obtain

analytical results for small lattice wave numbers

• We need to use Numerical GR for all other

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Thank You

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