numerical techniques for holography · numerical scheme • pseudospectral methods for discretizing...

Koushik Balasubramanian

YITP, Stony Brook UniversityNew Frontiers in Dynamical Gravity, 2014

Numerical Techniques for Holography

based on KB, Christopher P. Herzog arXiv:1312.4953

Saturday, March 29, 14

Motivation


Motivation

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


Motivation


• What can we learn about gravity?


Motivation



• What happens far from equilibrium?


Motivation



• What happens far from equilibrium?

• When is hydro not a good description? (Breakdown of gradient expansion)


Thanks to Computers


Why numerics?


Why numerics?

• I can’t think of any other way.


Why numerics?


• Numerical techniques are well-developed.


Why numerics?



• We can face nonlinear PDEs with courage.


Why numerics?




• Computers can stay awake longer than humans.


Why numerics?




• Computers can stay awake longer than humans.

• We can produce some nice screen-savers.


Outline


Outline

• I’ll start by showing some screen-savers


Outline


• Counterflow


Outline


• Counterflow

• Shockwave


Outline


• Counterflow

• Shockwave

• Lattice induced momentum-relaxation.


Outline


• Counterflow

• Shockwave


• linear regime-hydro & gravity


Outline


• Counterflow

• Shockwave


• linear regime-hydro & gravity

• nonlinear regime-hydro & gravity


Vorticity Profile as a function of time

Counterflow



Counterflow• 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)




hydrodynamics (Israel-Stewart type equations)• Flat metric.




hydrodynamics (Israel-Stewart type equations)• Flat metric.• Transport properties - ABJM

Plasma





Plasma • periodic bc





Plasma • periodic bc• Python/F2py (and Matlab)





Plasma • periodic bc• Python/F2py (and Matlab)Carrasco, Lehner, Myers (2012); Adams, Chesler, Liu (2013)






• Kolmogrov scaling; fractal-like structure of horizon






• Kolmogrov scaling; fractal-like structure of horizon• Forced/Driven Turbulence?






• Kolmogrov scaling; fractal-like structure of horizon• Forced/Driven Turbulence?• What happens when the

driving happens on short length scales?


Moving Ball

Temperature profile as a function of time


Moving Ball


gtt• Metric source


Moving Ball


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



Moving Ball



• How do we determine shock width and shock standoff distance?



Moving Ball



• How do we determine shock width and shock standoff distance?

• Breakdown of gradient expansion?



\nonumber

Shock Tube


Effects of Hall Viscosity


Losing Forward Momentum

Holographically

based on KB, Christopher P. Herzog arXiv:1312.4953


Momentum Relaxation• In most realistic systems, translation invariance

is broken by the presence of impurities.

• In the absence of impurities the DC conductivity is infinite

• Momentum dissipation leads to finite DC conductivity.

• Analogous to Stoke’s flow

�(!) ⇠ C0�(!)


Linear Response Theory

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

• Momentum relaxation time

�S = �

L

RdxO(x)eikx

1

⌧=

�2Lk2

✏+ p. lim!!0

=[GR(!, k)]

��L=0

!

• Near equilibrium, we can use linear response theory.


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 following definition:

gtt = (1 + �e

�m/tcos(kx)), O(x) ⌘ T

tt

@t

T̄tx

+1

⌧T̄tx

= 0


Relaxation Time

�• For small we can use linear response theory : (Kovtun, 2012)

• Hydrodynamic regime (small lattice wave number)

GROO =

k2(✏+ p)2

k2(c2s(✏+ p) + 4i⌘! � !2(✏+ p)+ ✏

(m = 0)

1

⌧=

2�2⌘k2

3✏0T0


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 =T0pgtt

, u = 0

1

⌧=

2(1�p1� �2)⌘k2

3✏0T0


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

to obtain relaxation time scale.

0 2 4 6 8 10

2.3k

� 8

� 7

� 6

� 5

� 4

� 3

� 2

� 1

0

log

D Im(G

(!))

2!

E Text

• Solve Linearized Einstein’s equations for small .

• Dotted line is the large wavenumber behavior (simple WKB approximation is not good enough).

• All dimensionful quantities are measured in units where

�

T = 34⇡

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


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 same as hydro.


Numerical Scheme• Bondi-Sachs coordinates

• Einstein’s equations have a nested structure.

• Gauge Choice: The location of apparent horizon is fixed.

• Error Monitoring: Check Bianchi constraint.

ds

2 = �✓e

2�V z � hABU

AU

B

z

2

◆dt

2�2e2�

z

2dt dz�2hABU

B

z

2dt dx

A+hAB

z

2dx

Adx

B.

��DAU

A � 2dt��

z=1= 0


Numerical Scheme

✓Dz +

4

zI◆�s = S� (z,↵s, ✓s,�s) (1)

(Dz)⇡As = S⇡A (z,↵s, ✓s,�s,�s) (2)

✓Dz +

2

zI◆UA

s = SUA

�z,↵s, ✓s,�s,�s,⇡

As

�(3)

(Dz) dt�s = Sdt�

�z,↵s, ✓s,�s,�s, U

As

�(4)

� = �0 �z3

2�3 + z4�s, ↵ = z2↵s, ✓ = z2✓s.

UA = �z@A(e2�0) + z2UA

s , ⇡A = � 2

z2@A�0 + ⇡A

s ,

V =1

z3(V0e

2�0 + z2Vs), dt↵ = ↵̇� z3

2V ↵0, dt✓ = ✓̇ � z3

2V ✓0

• Boundary Expansion

• Einstein’s Equations


Numerical Scheme

✓Dz +

1

zI◆dt↵s + C↵↵dt↵s + C↵✓dt✓s = Sdt↵ (. . . , dt�s) (1)

✓Dz +

1

zI◆dt✓s + C✓↵dt↵s + C✓✓dt✓s = Sdt✓ (. . . , dt�s) (2)

CHxxD

(2)x VH + CH

x D(1)x VH + CH

0 VH = SVH

�↵H , ✓H ,�H , UA

H ,�H , dt�H , dt↵H , dt✓H�

(1)

• Elliptic Equation (at Apparent horizon)

• Einstein’s equations


Boundary Data

@t↵s =1

z(dt↵)s +

1

2(z↵0

s + 2↵s)z3V (1)

@t✓s =1

z(dt✓)s +

1

2(z✓0s + 2✓s)z

3V (2)

@t� = S�

�VH , UA

H ,�H

�(3)

@tUA3 = SUA

3

�↵3, ✓3,�3, V3, U

A3 ,�0

�(4)

@tV3 = SV3

�↵3, ✓3,�3, V3, U

A3 ,�0

�(5)

• Boundary/Horizon Evolution Equations

• Boundary Conditions@z�s = 0 ,

⇡As = 3e�2�0UA

3 ,

@zUAs = UA

3 ,

dt↵s = 0 , dt✓s = 0 .


0 1 2 3 4 5 6 7 8

k2�2t

8⇡Ti

�8

�7

�6

�5

�4

�3

�2

�1

0

logD T

tx(t

)T

tx(0

)E

k = ⇡50

k = 4⇡50

k = 5⇡50

k = 6⇡50

Ref.

0 1 2 3 4 5 6

k2�2t

8⇡Ti

�6

�5

�4

�3

�2

�1

0

logD T

tx(t

)T

tx(0

)E

k = ⇡50

k = 4⇡50

k = 5⇡50

k = 6⇡50

Ref.

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.


�0.006

�0.004

�0.002

0.000

0.002

�T

tx

�0.003

�0.002

�0.001

0.000

0.001

�T

tt

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

k2�2t

8⇡Ti

�0.003

�0.002

�0.001

0.000

�T

xx

Difference in Stress Tensor

k = 4⇡/50

� = 0.2, v = 0.2Saturday, March 29, 14

0 1 2 3 4 5

k2�2t

8⇡Ti

�5

�4

�3

�2

�1

0

logD T

tx(t

)T

tx(0

)E

GravityHydroRef GRRef Hydro

Gravity vs Hydro

k =20⇡

50, � = 0.2, v = 0.2


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

logD T

tx(t

)T

tx(t

⇤)

E

� = 0.2

� = 0.3

� = 0.4

� = 0.5

Ref.

Large Behavior(Hydro)�

k = ⇡/50

1

⌧=

2(1�p1� �2)⌘k2

3✏0T0

f(�) = 2(1�p1� �2)


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

logD T

tx(t

)T

tx(t

⇤)

E

� = 0.2

� = 0.3

� = 0.4

Ref.

Large Behavior(Gravity)�

k = ⇡/50

1

⌧=

2(1�p1� �2)⌘k2

3✏0T0

f(�) = 2(1�p1� �2)


�40 �20 0 20 40�0.6�0.4�0.2

0.00.20.40.60.8

�T

⇥10�3

t = 2000

�40 �20 0 20 40�0.6�0.4�0.2

0.00.20.40.60.8

�T

⇥10�3

t = 4000

�40 �20 0 20 40

x

�0.6�0.4�0.2

0.00.20.40.60.8

�T

⇥10�3

t = 10000

Late time solution

T =T0pgtt

, u = 0

Late time solution agrees with the exac t ana l y t i c a l solution.

k = 4⇡/50


0 20 40 60 80 100

k2f (�)t

8⇡T0

�10

�8

�6

�4

�2

0

logD T

tx(t

)T

tx(0

)E

� = 0.2

� = 0.3

� = 0.4

Ref.

�̀ k

• No known analytical result!!!

• Relaxation seems slower for large lattice strength at large �, k


Summary

• 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 cases.


Thank You


numerical techniques for holography · numerical scheme • pseudospectral methods for discretizing...

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