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Numerical Techniques for Holography based on KB, Christopher P. Herzog arXiv:1312.4953 Koushik Balasubramanian YITP, Stony Brook University New Frontiers in Dynamical Gravity, 2014 Saturday, March 29, 14 Numerical Techniques for Holography

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SLIDE 1

Koushik Balasubramanian

YITP, Stony Brook University

New Frontiers in Dynamical Gravity, 2014

Numerical Techniques for Holography

based on KB, Christopher P. Herzog arXiv:1312.4953

Saturday, March 29, 14

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SLIDE 2

Koushik Balasubramanian

YITP, Stony Brook University

New Frontiers in Dynamical Gravity, 2014

Numerical Techniques for Holography

based on KB, Christopher P. Herzog arXiv:1312.4953

Saturday, March 29, 14

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SLIDE 3

Motivation

Saturday, March 29, 14

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SLIDE 4

Motivation

  • What can we learn about hydrodynamics

using gauge/gravity duality?

Saturday, March 29, 14

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SLIDE 5

Motivation

  • What can we learn about hydrodynamics

using gauge/gravity duality?

  • What can we learn about gravity?

Saturday, March 29, 14

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SLIDE 6

Motivation

  • What can we learn about hydrodynamics

using gauge/gravity duality?

  • What can we learn about gravity?
  • What happens far from equilibrium?

Saturday, March 29, 14

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SLIDE 7

Motivation

  • What can we learn about hydrodynamics

using gauge/gravity duality?

  • What can we learn about gravity?
  • What happens far from equilibrium?
  • When is hydro not a good description?

(Breakdown of gradient expansion)

Saturday, March 29, 14

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SLIDE 8

Thanks to Computers

Saturday, March 29, 14

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SLIDE 9

Why numerics?

Saturday, March 29, 14

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SLIDE 10

Why numerics?

  • I can’t think of any other way.

Saturday, March 29, 14

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SLIDE 11

Why numerics?

  • I can’t think of any other way.
  • Numerical techniques are well-developed.

Saturday, March 29, 14

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SLIDE 12

Why numerics?

  • I can’t think of any other way.
  • Numerical techniques are well-developed.
  • We can face nonlinear PDEs with courage.

Saturday, March 29, 14

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SLIDE 13

Why numerics?

  • I can’t think of any other way.
  • Numerical techniques are well-developed.
  • We can face nonlinear PDEs with courage.
  • Computers can stay awake longer than

humans.

Saturday, March 29, 14

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SLIDE 14

Why numerics?

  • I can’t think of any other way.
  • Numerical techniques are well-developed.
  • We can face nonlinear PDEs with courage.
  • Computers can stay awake longer than

humans.

  • We can produce some nice screen-savers.

Saturday, March 29, 14

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SLIDE 15

Outline

Saturday, March 29, 14

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SLIDE 16

Outline

  • I’ll start by showing some screen-savers

Saturday, March 29, 14

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SLIDE 17

Outline

  • I’ll start by showing some screen-savers
  • Counterflow

Saturday, March 29, 14

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SLIDE 18

Outline

  • I’ll start by showing some screen-savers
  • Counterflow
  • Shockwave

Saturday, March 29, 14

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SLIDE 19

Outline

  • I’ll start by showing some screen-savers
  • Counterflow
  • Shockwave
  • Lattice induced momentum-relaxation.

Saturday, March 29, 14

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SLIDE 20

Outline

  • I’ll start by showing some screen-savers
  • Counterflow
  • Shockwave
  • Lattice induced momentum-relaxation.
  • linear regime-hydro & gravity

Saturday, March 29, 14

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SLIDE 21

Outline

  • I’ll start by showing some screen-savers
  • Counterflow
  • Shockwave
  • Lattice induced momentum-relaxation.
  • linear regime-hydro & gravity
  • nonlinear regime-hydro & gravity

Saturday, March 29, 14

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SLIDE 22

Vorticity Profile as a function of time

Counterflow

Saturday, March 29, 14

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SLIDE 23

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

Saturday, March 29, 14

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SLIDE 24

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.

Saturday, March 29, 14

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SLIDE 25

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.
  • Transport properties - ABJM

Plasma

Saturday, March 29, 14

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SLIDE 26

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.
  • Transport properties - ABJM

Plasma

  • periodic bc

Saturday, March 29, 14

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SLIDE 27

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.
  • Transport properties - ABJM

Plasma

  • periodic bc
  • Python/F2py (and Matlab)

Saturday, March 29, 14

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SLIDE 28

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.
  • Transport properties - ABJM

Plasma

  • periodic bc
  • Python/F2py (and Matlab)

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

Saturday, March 29, 14

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SLIDE 29

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.
  • Transport properties - ABJM

Plasma

  • periodic bc
  • Python/F2py (and Matlab)

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

  • Kolmogrov scaling; fractal-like

structure of horizon

Saturday, March 29, 14

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SLIDE 30

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.
  • Transport properties - ABJM

Plasma

  • periodic bc
  • Python/F2py (and Matlab)

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

  • Kolmogrov scaling; fractal-like

structure of horizon

  • Forced/Driven Turbulence?

Saturday, March 29, 14

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SLIDE 31

Vorticity Profile as a function of time

Counterflow

  • 2+1 D Second order

hydrodynamics (Israel-Stewart type equations)

  • Flat metric.
  • Transport properties - ABJM

Plasma

  • periodic bc
  • Python/F2py (and Matlab)

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

  • Kolmogrov scaling; fractal-like

structure of horizon

  • Forced/Driven Turbulence?
  • What happens when the

driving happens on short length scales?

Saturday, March 29, 14

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SLIDE 32

Moving Ball

Temperature profile as a function of time

Saturday, March 29, 14

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SLIDE 33

Moving Ball

Temperature profile as a function of time

gtt

  • Metric source

Saturday, March 29, 14

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SLIDE 34

Moving Ball

Temperature profile as a function of time

  • Ideal hydro description is not

good (steepening of waves)

gtt

  • Metric source

Saturday, March 29, 14

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SLIDE 35

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

Saturday, March 29, 14

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SLIDE 36

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?

  • Breakdown of gradient

expansion?

gtt

  • Metric source

Saturday, March 29, 14

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SLIDE 37

\nonumber

Shock Tube

Saturday, March 29, 14

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SLIDE 38

Effects of Hall Viscosity

Saturday, March 29, 14

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SLIDE 39

Losing Forward Momentum Holographically

based on KB, Christopher P. Herzog arXiv:1312.4953

Saturday, March 29, 14

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SLIDE 40

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

σ(ω) ∼ C0δ(ω)

Saturday, March 29, 14

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SLIDE 41

Linear Response Theory

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

∆S = δL R dxO(x)eikx 1 ⌧ = 2

Lk2

✏ + p. lim

ω→0

=[GR(!, k)]

  • δL=0

!

  • Near equilibrium, we can use linear response

theory.

Saturday, March 29, 14

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SLIDE 42

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 + δe−m/t cos(kx)), O(x) ≡ T t

t

∂t ¯ Ttx + 1 τ ¯ Ttx = 0

Saturday, March 29, 14

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SLIDE 43

Relaxation Time

δ

  • For small we can use linear response

theory : (Kovtun, 2012)

  • Hydrodynamic regime (small lattice wave

number)

GR

OO =

k2(✏ + p)2 k2(c2

s(✏ + p) + 4i⌘! − !2(✏ + p) + ✏

(m = 0) 1 ⌧ = 22⌘k2 3✏0T0

Saturday, March 29, 14

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SLIDE 44

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 = T0 √gtt , u = 0 1 ⌧ = 2(1 − √ 1 − 2)⌘k2 3✏0T0

Saturday, March 29, 14

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SLIDE 45

Relaxation Time

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

to obtain relaxation time scale.

2 4 6 8 10

2.3k

− 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1

log D

Im(G(ω)) 2ω

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 δ

T =

3 4π

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

Saturday, March 29, 14

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SLIDE 46

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.

Saturday, March 29, 14

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SLIDE 47

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.

ds2 = − ✓ e2βV z − hABU AU B z2 ◆ dt2−2e2β z2 dt dz−2hABU B z2 dt dxA+hAB z2 dxA dxB .

  • −DAU A − 2dtχ
  • z=1 = 0

Saturday, March 29, 14

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SLIDE 48

Numerical Scheme

✓ Dz + 4 z I ◆ βs = Sβ (z, αs, θs, χs) (1) (Dz) πA

s

= SπA (z, αs, θs, χs, βs) (2) ✓ Dz + 2 z I ◆ U A

s

= SU A

  • z, αs, θs, χs, βs, πA

s

  • (3)

(Dz) dtχs = Sdtχ

  • z, αs, θs, χs, βs, U A

s

  • (4)

β = β0 − z3 2 χ3 + z4βs, α = z2αs, θ = z2θs. U A = −z∂A(e2β0) + z2U A

s ,

πA = − 2 z2 ∂Aβ0 + πA

s ,

V = 1 z3 (V0e2β0 + z2Vs), dtα = ˙ α − z3 2 V α0, dtθ = ˙ θ − z3 2 V θ0

  • Boundary Expansion
  • Einstein’s Equations

Saturday, March 29, 14

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SLIDE 49

Numerical Scheme

✓ Dz + 1 z I ◆ dtαs + Cααdtαs + Cαθdtθs = Sdtα (. . . , dtχs) ✓ Dz + 1 z I ◆ dtθs + Cθαdtαs + Cθθdtθs = Sdtθ (. . . , dtχs)

CH

xxD(2) x VH + CH x D(1) x VH + CH 0 VH = SVH

  • αH, θH, βH, U A

H, χH, dtχH, dtαH, dtθH

  • (1)
  • Elliptic Equation (at Apparent horizon)
  • Einstein’s equations

Saturday, March 29, 14

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SLIDE 50

Boundary Data

∂tαs = 1 z (dtα)s + 1 2(zα0

s + 2αs)z3V

(1) ∂tθs = 1 z (dtθ)s + 1 2(zθ0

s + 2θs)z3V

(2) ∂tχ = Sχ

  • VH, U A

H, χH

  • (3)

∂tU A

3

= SU A

3

  • α3, θ3, χ3, V3, U A

3 , β0

  • (4)

∂tV3 = SV3

  • α3, θ3, χ3, V3, U A

3 , β0

  • (5)
  • Boundary/Horizon Evolution Equations
  • Boundary Conditions

∂zβs = 0 , πA

s

= 3e−2β0U A

3 ,

∂zU A

s

= U A

3 ,

dtαs = 0 , dtθs = 0 .

Saturday, March 29, 14

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SLIDE 51

1 2 3 4 5 6 7 8

k2δ2t 8πTi

−8 −7 −6 −5 −4 −3 −2 −1

log D

Ttx(t) Ttx(0)

E

k = π

50

k = 4π

50

k = 5π

50

k = 6π

50

Ref.

1 2 3 4 5 6

k2δ2t 8πTi

−6 −5 −4 −3 −2 −1

log D

Ttx(t) Ttx(0)

E

k = π

50

k = 4π

50

k = 5π

50

k = 6π

50

Ref.

Numerical Simulations

δ = 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.

Saturday, March 29, 14

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SLIDE 52

−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

k2δ2t 8πTi

−0.003 −0.002 −0.001 0.000

∆Txx

Difference in Stress Tensor

k = 4π/50 δ = 0.2, v = 0.2

Saturday, March 29, 14

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SLIDE 53

1 2 3 4 5

k2δ2t 8πTi

−5 −4 −3 −2 −1

log D

Ttx(t) Ttx(0)

E

Gravity Hydro Ref GR Ref Hydro

Gravity vs Hydro

k = 20π 50 , δ = 0.2, v = 0.2

Saturday, March 29, 14

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SLIDE 54

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

k2f(δ)t 8πT0

−1.5 −1.0 −0.5 0.0

log D

Ttx(t) Ttx(t∗)

E

δ = 0.2 δ = 0.3 δ = 0.4 δ = 0.5 Ref.

Large Behavior(Hydro)

δ

k = π/50 1 ⌧ = 2(1 − √ 1 − 2)⌘k2 3✏0T0

f(δ) = 2(1 − √ 1 − δ2)

Saturday, March 29, 14

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SLIDE 55

0.0 0.2 0.4 0.6 0.8 1.0

k2f(δ)t 8πT0

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2

log D

Ttx(t) Ttx(t∗)

E

δ = 0.2 δ = 0.3 δ = 0.4 Ref.

Large Behavior(Gravity)

δ

k = π/50

1 ⌧ = 2(1 − √ 1 − 2)⌘k2 3✏0T0

f(δ) = 2(1 − √ 1 − δ2)

Saturday, March 29, 14

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SLIDE 56

−40 −20 20 40 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

∆T

×10−3 t = 2000 −40 −20 20 40 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

∆T

×10−3 t = 4000 −40 −20 20 40

x

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

∆T

×10−3 t = 10000

Late time solution

T = T0 √gtt , u = 0

Late time solution agrees with the exact analytical solution.

k = 4π/50

Saturday, March 29, 14

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SLIDE 57

20 40 60 80 100

k2f(δ)t 8πT0

−10 −8 −6 −4 −2

log D

Ttx(t) Ttx(0)

E

δ = 0.2 δ = 0.3 δ = 0.4 Ref.

`

δ

k

  • No known analytical

result!!!

  • Relaxation seems slower

for large lattice strength at large δ, k

Saturday, March 29, 14

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SLIDE 58

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.

Saturday, March 29, 14

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SLIDE 59

Thank You

Saturday, March 29, 14