View
4
Download
0
Category
Preview:
Citation preview
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
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
Saturday, March 29, 14
Motivation
• What can we learn about hydrodynamics using gauge/gravity duality?
Saturday, March 29, 14
Motivation
• What can we learn about hydrodynamics using gauge/gravity duality?
• What can we learn about gravity?
Saturday, March 29, 14
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
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
Thanks to Computers
Saturday, March 29, 14
Why numerics?
Saturday, March 29, 14
Why numerics?
• I can’t think of any other way.
Saturday, March 29, 14
Why numerics?
• I can’t think of any other way.
• Numerical techniques are well-developed.
Saturday, March 29, 14
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
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
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
Outline
Saturday, March 29, 14
Outline
• I’ll start by showing some screen-savers
Saturday, March 29, 14
Outline
• I’ll start by showing some screen-savers
• Counterflow
Saturday, March 29, 14
Outline
• I’ll start by showing some screen-savers
• Counterflow
• Shockwave
Saturday, March 29, 14
Outline
• I’ll start by showing some screen-savers
• Counterflow
• Shockwave
• Lattice induced momentum-relaxation.
Saturday, March 29, 14
Outline
• I’ll start by showing some screen-savers
• Counterflow
• Shockwave
• Lattice induced momentum-relaxation.
• linear regime-hydro & gravity
Saturday, March 29, 14
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
Vorticity Profile as a function of time
Counterflow
Saturday, March 29, 14
Vorticity Profile as a function of time
Counterflow• 2+1 D Second order
hydrodynamics (Israel-Stewart type equations)
Saturday, March 29, 14
Vorticity Profile as a function of time
Counterflow• 2+1 D Second order
hydrodynamics (Israel-Stewart type equations)• Flat metric.
Saturday, March 29, 14
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
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
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
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
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
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
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
Moving Ball
Temperature profile as a function of time
Saturday, March 29, 14
Moving Ball
Temperature profile as a function of time
gtt• Metric source
Saturday, March 29, 14
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
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
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
\nonumber
Shock Tube
Saturday, March 29, 14
Effects of Hall Viscosity
Saturday, March 29, 14
Losing Forward Momentum
Holographically
based on KB, Christopher P. Herzog arXiv:1312.4953
Saturday, March 29, 14
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�(!)
Saturday, March 29, 14
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.
Saturday, March 29, 14
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
Saturday, March 29, 14
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
Saturday, March 29, 14
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
Saturday, March 29, 14
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.
Saturday, March 29, 14
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
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
Saturday, March 29, 14
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
Saturday, March 29, 14
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
Saturday, March 29, 14
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 .
Saturday, March 29, 14
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.
Saturday, March 29, 14
�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
Saturday, March 29, 14
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)
Saturday, March 29, 14
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)
Saturday, March 29, 14
�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
Saturday, March 29, 14
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
Saturday, March 29, 14
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
Thank You
Saturday, March 29, 14
Recommended