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 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
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
Counterflow Vorticity Profile as a function of time Saturday, March 29, 14
Counterflow • 2+1 D Second order hydrodynamics (Israel-Stewart type equations) Vorticity Profile as a function of time Saturday, March 29, 14
Counterflow • 2+1 D Second order hydrodynamics (Israel-Stewart type equations) • Flat metric. Vorticity Profile as a function of time Saturday, March 29, 14
Counterflow • 2+1 D Second order hydrodynamics (Israel-Stewart type equations) • Flat metric. • Transport properties - ABJM Plasma Vorticity Profile as a function of time Saturday, March 29, 14
Counterflow • 2+1 D Second order hydrodynamics (Israel-Stewart type equations) • Flat metric. • Transport properties - ABJM Plasma • periodic bc Vorticity Profile as a function of time Saturday, March 29, 14
Counterflow • 2+1 D Second order hydrodynamics (Israel-Stewart type equations) • Flat metric. • Transport properties - ABJM Plasma • periodic bc • Python/F2py (and Matlab) Vorticity Profile as a function of time Saturday, March 29, 14
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) Vorticity Profile as a function of time Saturday, March 29, 14
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 Vorticity Profile as a function of time Saturday, March 29, 14
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? Vorticity Profile as a function of time Saturday, March 29, 14
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? Vorticity Profile as a function of time Saturday, March 29, 14
Moving Ball Temperature profile as a function of time Saturday, March 29, 14
Moving Ball • Metric source g tt Temperature profile as a function of time Saturday, March 29, 14
Moving Ball • Metric source g tt • Ideal hydro description is not good (steepening of waves) Temperature profile as a function of time Saturday, March 29, 14
Moving Ball • Metric source g tt • Ideal hydro description is not good (steepening of waves) • How do we determine shock width and shock standoff distance? Temperature profile as a function of time Saturday, March 29, 14
Moving Ball • Metric source g tt • Ideal hydro description is not good (steepening of waves) • How do we determine shock width and shock standoff distance? • Breakdown of gradient expansion? Temperature profile as a function of time Saturday, March 29, 14
Shock Tube \nonumber 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 σ ( ω ) ∼ C 0 δ ( ω ) • Momentum dissipation leads to finite DC conductivity. • Analogous to Stoke’s flow Saturday, March 29, 14
Linear Response Theory • Near equilibrium, we can use linear response theory. • Memory Function Formalism (c.f. Foster’s book) dx O ( x ) e ikx R ∆ S = δ L • Momentum relaxation time � � = [ G R ( ! , k )] � � ⌧ = � 2 L k 2 1 � δ L =0 ✏ + p. lim ! ω → 0 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. g tt = (1 + δ e − m/t cos( kx )) , O ( x ) ≡ T t t • Relaxation time scale can be computed using the following definition: T tx + 1 ∂ t ¯ ¯ T tx = 0 τ Saturday, March 29, 14
Relaxation Time • Hydrodynamic regime (small lattice wave number) • For small we can use linear response δ theory : (Kovtun, 2012) ( m = 0) k 2 ( ✏ + p ) 2 G R OO = s ( ✏ + p ) + 4 i ⌘! − ! 2 ( ✏ + p ) + ✏ k 2 ( c 2 ⌧ = 2 � 2 ⌘ k 2 1 3 ✏ 0 T 0 Saturday, March 29, 14
Relaxation Time • At late time, the flow relaxes to the following steady state solution: T 0 T = , u = 0 √ g tt • We can obtain an expression for the relaxation rate at late times using linear perturbation theory around this steady state solution √ 1 − � 2 ) ⌘ k 2 ⌧ = 2(1 − 1 3 ✏ 0 T 0 Saturday, March 29, 14
E D Relaxation Time • For large k , we can use gauge/gravity correspondence to obtain relaxation time scale. • Solve Linearized Einstein’s equations for small . δ 0 − 1 • Dotted line is the large Text − 2 wavenumber behavior Im ( G ( ω )) − 3 (simple WKB approximation 2 ω is not good enough). − 4 log − 5 • All dimensionful quantities − 6 are measured in units where − 7 − 8 3 T = 0 2 4 6 8 10 2 . 3 k 4 π The markers show values obtained by solving the full nonlinear equations. Saturday, March 29, 14
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