Binary Black Holes
A GENERIC Approach
Ref Bari
Advisor: Prof. Brendan Keith
Binary Black Holes
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
\textbf{>} \text{Preliminary Dissipation Model}
\textbf{>} \text{GENERIC Form of } M\nabla S
Binary Black Holes
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
\textbf{>} \text{Preliminary Dissipation Model}
\textbf{>} \text{GENERIC Form of } M\nabla S
Binary Black Holes
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
\textbf{>} \text{Preliminary Dissipation Model}
\textbf{>} \text{GENERIC Form of } M\nabla S
\textbf{>} \text{What orbits are possible around black holes?}
\textbf{>} \text{What are the constraints on } (p,e) \text{ and }(r,\phi) \text{ for each orbit?}
\textbf{>} \text{What is the mapping between } (p,e) \text{ and }(r,\phi) \text{?}
\textbf{>} \text{How can we verify that both descriptions are equivalent?}
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
\text{Transformation between } (p,e) \to (r,\phi)
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Stable Circular Orbit
\text{Transformation between } (p,e) \to (r,\phi)
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Elliptical Orbit
\text{Transformation between } (p,e) \to (r,\phi)
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Hyperbolic Orbit
\text{Transformation between } (p,e) \to (r,\phi)
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Unstable Circular Orbit
\text{Transformation between } (p,e) \to (r,\phi)
Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)
Plunge Orbit
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Circular
Elliptical
Hyperbolic
Plunge
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Elliptical
Hyperbolic
Plunge
Circle
\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Elliptical
Hyperbolic
Plunge
Circle
\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Elliptical
Hyperbolic
Plunge
Circle
\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Hyperbolic
Plunge
Circle
\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)
Ellipse
0.95<\tilde{E}^2<1
\tilde{J}^2>12M^2
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, \left(1-\frac{2M}{r}\right){p_r}, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,\dot{p_r},0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, p_{r,0}, 0, L)
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Circle
\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)
Ellipse
0.95<\tilde{E}^2<1
\tilde{J}^2>12M^2
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, \left(1-\frac{2M}{r}\right){p_r}, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,\dot{p_r},0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, p_{r,0}, 0, L)
Ellipse
Unbound
:(
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
Circle
\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)
Ellipse
0.95<\tilde{E}^2<1 \to 0\leq e < 1
\tilde{J}^2>12M^2 \to p>6+2e
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, \left(1-\frac{2M}{r}\right){p_r}, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,\dot{p_r},0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, p_{r,0}, 0, L)
Ellipse
Constraints:
0.95<\tilde{E}^2<1
\tilde{J}^2>12M^2
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Mission Statement
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
H(\vec{x})
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{Neural ODE}
x(t)
\textbf{Neural ODE}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{ODE Solver}
x(t)
\textbf{Neural Net}
+
\dot{x}=f(x)
x(t)
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{ODE Solver}
x(t)
\textbf{Neural Net}
+
\dot{x}=f_{NN}(\vec{x}(t,\eta))
x(t)
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{ODE Solver}
x(t)
\textbf{Neural Net}
+
\dot{x}=f_{NN}(\vec{x}(t,\eta))
x(t)
\text{min}_{\eta} \mathcal{L} = \sum_{i=1}^N |x(t)-\hat{x}(t)|^2
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x,\mathcal{F}(x))
x(t)
\textbf{Neural ODE}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x,\mathcal{F}(x))
x(t)
\textbf{Neural ODE}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\textbf{Discover Hamiltonian (in GENERIC Form) of EMRIs}
\textbf{from Gravitational Waves using Neural ODEs}
H
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\textbf{Discover Hamiltonian (in GENERIC Form) of EMRIs}
\textbf{from Gravitational Waves using Neural ODEs}
H=H_{\text{conservative}}+H_{\text{dissipative}}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\textbf{Discover Hamiltonian (in GENERIC Form) of EMRIs}
\textbf{from Gravitational Waves using Neural ODEs}
H=H_{\text{conservative}}+H_{\text{dissipative}}
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Introduction to GENERIC
G
E
N
E
R
I
C
Introduction to GENERIC
G
E
N
E
R
I
C
eneral
quation for
on
quilibrium
eversible
rreversible
oupling
Introduction to GENERIC
dU = dW + dQ
Introduction to GENERIC
dU = \underbrace{dW} +\underbrace{dQ}
"reversible"
"irreversible"
dU = \underbrace{dW} +\underbrace{dQ}
Introduction to GENERIC
dU = \underbrace{dW} +\underbrace{dQ}
"reversible"
"irreversible"
dU = -pdV+TdS
Introduction to GENERIC
dU = \underbrace{dW} +\underbrace{dQ}
"reversible"
"irreversible"
\dot{x} = \dot{x}_{\text{reversible}} + \dot{x}_{\text{irreversible}}
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\underbrace{\begin{pmatrix}
\dot{p}\\
\dot{q}
\end{pmatrix}}_{\dot{x}} =
\underbrace{\begin{pmatrix}
0 & -1\\
+1 & 0
\end{pmatrix}}_{L}
\underbrace{\begin{pmatrix}
\frac{\partial H}{\partial p}\\
\frac{\partial H}{\partial q}
\end{pmatrix}}_{\nabla E}
Conservative Dynamics
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\underbrace{\begin{pmatrix}
\dot{p}\\
\dot{q}
\end{pmatrix}}_{\dot{x}} =
\underbrace{\begin{pmatrix}
? & ?\\
? & ?
\end{pmatrix}}_{M}
\underbrace{\begin{pmatrix}
\frac{\partial S}{\partial p}\\
\frac{\partial S}{\partial q}
\end{pmatrix}}_{\nabla S}
Dissipative Dynamics
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\frac{dU}{dt} = 0
M \nabla E =0
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\frac{dU}{dt} = 0
M \nabla E =0
L \nabla S =0
\frac{dS}{dt} \geq 0
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\frac{dU}{dt} = 0
M \nabla E =0
L \nabla S =0
\frac{dS}{dt} \geq 0
L^T=-L
M^T=M, M\geq0
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
- Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Examples of GENERIC
Examples of GENERIC
x=\{q,p,S\}
State Variables
x=\{q,p,S\}
State Variables
Examples of GENERIC
x=\{q,p,S\}
State Variables
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
Examples of GENERIC
x=\{q,p,S\}
State Variables
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
S = S
Energy
Entropy
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
S = S
Energy
Entropy
Examples of GENERIC
x=\{q,p,S\}
State Variables
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
S = S
Energy
Entropy
\nabla E =(kq,p/m,T)
\nabla S = (0,0,1)
Entropy
\nabla E =(kq,p/m,T)
\nabla S = (0,0,1)
Examples of GENERIC
x=\{q,p,S\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
\nabla E =(kq,p/m,T)
\nabla S = (0,0,1)
L={\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
M=\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
\dot{q}=p
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
\dot{q}=p
\dot{p}=-q-\gamma p
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
\dot{S}=\gamma p^2
\dot{q}=p
\dot{p}=-q-\gamma p
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
S = S
\underbrace{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}_{\nabla E}+
\underbrace{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}_{\nabla S}
\underbrace{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}_{L}
\underbrace{\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}}_{M}
\dot{S}=\gamma p^2
\dot{q}=p
\dot{p}=-q-\gamma p
\dot{x} = L \nabla E + M \nabla S
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
L=\begin{pmatrix}\mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0\end{pmatrix}
L=
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
L=\frac{1}{m}\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ -1 & 0 & p_1 A_{\mathrm{c}} & -p_2 A_{\mathrm{c}} \\ 0 & -p_1 A_{\mathrm{c}} & 0 & 0 \\ 0 & p_2 A_{\mathrm{c}} & 0 & 0\end{array}\right)
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
L=\begin{pmatrix}\mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0\end{pmatrix}
L=
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
L=\frac{1}{m}\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ -1 & 0 & p_1 A_{\mathrm{c}} & -p_2 A_{\mathrm{c}} \\ 0 & -p_1 A_{\mathrm{c}} & 0 & 0 \\ 0 & p_2 A_{\mathrm{c}} & 0 & 0\end{array}\right)
Examples of GENERIC
M=\begin{pmatrix}\begin{array}{cccccc}\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \kappa \frac{\theta_b}{\theta_a} & -\kappa \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\kappa & \kappa \frac{\theta_a}{\theta_b}\end{array}\end{pmatrix}
M=\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
M=\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha & -\alpha \\ 0 & 0 & -\alpha & \alpha\end{array}\right)
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
L=\begin{pmatrix}\mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0\end{pmatrix}
L=
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
L=\frac{1}{m}\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ -1 & 0 & p_1 A_{\mathrm{c}} & -p_2 A_{\mathrm{c}} \\ 0 & -p_1 A_{\mathrm{c}} & 0 & 0 \\ 0 & p_2 A_{\mathrm{c}} & 0 & 0\end{array}\right)
M=\begin{pmatrix}\begin{array}{cccccc}\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \kappa \frac{\theta_b}{\theta_a} & -\kappa \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\kappa & \kappa \frac{\theta_a}{\theta_b}\end{array}\end{pmatrix}
M=\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
M=\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha & -\alpha \\ 0 & 0 & -\alpha & \alpha\end{array}\right)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
- Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
GENERIC for Black Holes (No Dissipation)
H = \frac{p^2}{2}
GENERIC for Black Holes (No Dissipation)
H = \frac{p^2}{2}=\text{Don't be lazy, Ref! Write it on the board.}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
\begin{pmatrix}
\dot{t} \\
\dot{r} \\
\dot{\theta} \\
\dot{\phi} \\
\dot{p_t} \\
\dot{p_r} \\
\dot{p_\theta} \\
\dot{p_\phi}
\end{pmatrix} =
\begin{pmatrix}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\
\end{pmatrix}
\begin{pmatrix}
{\partial H} / {\partial {t}} \\
\partial H / \partial {r} \\
\partial H / \partial {\theta} \\
\partial H / \partial {\phi} \\
\partial H / \partial {p_t} \\
\partial H / \partial {p_r} \\
\partial H / \partial {p_\theta} \\
\partial H / \partial {p_\phi}
\end{pmatrix}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
\begin{pmatrix}
\dot{t} \\
\dot{r} \\
\dot{\theta} \\
\dot{\phi} \\
\dot{p_t} \\
\dot{p_r} \\
\dot{p_\theta} \\
\dot{p_\phi}
\end{pmatrix} =
\begin{pmatrix}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\
\end{pmatrix}
\begin{pmatrix}
{\partial H} / {\partial {t}} \\
\partial H / \partial {r} \\
\partial H / \partial {\theta} \\
\partial H / \partial {\phi} \\
\partial H / \partial {p_t} \\
\partial H / \partial {p_r} \\
\partial H / \partial {p_\theta} \\
\partial H / \partial {p_\phi}
\end{pmatrix}
L
\nabla E
\dot{x}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
\dot{t} = \left(1-\frac{2M}{r}\right)^{-1}E
\dot{r} = \left(1-\frac{2M}{r}\right){p_r}
\dot{\theta} = r^{-2}{p_\theta}
\dot{\phi} = r^{-2}\sin^{-2}\theta{L}
\dot{p_t} =\dot{E}=0
\dot{p_r} = -\frac{1}{2}\left[\left(1-\frac{2M}{r}\right)^{-2}\left( \frac{2M}{r^2}\right) (p_t)^2 + \frac{2M}{r^2}(p_r)^2-2r^{-3}(p_{\theta})^2-2r^{-3}\sin^{-2}\theta (p_{\phi})^2\right]
\dot{p_\theta} =-\frac{1}{2}\left[ -2r^{-2}\sin^{-3}\theta\cos\theta\right](p_\phi)^2
\dot{p_\phi} =\dot{L}=0
\dot{x}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
GENERIC for Black Holes (Gravitational Waves)
M\nabla S = ?