deletions | additions       diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex  index 9d1ce49..981b8ee 100644  --- a/Trajectory spawning.tex  +++ b/Trajectory spawning.tex 
...
First, just to orient ourselves, here are some equations from Shangs's talk.  \begin{equation}  V_{basin} = V_{ref} + S_{ref} \left< \frac{1}{\rho} \right>^{-1}  \label{eqn:basin_volume_from_traj}  \end{equation}  where $S_{ref}$ is the area of the spherical reference surface near the minimum.  \begin{equation} 
...
\begin{equation}  J(x, t+dt) = J(x,t) \left[ 1 + Tr \left( H(x(t) \right) dt \right]  \end{equation}  Where $H$ is the Hessian. I suppose that $J(x, t=0)$ is simply set equal to 1.  For a trajectory starting from $x(t=0)$ the a  small surface element of area $dS$ sweeps out a volume, $dS \int J(x(t)) dt$ (should check this). The extra term in the definition of $\rho$ above is for the projection onto the reference surface. This means that  $dS J(x(t),t))$ is the area of the surface element at time $t$, or $J(x(t),t)$ is the factor by which the surface element has increased or decreased by time $t$.  The idea was to spawn new trajectories when the density of trajectories becomes very low. This can be implemented by spawning new trajectories when alpha is larger than a factor $\alpha > 1$. At this point the local density of trajectories has decreased by a factor of $\alpha$.   At time $t$ into trajectory $i$, we can spawn $N$ new trajectories from a surface of size $dS_i J(x_i(t), t)$ that is perpendicular to the trajectory. If we want to keep the density constant we can choose $N \sim \alpha^{-1}$. The starting points for these trajectories are chosen somehow (randomly) from he surface. At this point trajectory $i$ dies and is replace by $N$ new ones. Those propagate forward and potentially spawn new trajectories.  This is describing a tree structure. Each trajectory has the potential to spawn children and the children can spawn new children. There is never any interaction between trajectories, everything is computed from local quantities, so the computation is still fairly trivially parallelizable.  How do we reconstruct the total  volume from all these spawning events? To do so, we use a local variant of eq. \ref{eqn:basin_volume_from_traj}.  \begin{equation}  \rho_i V_{DS}  = \left| \vec{n}(x_{i}(0)) \cdot \vec{\nabla} U(x_i(0)) \right| S_{ref} F{\{\rho\}}  \label{eqn:DS_volume_from_traj}  \end{equation} where $F() = \left< \frac{1}{\rho} \right>^{-1}