deletions | additions       diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex  index 2f1c73d..cd0a51b 100644  --- a/Trajectory spawning.tex  +++ b/Trajectory spawning.tex 
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Note that $\vec{\nabla} \cdot \vec{b}$ is the divergence of $\vec{b}$ (written in Eric's notes as $Tr \vec{\nabla} \vec{b}$. This is related to the trace of the Hessian $H$ (they are simply proportional if $\vec{b}$ is proportional to the gradient). $J(x, t=0)$ is simply set equal to 1.   For a trajectory starting from $x$ a small surface element of area $d\sigma$ (normal to the trajectory) sweeps out a volume, $|\vec{b}(x)| d\sigma \int J(x, t) dt$ (should check this).  The extra term in the definition of $\rho$ above is for the projection onto the reference surface. If $|\vec{b}(x)| $ds = |\vec{b}(X(x,t))|  dt$ has units length (measuring and measures  the distance along the trajectory), trajectory,  then$dS J(x,t))$ is the area of the surface element at time $t$, or  \begin{equation}  \chi(x,t) = J(x,t) \frac{\left|\vec{b}(X(x,t))\right|}{\left|\vec{b}(x)\right|}  \end{equation}  is the factor by which the surface element has increased or decreased by time $t$. (the factors of $b$ seem a bit wrong here. Maybe it be $\chi(x,t) = J(x,t) |b(X(x,t)|/|b(x)|$) (I'm not totally sure about this)  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 $\chi$  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 starting from $x_i$, we can spawn $N$ new trajectories from a surface of size $d\sigma_i \chi(x_i, t)$ 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.