deletions | additions       diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex  index 21f285f..06e4c11 100644  --- a/Trajectory spawning.tex  +++ b/Trajectory spawning.tex 
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Shang gave his talk today about the trajectory sampling method that Eric Vanden Eijden came up with. One problem he had was that a uniform distribution on the central sphere leads to very poor coverage of space because the trajectories diverge very fast in some regions and bunch up in other regions. This section will be about the idea of adaptively spawning new trajectories in regions where the trajectories are diverging.  note: $\vec{b}(x)$ is a vector field which is aligned with the gradient at $x$. I will use $b$ instead of the gradient because I don't fully understand the relationship between them. $b$ has units length / time. The gradient has units energy / length. At this point I think they are just related by a constant, but part of me also thinks constant.   more notes: The way Eric has it set up $\vec{b}$ is a velocity field, and  the length volume elements throughout the volume flow along the velocity field toward the central reference surface. You can think about this as the flow  of $b$ should not change some material that is initially at constant density  throughout space the volume. Then you measure the total amount of material that flows through the surface. The amount of material that flows through a small surface element $\sigma(x)$ is $\int_0^{\infty} J(x,t) dt$. Since the velocity field is never zero outside the reference surface the time for all the material to flow is always finite. The maximum time is determined by the magnitude of the velocity. Thus changing $\vec{b}$ by a constant factor just rescales the time axis. So if  $\vec{b} \sim \vec{\nabla} U = -\vec{\nabla} U$ then time really does have units length$^2$  / | \nabla U |$. energy. If you double the energy, then you should half your step in time keep the steps in space constant.  First, just to orient ourselves, here are some equations from Shangs's talk and Eric's notes.  \begin{equation}