Uniform density runs, sorted on initial position

I went back to the original 4 cases where I varied the initial conditions.  The initial density profile is uniform, and the particles lie exactly on a grid.  Each run has 10k particles, and I have repeated it 4 times to get error bars on the final density profile.  Here is a plot to remind you of what the different initial conditions were:

Runs 1 and 3 behaved very similarly in their evolution, forming a single object that winds up and collapses into the final virialized object.  Runs 2 (with the inflection point in the initial velocities) and 4 (which was initially expanding rather than contracting) were similar to one another in that they each formed 2 distinct objects that later merged to form the final product.  What you will see in this post is that the fate of the innermost and outermost particles depends greatly on the formation history, although the final density profile of the remnant does not. 

Here are the final phase space configurations (transformed so you can see the interior parts too) of the 4 runs.  Reddish brown particles started farthest out, and blue and green particles began life farthest in:

Here we see that for initial conditions 2 and 4, the ones in which there was a merger event, the innermost particles are driven to the far exterior of the object, and the innermost particles of the remnant started life in the outermost ring.   In contrast, for the monolithic cases, the innermost particles of the remnant come from a mixture of intermediate and inner rings, and just as we saw in the power law cases, the distribution of the inner particles is not isotropic in phase space.

Here are plots the initial vs. final positions and velocities of the particles:

These show that there is a very specific band of the initial particles that wind up in the middle of the remnant, in the merger cases.  Also, the "tail" that we were seeing int he velocity plots of the power law cases does not seem to appear here.

Finally, just to complete the story, here are plots of the final density profile of the 4 different runs.

 

Here it is evident that the monolithic cases scale as $r^{-1/2}$ over more decades than the cases with merger events, but even the cases where the inner particles come mostly from the outer layers scale quite convincingly (at least by eye) like $r^{-1/2}$.