I found it very tricky to get these working. They are not all working yet. But here are a few that did, with 10k particles, 6 runs of each. The initial velocities are set so that there is a coherent collapse going on, the same as for the equally spaced particles, namely:
V=shot.V(n) = -V*sin(shot.X(n));
These runs take a long time somehow. Anyway, what happened is sort of interesting, I think. Here is a plot of the initial conditions, these are all power law densities $\rho = A x^a$. The slopes I picked to test out were a= -0.5, -0.8, -0.3, -2.0, +0.5 .
I have not plotted the -2.0 because I have not succeeded in getting that set of initial conditions to run at all yet, I have not diagnosed what the problem could be. So, the final density profiles are here:
So, my interpretation of this is as follows: -0.8 seems to stay put at its initial slope, (as obviously does -0.5), whereas -0.3 and +0.5 seem to drift to the -1/2 attractor. However it seems unlikely that there is anything special about the particular value -0.8. The big difference I see is that since I fix the initial spacial interval (which thus determines the normalization A), for a fixed number of particles the -0.8 initial configuration extends much farther into the center than the -0.3 and +0.5. But all the runs collapse inward so that at the final configuration, they all have lots of particles interior to $10^{-3}$. My guess is that if there are already particles in the interior, and they follow a different power law slope, then apparently they stay there, but if there are no particles in there to begin with, when they get there they do the $r^-1/2$ behavior. I am pretty sure that adding a bunch more particles to the -0.3 run would make the initial configuration extend farther inward, and then we could test this. I might make it -0.2 too, so it's easier to distinguish from -0.5 at the end.
Incidentally, I also did a -0.8 run where the initial conditions were expanding rather than contracting (but the initial density configuration looks identical to what you see in the first figure). This is what you get:
So the stability of the inner slope does not seem to care about what the initial velocities are doing, as long as there is some long range coherence (and not thermal behavior between the particles). Ignore the spike, my code was having trouble recentering one of the 6 runs. I might try running some warm initial conditions on these non-uniformly-spaced runs, to see if it still flattens itself out in the central region.
