Sorting by inital positions

I have added a feature to Walter's code, namely each particle carries about a memory of where it started it's life.   I have done several runs, but all of the following plots refer to initial conditions that were a power law density profile with $\rho \propto x^{-1/2}$.   I have defined 5 logarithmic bins in the particle initial positions.  Below is the phase space diagram, with different colors representing particles that started in different initial positions, the reddish brown ones began life the bin farthest out, and the dark blue began life in the innermost bin.  Each plot is zoomed in from it's predecessor (you can click on the plots to view them larger):

Just for illustration, here are the same types of plots, but for the initial conditions (note the aspect ratio is no longer square):

To get a better idea of how the different colors are distributed around phase space, I did a transformation on these data to fit all the information onto a single plot.  Below I have defined a distance in phase space which is $r=\sqrt{x^2 + v^2}$.  Then instead of plotting $(r,\theta)$, I plot $(r^{0.1}, \theta)$ so that we can see the inner parts. 

Here are the initial conditions in this view:

These plots demonstrate that the outermost bins retain their identity, but the inner bins get mixed in the ordering of their positions.  All of the velocities increase, but the inner bins tend to have only high velocities and small positions.  The phase space orbits for the outer particles and inner particles seem qualitatively different.  This result is robust to the phase of the winding:  I checked that it holds true for the last several time dumps, and also for different runs of the same initial conditions (different runs have different numbers of particles).

Another useful view is to look at scatter plots of $|x_i|$ versus $|x_f|$ and $|v_i|$ versus $|v_f|$.  Here they are, the black lines show $x_f=x_i$ and the analogous for the velocity.

You can see that the positions on average are collapsing inward, as expected, but the innermost particles do not collapse inward as much as some of their outer counterparts, in fact the very innermost ones seem to be thrown out of the central region.  The velocities are all increasing, with the innermost particles seeing the largest increase in velocity.  This basic structure is set up very early in the run, here is the first time dump after the start of the run, at 20 ATUs.

I was puzzled about that tail in the outer particles, I thought maybe looking at these plots in a slightly different way might shed some light.  Here are $x_i$ versus $x_f$ and $v_i$ versus $v_f$ plots, where again I just zoom in rather than doing the polar trick.  The axes have the same ranges as in the beginning of this post (except v_initial, which I scale down in the same proportions):

So you can see the velocity tail in a different form, in the first of these, I am still thinking aobut why it's there.  The innermost bins are doing something qualitatively different than the outer three, in my opinion.

There are still many more plots to make, I will start checking some of the other power laws.  Also, if I want to do warm runs, I need to repeat the modification to Walter's code, adding one more field (and since it's the last available bit, that is it, can't add any more data to carry around without doing something hard).    I will try and make some more plots this weekend, have good one.