Mean particles

The results of the previous post on mean points shows that average sets of points tend to not be next to each other. If we introduce time into the system, then this can be seen as a repulsion of the points. So let's try this out. 

Looking at just points in a 1+1D world, where the spatial dimension is in a circle. We can evolve a set of points as follows:

1. Pick any set of starting points, these are also the current mean point set.
   
2. Create a large population of sets, each set is equal to this starting set. 
3. For each time step, add a random offset to each point set, and rotate it around the circle to be as close as possible to the mean set. Accumulate the deltas (relative angle of these rotated points to the mean points).
4. add the total delta onto the mean point set. (i.e. generate a new mean set)
5. go to 3.

 The results are shown with time going from bottom to top, with starting positions 1,1.1,7,8  (angles out of 10 ticks around the circle):

Notice that points appear to repel each other, and the two points closest to each other repel the quickest.

The size of the random offset applied at each time frame affects how fast the particles communicate and 'see each other', with no repelling until the random dispersion of each point starts to interfere with its neighbour.

Since we use a large population of random sets, it basically represents a uniform dispersion of the sets over time, with a chosen dispersion rate. The dispersion rate does not have any effect other than changing the time scale on the resulting trajectories. So the results are effectively parameter-free. They are simply the mean trajectories of a set of n points, given the circle's rotational symmetry, and a maximum communication speed. As such, the trajectories are otherwise entirely defined by the initial position of the points.

The behaviour isn't particularly interesting though, but we can make it more so by going from first order to second order dynamics. This is done by adding the change in the mean points on to all the candidates at each time step, reflecting the nature of particles to maintain their velocity. The result has a little more oscillation:

This is basically degeneracy pressure in action, just without complex numbers. However, there is an additional behaviour in this system: unlike degeneracy pressure in absolute coordinates, which converges (if it loses energy over time) to equidistant points, with circular symmetry included the points stabilise to the non-uniform dispersal seen in the previous post. At least it would if the communication distance was wide enough.

This makes it a system that has repulsion and a sort of attraction (as the converged set of points cluster on one side of the circle). This is not visible on the above graph due to the communication distance being lower, but it would be interesting to investigate this behaviour further.  

I expect the 2D version (on a sphere) to show even more unusual behaviour, it could well display orbiting behaviour around the lowest energy point locations (seen in the previous post).

Also note that the set of particles conserves momentum. At least, it appears to.

Particles:

The above reminds me somewhat of the path integral methods of quantum mechanics. The average over all paths being very similar to the integration over all paths in the path integral, one difference is that the points in quantum mechanics are complex (in the simplest case), and the other being that I am applying the circle rotational symmetry constantly. 

Next I would like to get closer to the path integral solution, to see what different the rotational symmetry part will make.