The Preliminary Results and Some Follow Up Ideas
Based on the analysis on the parameters, we can hypothesize certain conditions for which an orbital system would display synchrony or asynchrony. Combined, we can confidently say that the optimal parameters for synchrony in the system are a small value for the dissipation in energy. There might be an optimal parameter for the asymmetry in energy as well.
While the number of particles does appear to have an effect on the order parameter, we infer that the effect observed is more an effect of time than of number of particles present. When there are more particles in the system, the system synchronizes faster; there is less time in between collisions. Therefore, when the order parameter is averaged over time, the systems with more particles are more stable because they have experienced more collisions.
The results of the phase plot for the two particle system does show some promising results. There does appear to be evidence of some fixed points. This indicates synchrony because the particles are close to one another.
There should be further investigation into what kinds of real physical situations would produce the loss of energy in a collision happening in space.
The two particle system merits some real investigation. If the two particle system shows such a strong connection to the Kuramoto model, there is a high possibility that there could be a strong connection with the Kuramoto model for a many particle system. The next step would be to analyze the steady states for a many particle system. If there is a similar steady state solution, we can say with more confidence that synchrony played a role in the formation of astrophysical orbital systems.
Additionally, there could be modifications made to the simulation to make it more physically realistic. We made some simplifications throughout for the purpose of greater efficiency in computation. While we infer that these simplifications are reasonable, they were not tested to be such. Increased physical accuracy would support the theory further.
