Effect of Physically Realistic Potential Energy Form on Spatial Pattern Complexity in a Collective Motion Model

Collective motion models most often use self-propelled particles, which are known to produce organized spatial patterns via their collective interactions. However, there is less work considering the possible organized spatial patterns achievable by non-self-propelled particles (nondriven), i.e., those obeying energy and momentum conservation. Moreover, it is not known how the potential energy interaction between the particles affects the complexity of the patterns. To address this, in this paper, a collective motion model with a pairwise potential energy function that conserved the total energy and momentum of the particles was implemented. The potential energy function was derived by generalizing the Lennard–Jones potential to reduce to gravity-like and billiard-ball-like potentials at the extremes of its parameter range. The particle model was simulated under a number of parameterizations of this generalized potential, and the average complexity of the spatial pattern produced by each was computed. Complexity was measured by tracking the information needed to describe the particle system at different scales (the complexity profile). It was found that the spatial patterns of the particles were the most complex around a specific ratio in the parameters. This parameter ratio described a characteristic shape of the potential energy function that is capable of producing complex spatial patterns. It is suggested that the characteristic shape of the potential energy produces complex behavior by balancing the likelihood for particles to bond. Furthermore, these results demonstrate that complex spatial patterns are possible even in an isolated system.