Voronoi Neighborhood Model

The applet above demonstrates the new model, which is closely related to the Vicsek model. A similar view to the previous applet is seen when the "Particle" button is selected. The major difference is that the interaction is no longer based on a fixed interaction length. Following the research on starlings, the birds interact with their nearby neighbors in a density-independent way. In particular, we define the neighbors through the use of the cells in a Voronoi tesselation¹ , seen when the "Voronoi" button is selected.

Each cell contains one bird and all points of the space which are closer to that bird than any other. A bird's neighbors are simply defined as those birds in adjacent cells. The size and shape of these cells are not as important as this adjacency relation, so it is often simpler to work with a closely related diagram....
The Delaunay triangulation1 seen in the "Delaunay" view is a diagram of the connections between neighbors. Each vertex represents a bird, and the line segments connect birds who interact with one another (i.e. those that belong to neighboring Voronoi cells). The implementation of the model calculates the Delaunay triangulation directly without reference to the Voronoi tesselation, using some new algorithms.

Unlike the Vicsek model the birds don't collect into dense flocks in the ordered phase, and the model should be considered as a model of the interior of the flock. Note the sudden changes in direction in the ordered phase which might evoke the sudden fluctuations seen in the videos of starlings.

As in the Vicsek model, there is a phase transition as the noise is varied. We have calculated quantities called critical exponents which determine most large scale properties of the flock near the phase transition. This applet also implements a "repulsion" interaction between the birds, which causes them to keep a safe distance from each other. But the presence of the repulsion does not change the critical exponents - an example of universality.