Detecting phase transitions in collective motion using manifold's curvature

If a given behavior of a multi-agent system corresponds to restrict the phase variable to a invariant manifold, then we define a phase transition as change of physical characteristics such as speed, coordination and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the transition is physically represented on. Here, we propose a method of detecting phase transitions and splitting the manifold into transitions free sub-manifolds. Therein, we utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion to higher dimensions using the shape operator. Then we attest that, the same transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the transitions detection method using one particle simulation and three real world examples.