Stochastic processes for collective dynamics of self-propelled particles

In the last years, collective motions have fascinated and attracted the attention of the scientific community. Indeed, this type of phenomena is ubiquitous in nature and often it is of rare beauty, just think about the dance of a large swarm of starlings during last hours of the day. It is also impressive how ordered patterns and coherent behaviour emerge from the interactions among individuals. In this Thesis, we propose a new microscopic model based on a system of interacting stochastic differential equations of Langevin-type for describing the collective dynamics of a self-propelled particle swarm. The concept of self-propelled particles is of crucial importance in modeling the behaviour of animal groups or of cell populations. Indeed, agents of biological systems are capable of persistent and active motion, and they can freely move by virtue of their internal metabolism. There is a rich literature about swarming behaviour, and we formulated our model by carefully choosing mathematical tools and assumptions. In many of available models the swarm dynamics is settled in the overdamped regime. Here we formulate a stochastic dynamical process by adopting an underdamped approach. In our model we combined three well known forces. First a Morse-type potential that accounts for social interactions within the swarm. A friction force that has the role of a restoration term for the microscopic dynamics. Finally, an alignment term that stimulates particles to assume the velocity of their neighbours via a weighted average procedure. This alignment term has been proven to have a central role in leading the swarm dynamics, as it has been shown in several papers in literature (see, for example, “Novel type of phase transition in a system of self-driven particles” by Vicsek and co-workers (Phys. Rev. Lett., 1995), or “On the mathematics of emergence” by Cucker & Smale (Japan. J. Math., 2007)). In particular, we first derive the equation of motion for the center of mass, and later we identify three different dynamical regimes for the swarm dynamics that correspond to the balancing between the alignment and friction effects. Moreover, numerical simulations have been performed for supporting the theoretical results. To conclude, we studied the role and the effect of the finiteness of the swarm ensemble. We found that such finiteness is responsible for the emerging of random fluctuations in the motion of the center of mass of the swarm, which is ruled by a stochastic motion as well: the statistical characterisation of this stochastic motion has been derived.