Titolo della tesi: Renormalization group approach to active biological systems:the swarming phase
This thesis aims to describe the critical dynamical properties observed in wild swarms of midges. From the experimental point of view, it is possible to measure velocity-velocity correlation functions, and the experimental evidence is that swarms of midges are strongly correlated, meaning that the correlation length of the system scales with its size. This fact hints that this particular biological system lives at the edge between order and disorder, in the same way as a Heisenberg model near the critical point is strongly correlated. Moreover, the relaxation time and the correlation length of swarms of midges are linked by the relation $\tau \sim \xi ^z$, with $z_{\mathrm{exp}} \simeq 1.2$. This phenomenon, which is that the characteristic relaxation time grows with the correlation length, is known as critical slowing down, and it is typical of critical systems. Furthermore, this system satisfies a dynamical scaling hypothesis, corroborating the idea that swarms of insects are posed near a critical phase transition. The most basic model of collective behavior for self-propelled particles, the Vicsek model, reveals to be inadequate to describe the behavior of insect swarms.
A more promising model that can capture the essential features observed in experiments is the Inertial Spin Model. Here I study the Inertial Spin Model in its critical, near ordering, phase; I focus on the coarse-grained equations of the Inertial Spin model, which consist of a field theory for three fluctuating fields: velocity, spin and density. The coupling between density and velocity fields renders the problem cumbersome, for this reason, we work using the constant density approximation. Even in this simpler regime, studying the inertial spin model with the renormalization group results to be quite complicated, involving the computation of more than sixty Feynman diagrams. The dynamical critical exponent, obtained by studying the Inertial Spin Model, is compatible with the experimental one.