Renormalization group approach to active biological systems

The renormalization group is a key set of ideas and quantitative tools of statistical physics that allow for the calculation of quantities that encompass the collective behaviour of different kinds of systems. According to the renormalization group theory, different collective behaviours can be organised into universality classes, based only on general properties of the underlying system, such as symmetries and conservation laws. Extension of the predictive power of the renormalization group, and its most fruitful consequence, namely universality, to collective biological systems would greatly strengthen the effort to put biophysics on a firm basis. Living systems are very different from equilibrium systems statistical physics used to deal with. The ability to convert free energy from the environment into systematic movement, known as activity, drives living systems far from thermal equilibrium. Furthermore, the wide range of scales over which biological processes take place, from the chemical reactions of metabolism at the cell scale up to social interactions on inter-individual scales, makes the behaviour of living systems infinitely more complex. Given these premises, one may question whether the collective behaviour of a living system can solely be described by symmetries and conservation laws, challenging the idea of universality. Here, by focusing on the case of natural swarms of insects, I provide one of the first successful tests of universality in active biological systems, by calculating the dynamic critical exponent of swarms using the renormalization group. Swarms of midges in the field perform collective behaviour, as recent experiments highlighted. Despite the lack of global order, swarms exhibit strong scale-free connected correlations between midges' velocities. These findings, together with the observation of scaling laws, both static and dynamic, suggested that swarms behave as systems near a critical point, in which order is low but correlations are large and the capability to respond collectively to external perturbations is strong. By focusing only on a few general properties of swarming midges, I developed a field theory aiming to describe their collective behaviour. Although based on experimental evidence, this approach entirely relies on the symmetries of the system, mainly the rotational symmetry, and the associated conservation laws entailed by the Noether theorem. The novelty of the field theory here developed is the combined presence of activity and inertia. Activity is represented by the ability of individuals to self-propel and thus accounts for the fact that the order parameter is the local direction of motion. Inertial behaviour, on the other hand, stems from the presence of a mode-coupling interaction with the conserved generator of the rotational symmetry of the direction of motion of midges and accounts for the observation of non-exponential relaxation of the temporal velocity-velocity correlation function. I then perform a perturbative renormalization group analysis of this theory at first order in ε=4-d, showing that a new fixed point where both activity and inertia are relevant emerges. At this fixed point, the dynamic critical exponent is found to be z = 1.35, in very good agreement with experiments on natural swarms (z=1.37 ± 0.11). On the path to this result, particular care has been taken to the effects of density fluctuations, which I show must be small in all active systems exhibiting collective behaviour in the absence of global order. This agreement between theory and experiments suggests that statistical physics, particularly the renormalization group, can play a decisive role in describing collective behaviours in biological systems. Finally, within the present work, I focus also on the development of a field-theoretical framework for models of flocking with discrete symmetry. Intending to understand the universal features of active systems near the flocking transition, I provide a comprehensive hydrodynamic approach to these systems, investigating their behaviour at the onset of flocking.