Study of the critical dynamics of active matter models to explain anomalous relaxation in natural swarms

A major challenge of modern physics and mathematics is the theoretical modeling of living systems. Recently, methods of condensed matter physics have been successfully applied to biological systems, providing an accurate description of the statistical laws governing active matter. Systems of interest in active matter are made of a large number of entities interacting locally and leading to large-scale collective behavior. Manifestations of this appear at multiple scales of living systems, from cell migration to swarming of insects and flocking of birds. Interestingly, though these systems are microscopically very different, they exhibit common macroscopic features. The study of active matter through the lens of statistical physics aims at developing a unified description of these behaviors. A fruitful approach uses the theory of critical phenomena, developed in condensed matter physics to study phase transitions in inanimate systems. This theory has allowed a systematic description of equilibrium phenomena through the introduction of concepts like long-range correlations, scaling laws, and renormalization. A major result of this theory states that an equilibrium system close to a second-order phase transition exhibits a macroscopic behavior independent of the microscopic details. This notion of universality has inspired the use of the same methods to study non-equilibrium collective behaviors in biology. Hydrodynamics, active matter models, and non-equilibrium statistical physics are paramount to the study of self-organization in biological systems. Several questions are still open. To what extent can the search for universality be applied to biological collective behavior? How should equilibrium theories be transformed in order to study non-equilibrium living systems? Only a combination of theory and experiments can provide answers to these problems. In this thesis I try to tackle these questions, focusing on the collective behavior of natural swarms. Recent field experiments unveiled long-range correlations in swarms of midges in absence of collective motion [1], thus suggesting that the system is disordered but with a near-critical phenomenology [2]. Moreover, experiments confirmed the emergence of dynamical scaling laws in the spatio-temporal correlation functions of different swarms [3], providing additional evidence about the universality of their behavior. Dynamical scaling stems from the study of time-dependent critical phenomena of equilibrium systems [4]. It affirms that, close to criticality, the correlation length ξ is the only relevant length scale of the system, ruling also its dynamical relaxation [4]. This phenomenon is known as critical slowing down, and it is expressed by the power law τ ∼ ξ^z, where τ is the characteristic time scale and z is called dynamical critical exponent [5]. When this property holds, the exponent determines the dynamical universality class of the system, containing all the information on its macroscopic dynamical behavior. Natural swarms obey this law with a dynamical critical exponent z ≃ 1.2, a value not found in any other statistical model [3]. Moreover, the relaxation reflects an anomalous underdamped decay of velocity correlation functions, which is not compatible with standard models. This thesis proposes to combine classic and novel active matter models with standard statistical field theory tools, with the purpose of rationalizing this experimental finding. The validity of scaling laws suggests that a description in terms of out-of-equilibrium critical phenomena is legitimate, therefore our study will employ Renormalization Group (RG) techniques and numerical simulations of active matter models in a near-critical regime. The investigation focuses on two active matter models. The first is the Vicsek model that describes self-propelled dissipative dynamics in the velocities. An analytical calculation on the respective hydrodynamic incompressible field theory reveals that activity lowers the value of the dynamical critical exponent with respect to the equilibrium universality class through a mechanism of crossover. Numerical simulations confirm that this result is valid also for compressible systems. The second studied theory is the Inertial Spin Model, which formulates a second-order dynamics in the velocities able to qualitatively reproduce the swarms’ relaxation. A fixed-network RG calculation unveils the role of inertia in determining the critical dynamics of weakly damped systems: through a dynamical crossover they can exhibit a z = 1.5 critical exponent, a value lower than the dissipative case. The information acquired with these studies is combined in a theoretical model that includes self-propulsion and inertial dynamics, ingredients that both lower the value of the critical exponent finally arriving at consistency with experimental data.