We study the critical behaviour of a model with non-dissipative couplings aimed at describing the collective behaviour of natural swarms, using the dynamical renormalization group. At one loop, we find a crossover between a conservative yet unstable fixed point, characterized by a dynamical critical exponent z = d/2, and a dissipative stable fixed point with z = 2, a result we confirm through numerical simulations. The crossover is regulated by a conservation length scale that is larger the smaller the effective friction, so that in finite-size biological systems with low dissipation, dynamics is ruled by the conservative fixed point. In three dimensions this mechanism gives z = 3/2, a value significantly closer to the experimental result z ≈ 1 than the value z ≈ 2 found in fully dissipative models, either at or off equilibrium. This result indicates that non-dissipative dynamical couplings are necessary to develop a theory of natural swarms fully consistent with experiments.Collective behaviour in biological groups emerges when local interactions give rise to correlations that significantly exceed the scale of the individuals. According to this somewhat restricted and yet compelling definition, collective behaviour seems the ideal hunting ground for statistical physics, and in particular for its most powerful theoretical tool, the Renormalization Group (RG) [1]. In statistical physics, a taxing but crucial requirement a successful theory of a collective phenomenon must meet is to reproduce the right form of the correlation functions and, most importantly, the correct values of the critical exponents [2], the calculation of which is the RG task. It is not surprising, then, that the RG can be applied to collective biological systems; the hydrodynamic theory of flocking of Toner and Tu is a pioneering step in this direction [3]. Here we adopt an RG approach to the collective dynamics of swarms.Experiments on large natural swarms in the field [4], show two things: i) dynamic correlations of the velocities have an inertial form incompatible with the classic exponential relaxation of overdamped systems, and ii) critical slowing down, namely the relation linking relaxation time and correlation length, τ ∼ ξ z , holds with a dynamical critical exponent z ≈ 1, very unusual for purely dissipative dynamics. Both facts urge for a theoretical explanation. The simplest, and yet most far-reaching model of collective behaviour in biological systems was introduced by Vicsek and co-workers [5]: it describes individuals as self-propelled particles moving at fixed speed and aligning their velocities to those of their neighbours through a so-called 'social force' [6,7]. Vicsek's model is analogous to a ferromagnetic system (velocities playing the role of local magnetizations) with fully dissipative Langevin dynamics; however, at variance with equilibrium ferromagnets, the interaction network in the Vic-sek model changes in time, due to the self-propulsion of the individuals. In its great flexibility, the Vicsek model describes both ...
We introduce a new ferromagnetic model capable of reproducing one of the most intriguing properties of collective behaviour in starling flocks, namely the fact that strong collective order coexists with scale-free correlations of the modulus of the microscopic degrees of freedom, that is the birds' speeds. The key idea of the new theory is that the single-particle potential needed to bound the modulus of the microscopic degrees of freedom around a finite value, is marginal, that is, it has zero curvature. We study the model by using mean-field approximation and Monte Carlo simulations in three dimensions, complemented by finite-size scaling analysis. While at the standard critical temperature, T c , the properties of the marginal model are exactly the same as a normal ferromagnet with continuous symmetry-breaking, our results show that a novel zero-temperature critical point emerges, so that in its deeply ordered phase the marginal model develops divergent susceptibility and correlation length of the modulus of the microscopic degrees of freedom, in complete analogy with experimental data on natural flocks of starlings.
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