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 ...

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Motivated by the collective behaviour of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. By performing a dynamical renormalization group calculation at one loop, we show that the violation of momentum conservation generates a crossover between a conservative yet IR-unstable fixed point, characterized by a dynamic critical exponent z = d/2, and a dissipative IR-stable fixed point with z = 2. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent κ = 4/d. Such crossover is regulated by a conservation length scale, R0, which is larger the smaller the dissipation: beyond R0 the dissipative fixed point dominates, while at shorter distances dynamics is ruled by the conservative fixed point and critical exponent, a behaviour which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents z = 3/2 and z = 2 in the critical slowing down of the system, confirming the renormalization group results. From the biophysical point of view, our calculation indicates that in finite-size biological groups mode-coupling terms in the equation of motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step towards reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetrybreaking terms violating number conservation, as quantum magnets or photon gases. arXiv:1905.01228v1 [cond-mat.stat-mech]