The renormalization group is a key set of ideas and quantitative tools of statistical physics that allow for the calculation of universal quantities that encompass the behaviour of different kinds of collective systems. Extension of the predictive power of the renormalization group to collective biological systems would greatly strengthen the effort to put physical biology on a firm basis. Here we present a step in that direction by calculating the dynamical critical exponent z of natural swarms of insects using the renormalization group to order ϵ = 4 − d. We report the emergence of a novel fixed point, where both activity and inertia are relevant. In three dimensions, the critical exponent at the new fixed point is z = 1.35, in agreement with both experiments (1.37 ± 0.11) and numerical simulations (1.35 ± 0.04). Our results probe the power of the renormalization group for the quantitative description of collective behaviour, and suggest that universality may also play a decisive role in strongly correlated biological systems.

We study the critical behavior of a model with nondissipative couplings aimed at describing the collective behavior of natural swarms, using the dynamical renormalization group under a fixed-network approximation. At one loop, we find a crossover between an unstable fixed point, characterized by a dynamical critical exponent $z=d/2$, and a stable fixed point with $z=2$, a result we confirm through numerical simulations. The crossover is regulated by a length scale given by the ratio between the transport coefficient and the effective friction, so that in finite-size biological systems with low dissipation, dynamics is ruled by the unstable fixed point. In three dimensions this mechanism gives $z=3/2$, a value significantly closer to the experimental window, $1.0\le z\le 1.3$, than the value $z\approx 2$ numerically found in fully dissipative models, either at or off equilibrium. This result indicates that nondissipative dynamical couplings are necessary to develop a theory of natural swarms fully consistent with experiments.

Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. Under a fixed-network approximation, we perform a dynamical renormalization group calculation at one loop in the near-critical disordered region, and we show that the violation of momentum conservation generates a crossover between an unstable fixed point, characterized by a dynamic critical exponent $z=d/2$, and a 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 $\kappa =4/d$. The crossover is regulated by a conservation length scale ${\mathcal{R}}_0$, given by the ratio between the transport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond ${\mathcal{R}}_0$, the stable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and critical exponent, a behavior 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 slowdown 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 toward 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 symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.

We study the crossover between equilibrium and off-equilibrium dynamical universality classes in the Vicsek model near its ordering transition. Starting from the incompressible hydrodynamic theory of Chen et al. [Critical phenomenon of the order-disorder transition in incompressible active fluids, New J. Phys. 17, 042002 (2015)], we show that increasing the activity leads to a renormalization group (RG) crossover between the equilibrium ferromagnetic fixed point, with dynamical critical exponent $z=2$, and the off-equilibrium active fixed point, with $z=1.7$ (in $d=3$). We run simulations of the classic Vicsek model in the near-ordering regime and find that critical slowing down indeed changes with activity, displaying two exponents that are in remarkable agreement with the RG prediction. The equilibrium to off-equilibrium crossover is ruled by a characteristic length scale, beyond which active dynamics takes over. The larger the activity is, the smaller is such a length scale, suggesting the existence of a general trade-off between activity and the system's size in determining the dynamical universality class of active matter.

Flocking systems are known to be strongly out of equilibrium. Energy input occurs at the individual level to ensure self-propulsion, and the individual motility, in turn, contributes to ordering, enhancing information propagation and strengthening collective motion. However, even beyond ordering, a crucial feature of natural aggregations is response. How, then, do off-equilibrium features affect the response of the system? In this work, we consider a minimal model of flocking and investigate response behavior under directional perturbations. We show that equilibrium dynamical fluctuation-dissipation relations between response and correlations are violated, both at the local and at the global levels. The amount of violation peaks at the ordering transition, exactly as for the entropy production rate. Entropy is always produced locally and connected to the local fluctuation-dissipation violation via Harada-Sasa relationships. However, cooperative mechanisms close to the transition spread off-equilibrium effects to the whole system, producing an out-of-equilibrium response on the global scale. Our findings elucidate the role of activity and interactions in the cost repartition of collective behavior and explain what is observed in experiments on natural living groups.

A realistic description of active particles should include interactions with the medium, commonly a momentum-conserving simple fluid, in which they are suspended. In this work, we consider a multispecies suspension of self-diffusiophoretic Janus colloids interacting via chemical and hydrodynamic fields. Through a systematic coarse-graining of the microscopic dynamics, we calculate the multicomponent contribution to the hydrodynamic stress tensor of the incompressible Stokesian fluid in which the particles are immersed. For a single species, we find that the strength of the stress produced by the gradients of the number density field is determined by the particles' self-propulsion and chemotactic alignment, and can be tuned to be either contractile or extensile. For a multispecies system, we unveil how different forms of activity modify the stress tensor and how it can acquire nonreciprocal couplings due to phoretic effects.