Discrete Laplacian thermostat for flocks and swarms: the fully conserved Inertial Spin Model

Abstract

Experiments on bird flocks and midge swarms reveal that these natural sys-tems are well described by an active theory in which conservation laws playa crucial role. By building a symplectic structure that couples the particles’velocities to the generator of their internal rotations (spin), the Inertial SpinModel (ISM) reinstates a second-order temporal dynamics that captures manyphenomenological traits of flocks and swarms. The reversible structure ofthe ISM predicts that the total spin is a constant of motion, the central con-servation law responsible for all the novel dynamical features of the model.However, fluctuations and dissipation introduced in the original model to makeit relax, violate the spin conservation law, so that the ISM aligns with thebiophysical phenomenology only within finite-size regimes, beyond which theoverdamped dynamics characteristic of the Vicsek model takes over. Here, we introduce a novel version of the ISM, in which the irreversible terms neededto relax the dynamics strictly respect the conservation of the spin. We performa numerical investigation of the fully conservative model, exploring both thefixed-network case, which belongs to the equilibrium class of Model G, andthe active case, characterized by self-propulsion of the agents and an out-of-equilibrium reshuffling of the underlying interaction network. Our simulationsnot only capture the correct spin wave phenomenology of the ordered phase,but they also yield dynamical critical exponents in the near-ordering phase thatagree very well with the theoretical predictions.