Flocking with discrete symmetry: the 2d Active Ising Model

We study in detail the active Ising model, a stochastic lattice gas where collective motion emerges from the spontaneous breaking of a discrete symmetry. On a 2d lattice, active particles undergo a diffusion biased in one of two possible directions (left and right) and align ferromagnetically their direction of motion, hence yielding a minimal flocking model with discrete rotational symmetry. We show that the transition to collective motion amounts in this model to a bona fide liquid-gas phase transition in the canonical ensemble. The phase diagram in the density/velocity parameter plane has a critical point at zero velocity which belongs to the Ising universality class. In the density/temperature "canonical" ensemble, the usual critical point of the equilibrium liquid-gas transition is sent to infinite density because the different symmetries between liquid and gas phases preclude a supercritical region. We build a continuum theory which reproduces qualitatively the behavior of the microscopic model. In particular we predict analytically the shapes of the phase diagrams in the vicinity of the critical points, the binodal and spinodal densities at coexistence, and the speeds and shapes of the phase-separated profiles.Comment: 20 pages, 25 figure

Sedimentation, trapping, and rectification of dilute bacteria

The run-and-tumble dynamics of bacteria, as exhibited by \textit{E. coli}, offers a simple experimental realization of non-Brownian, yet diffusive, particles. Here we present some analytic and numerical results for models of the ideal (low-density) limit in which the particles have no hydrodynamic or other interactions and hence undergo independent motions. We address three cases: sedimentation under gravity; confinement by a harmonic external potential; and rectification by a strip of `funnel gates' which we model by a zone in which tumble rate depends on swim direction. We compare our results with recent experimental and simulation literature and highlight similarities and differences with the diffusive motion of colloidal particles

From Phase to Micro-Phase Separation in Flocking Models: The Essential Role of Non-Equilibrium Fluctuations

We show that the flocking transition in the Vicsek model is best understood as a liquid-gas transition, rather than an order-disorder one. The full phase separation observed in flocking models with Z2 rotational symmetry is, however, replaced by a microphase separation leading to a smectic arrangement of traveling ordered bands. Remarkably, continuous deterministic descriptions do not account for this difference, which is only recovered at the fluctuating hydrodynamics level. Scalar and vectorial order parameters indeed produce different types of number fluctuations, which we show to be essential in selecting the inhomogeneous patterns. This highlights an unexpected role of fluctuations in the selection of flock shapes.Comment: 5 p., 5 fig.. Supplementary material: 7 movie