Flocking with discrete symmetry: The two-dimensional active Ising model

Abstract

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 two-dimensional 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.

Signatures of a macroscopic switching transition for a dynamic microtubule

“…4(b)) supporting yet again a discontinuous transition like scenario. Note that the extent of dip enhances with increasing T which is the expected standard finite system size behaviour4042. We have checked that similar characteristics extend to k ′/ k  = 0 limit–see Figure.…”

“…We now study the Binder cumulant (BC) defined as 1 − [〈 v 4 〉/3〈 v 2 〉 2 ], a well-known quantity that is used to distinguish between discontinuous and continuous transitions404142. BC is supposed to have a smooth crossover at a continuous transition.…”

Gaussian theory for spatially distributed self-propelled particles

“…In addition to the hysteresis effect, discontinuous transitions are well visible. Both the discontinuous transitions and the hysteresis effect are the results of spatio-temporal band structures found in the system [32,[71][72][73][74][75][76][77][78]. Figure 2 presents the spatio-temporal structures that are observed in steady state of the system.…”

“…To obtain the spatio-temporal structures one can either solve the Fokker-Planck Eq. (7) [78,83], or the continuum equations [47,61,75,76]. In this article we compare the solutions to different continuum equations which have been obtained from the GA and the truncation method by integrating the partial differential equation in subsection VI B.…”

Estimation of the critical behavior in an active colloidal system with Vicsek-like interactions

“…A different approach has recently been proposed 46 , where the critical point of an active Lennard-Jones system has been determined by fitting various power laws and assuming an exponential dependence on parameters. In other models, the critical point in the active case turned out to be at infinite density and could thus not be determined directly via simulation 47 . In this paper, we will discuss the determination of the critical point in a system of self-propelled particles using the subsystem-block-distribution analysis.…”