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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.
A numerical study of finite temperature features of thermodynamical observables is performed for the lattice 2d Ising model. Our results support the conjecture that the Finite Size Scaling analysis employed in the study of integrable perturbation of
We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the Replica Exchange Monte Carlo (Parallel Temperin
We study the dynamical behavior of a square lattice Ising model with exchange and dipolar interactions by means of Monte Carlo simulations. After a sudden quench to low temperatures we find that the system may undergo a coarsening process where strip
The effect of quenched (frozen) orientational disorder on the collective motion of active particles is analyzed. We find that, as with annealed disorder (Langevin noise), active polar systems are far more robust against quenched disorder than their e
We study in detail the hydrodynamic theories describing the transition to collective motion in polar active matter, exemplified by the Vicsek and active Ising models. Using a simple phenomenological theory, we show the existence of an infinity of pro