Mean-Field Analysis of a Dynamical Phase Transition in a Cellular Automaton Model for Collective Motion

A cellular automaton model is presented for random walkers with biologically motivated interactions favoring local alignment and leading to collective motion or swarming behavior. The degree of alignment is controlled by a sensitivity parameter, and a dynamical phase transition exhibiting spontaneous breaking of rotational symmetry occurs at a critical parameter value. The model is analyzed using nonequilibrium mean-field theory: Dispersion relations for the critical modes are derived, and a phase diagram is constructed. Mean-field predictions for the two critical exponents describing the phase transition as a function of sensitivity and density are obtained analytically.