Phase Transitions in Self-Driven Many-Particle Systems and Related Non-Equilibrium Models: A Network Approach

We investigate the conditions that produce a phase transition from an ordered to a disordered state in a family of models of two-dimensional elements with a ferromagnetic-like interaction. This family is defined to contain under the same framework, among others, the XY-model and the Self-Driven Particles Model introduced by Vicsek et al. Each model is distinguished only by the rules that determine the set of elements with which each element interacts. We propose a new member of the family: the vectorial network model, in which a given fraction of the elements interact through direct random connections. This model is analogous to an XY-system on a network, and as such can be of interest for a wide range of problems. It captures the main aspects of the interaction dynamics that produce the phase transition in other models of the family. The network approach allows us to show analytically the existence of a phase transition in this vectorial network model, and to compute its relevant parameters for the case in which all elements are randomly connected. Finally we study numerically the conditions required for a phase transition to exist for different members of the family. Our results show that a qualitatively equivalent phase transition appears whenever even a small amount of long-range interactions are present (or built over time), regardless of other equilibrium or non-equilibrium properties of the system.