Long-range order in a system of thermal Brownian particles*

We present a model that exhibits an order-disorder phase transition in two spatial-dimensions. The model considers a collection of $N$ thermal Brownian particles moving in a square of length $L$ subjected to periodic boundary conditions and to velocity-alignment forces. The alignment force affects only the velocity direction in a way that it makes it equal to the velocity direction of the nearby group. Our results contrast with those obtained from the well known model of Vicsek {\it et al.} [Phys. Rev. Lett. {\bf 75}, 1226 (1995)] where such a transition occurs out of equilibrium.