The Brownian motion model introduced by Dyson (7) for the eigenvalues of unitary random matrices N N is interpreted as a system of N interacting Brownian particles on the circle with electrostatic inter-particles repulsion. The aim of this paper is to dene the nite particle system in a general setting including collisions between particles. Then, we study the behaviour of this system when the number of particles N goes to innity (through the empirical measure process). We prove that a limiting measure-valued process exists and is the unique solution of a deterministic second-order PDE. The uniform law on ( ;) is the only limiting distribution of t when t goes to innity and t has an analytical density.