Stochastic description of the stationary Hall effect

The properties that characterize the stationary Hall effect in a Hall bar are derived from the Langevin equations describing the Brownian motion of an ensemble of interacting moving charges in a constant externally applied electromagnetic field. It is demonstrated that a non-uniform current density (a) superimposes on the injected one, (b) is confined in a boundary layer located near the edges over the Debye-Fermi length scale, (c) results from coupling between diffusion and conduction, and (d) arises because of charge accumulation at the edges. The theory can easily be transposed to describe the Hall effect in metals, semi-conductors, and plasmas and agrees with standard and previously published results.