  The properties which 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 the 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.
