  Phase randomizing processes in mesoscopic systems can be described in a
phenomenological way within the Landauer-B\"{u}ttiker formalism by attaching
extra voltage probes to the sample. In this paper, it is shown that a
perturbation treatment of this idea allows for the incorporation of such
effects without the need of giving up the efficiency of recursive techniques
commonly used for calculating the transmission coefficients. The technique is
applied to a 4-probe ring, where a Hall effect can be observed that originates
from quantum interference rather than a Lorentz force acting on the electrons.
The influence of inelastic scattering on both the Hall resistance and the
Aharonov-Bohm oscillations in the longitudinal resistance are examined.
