  Using a first-principles classical many-body simulation of a Hall bar, we
study the necessary conditions for the formation of the Hall potential: (i)
Ohmic contacts with metallic reservoirs, (ii) electron-electron interactions,
and (iii) confinement to a finite system. By propagating thousands of
interacting electrons over million time-steps we capture the build-up of the
self-consistent potential, which resembles results obtained by
conformal-mapping methods. As shown by a microscopic model of the current
injection, the Hall effect is linked to specific boundary conditions at the
particle reservoirs.
