  Brownian dynamics simulations require the connection of a small discrete
simulation volume to large baths that are maintained at fixed concentrations
and voltages. The continuum baths are connected to the simulation through
interfaces, located in the baths sufficiently far from the channel. Average
boundary concentrations have to be maintained at their values in the baths by
injecting and removing particles at the interfaces. The particles injected into
the simulation volume represent a unidirectional diffusion flux, while the
outgoing particles represent the unidirectional flux in the opposite direction.
The classical diffusion equation defines net diffusion flux, but not
unidirectional fluxes. The stochastic formulation of classical diffusion in
terms of the Wiener process leads to a Wiener path integral, which can split
the net flux into unidirectional fluxes. These unidirectional fluxes are
infinite, though the net flux is finite and agrees with classical theory. We
find that the infinite unidirectional flux is an artifact caused by replacing
the Langevin dynamics with its Smoluchowski approximation, which is classical
diffusion. The Smoluchowski approximation fails on time scales shorter than the
relaxation time $1/\gamma$ of the Langevin equation. We find the unidirectional
flux (source strength) needed to maintain average boundary concentrations in a
manner consistent with the physics of Brownian particles. This unidirectional
flux is proportional to the concentration and inversely proportional to
$\sqrt{\Delta t}$ to leading order. We develop a BD simulation that maintains
fixed average boundary concentrations in a manner consistent with the actual
physics of the interface and without creating spurious boundary layers.
