Fixed-density boundary conditions in overdamped Langevin simulations of diffusion in channels

We consider the numerical integration of Langevin equations for particles in a channel, in the presence of boundary conditions fixing the concentration values at the ends. This kind of boundary condition appears, for instance, when considering the diffusion of ions in molecular channels, between the different concentrations at both sides of the cellular membrane. For this application the overdamped limit of Brownian motion (leading to a first order Langevin equation) is most convenient, but in previous works some difficulties associated with this limit were found for the implementation of the boundary conditions. We derive here an algorithm that, unlike previous attempts, does not require the simulation of particle reservoirs or the consideration of velocity variables or adjustable parameters. Simulations of Brownian particles in simple cases show that results agree perfectly with theory, both for the local concentration values and for the resulting particle flux in nonequilibrium situations. The algorithm is appropriate for the modeling of more complex ionic channels and, in general, for the treatment of analogous boundary conditions in other physical models using first order Langevin equations.