Unidirectional Flux In Brownian And Langevin Simulations Of Diffusion
Brownian and Langevin simulations of ions in solution require the maintenance of average fixed concentrations at the interface between the simulation volume and the surrounding continuum. This requires the injection of new trajectories into the simulation, which creates a unidirectional flux at the interface. The Wiener path integral splits the net diffusion flux into infinite unidirectional fluxes, whose difference is finite, as in classical diffusion theory. The infinite unidirectional flux is an artifact of the diffusion approximation to Langevin’s equation, which fails on time scales shorter than the relaxation time 1/γ. The probability of Brownian trajectories that cross a point in one direction per unit time Δt equals that of Langevin trajectories if γΔt = 2. This result is relevant to Brownian dynamics simulation of particles in a finite volume inside a large bath.