Efficient Brownian dynamics simulation of particles near walls. I. Reflecting and absorbing walls

In this paper a method of numerically handling boundary conditions within Brownian dynamics simulations is discussed. The usual naive treatment of identifying reflection or absorption processes by checking for boundary crossings yields O(sqrt[deltat]) discretization errors. The method we propose here yields O(deltat) errors, similar to the case of Brownian dynamics without wall interaction. The main idea is to ensure that the zeroth (in the case of absorption), first, and second moments of the particle's displacement steps are correct up to order deltat. To fulfill this requirement near a wall, one has to include nontrivial corrections, because the stochastic contribution does not average out when the distance to the wall is of the order of the step length. We demonstrate here that the method substantially reduces the discretization error for the simple cases of an absorbing and a reflecting wall. Our method comprises an improvement over earlier methods proposed by Lamm and Schulten [J. Chem. Phys. 78, 2713 (1983)] and Ottinger [J. Chem. Phys. 91, 6455 (1937)]. Their methods heavily depend on full, explicit, analytical expressions for solutions of the diffusion equation near a wall, which they use to make a correction after a stochastic step has been made. Our method only involves the, usually much simpler, lowest moments (up to the second) of the probability density distributions for the displacement of the particle in one time step. This means the method only uses the initial particle position to determine a valid step, and there is no need for corrections afterwards. Because much less information is needed (three moments instead of full probability densities), in many cases information can be stored simply in interpolation functions and there is no need to evaluate complicated analytical expressions at every time step. This makes the method more efficient and easy to generalize to other situations than the relatively simple case of a flat wall. Moreover, because analytic expressions are not needed, other methods to determine the needed moments can be used. This makes our method much more flexible.