Brownian Simulation of Langevin’s

It was argued in Sect. 2.6 that mathematical Brownian motion (MBM) is the overdamped limit of the Langevin displacement process. It is tempting, therefore, to coarse-grain the two-dimensional phase space simulation of the overdamped one-dimensional Langevin equation into the one-dimensional configuration space of an MBM. This simplification, however, comes at a price: It inherits the artifacts of the MBM, such as the infinite rate of level crossing, with fatal consequences. An MBM that crosses a boundary (a point, a curve, or a surface) recrosses it infinitely many times in any time interval. Therefore, the number of recrossings of a boundary increases indefinitely as the step size of the simulation is decreased. Consequently, it becomes impossible to determine when a simulated trajectory is on one side of the boundary or the other. This phenomenon shows up, for example, in the simulation of ions in a given (small) volume in solution: Simulated ionic trajectories have to enter and leave the simulation domain an unbounded number of times as the step size of the simulation decreases, leaving no room for determining the convergence of the simulation. Quoting Einstein (1956) in this context, “the movements of one and the same particle after different intervals of time must be considered as mutually independent processes, so long as we think of these intervals of time as being chosen not too small.” This means that the MBM idealization should be taken with a grain of salt; the time step in a simulation cannot be refined beyond a certain limit. This limit has to be determined from the more refined Langevin model of the Brownian movement in the limit of large damping.