What Exactly does a Random Walk Simulation Simulate?

Abstract

One of the most common statements about random walks or Brownian motion is that they are statistically self-similar. Self-similarity is commonly depicted as an individual trajectory at different magnifications or plots of square displacement versus time at different magnifications. The first question is, what exactly does self-similarity mean here? That the trajectory itself and certain functionals of it are self-similar? That all functionals of it are self-similar? That all functionals of a random walk are self-similar in the limit as the random walk approaches Brownian motion? This problem is a form of the standard question of how a random walk (finite step length) approaches the limit of Brownian motion (infinitesimal step length). This problem is well understood mathematically but for applications it is useful to understand how the approach to the Brownian limit affects simulations. When does the histogram of a functional of a random walk approximate the corresponding distribution for Brownian motion? Second, what exactly does a random walk simulation yield? In the simulation the step size is chosen from a Rayleigh distribution with mean-square radius = 2d D₀ Δt, and the angle is random. This is exactly what the propagator for Brownian motion yields for a time interval Δt. Thus the usual random walk simulation is periodically sampled Brownian motion. Periodic sampling — in engineering terminology, downsampling — is a form of low-pass filtering. Likewise, an experimental single-particle tracking trajectory is to a first approximation periodically sampled Brownian motion, but there are deviations. At short distances, the dynamics is Newtonian, not Brownian, and at longer distances, complex behavior may occur, such as the Alder-Wainwright long-time tails from hydrodynamics, and the collective motions found by Vattulainen and collaborators. Supported in part by NIH grant GM038133.