Figure 7.5 — Simulation of white noise and random walk

Fig. 7.5 — Simulation of white noise and random walk
Figure 7.5 — Simulation of white noise and random walk. As the time step 􏰀t decreases, one must employ larger values of the noise Wi to approximate the solution of the free diffusion equation [Eq. (7.21)] accurately: (a) Δt = 1, (b) 0.5, and (c) 0.1. The corresponding solutions of the finite-difference free diffusion equation [Eq. (7.22)] in (d)–(f ) for ri (lines) behave similarly: although these solutions differ because they are specific realisations of a random process, their statistical properties do not change, as can be seen by comparing the shaded areas, which show the regions within one standard deviation of the mean of 10000 realisations.
Adapted with permission from Volpe and Volpe, Am. J. Phys. 81, 224–30. Copyright 2013, American Association of Physics Teachers.