Random Walks, Brownian Motion, and Drift

The major conclusion of the Einsteinian relation is that the mean square displacement of the particle is proportional to the time elapsed. 

All experiments dealing with large-scale dynamics of suspended particles in solutions are bounded by the above two limits. The Einsteinian law is manifest when collisions by the solvent molecules due to their thermal motion dominate the dynamics of the particle. On the other hand, if the externally imposed forces on the particle dominate, then the Newtonian limit is approached. In general, the behavior is diffusion-drift.  

---

In this section, we begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

Figure 1 is a schematic representation of Frenkel s notion an atom or ion can get dislodged from its normal site to form etn interstitial-vacancy pair. He further proposed that they do not always recombine but instead may dissociate and thus contribute to diffusional transport and electrical conduction. They were free to Wcuider about in a "random walk" mcuiner essentially equivalent to that of Brownian motion. . . this meant they should exhibit a net drift in an applied field. 

The intuitive meaning of this formula is as follows. Let X (t) be the position of a particle performing a random Ldvy walk with (3.107), then X(t) = at + B t) + N it). The particle starts at zero and then follows the Brownian motion B t) with the drift velocity a until the random time at which a jump of size Zq takes place. Between random times and T2 we have again the Brownian motion with a drift and then another jump of the same size Zq at time T2. The last term in (3.107) is related to a Poisson process N (t) with the rate k in the set of values nzo with... 

---