UCF Math Department's
Probability and Statistics Seminar
Spring 2009

Fridays in MAP 213 from 2:30 PM - 3:30 PM, unless otherwise noted.

Organized by Jason Swanson

Past Seminars

Schedule and Abstracts

A pollen grain suspended on the surface of still water can be seen to move about in a “jittery” fashion. The usual explanation for this phenomenon is that the grain is being bombarded by the surrounding water molecules, causing it to perform a very rapid random walk with very small steps. If the water is not still, then each step in this random walk will have a component that moves in the direction of the current, plus a random component due to the molecular bombardment. A crude, first-order approximation to the trajectory of the pollen grain is given by the law of large numbers, which simply estimates the walk by its mean. In this case, the mean path is the one that flows with the current. If the water is still, then there is no current, and the first-order approximation is that the pollen grain simply does not move.

For a better approximation, we appeal to the central limit theorem (and its functional equivalents) to create so-called “fluctuation” limits, which are second-order approximations that describe the fluctuations about the mean. The second-order approximation of the random walk in still water is Brownian motion, which is the classical stochastic process used to describe the jittery motion of the pollen grain. It is worth pointing out that in this case, a useful and realistic model does not arise until we consider the second-order approximations.

Now imagine a cloud of pollen grains moving about on the surface of still water. The first-order approximation to the behavior of this cloud is well-known. If u(x,t) denotes the density of the cloud at time t, at the point x, then u satisfies the diffusion equation, which is a classical (i.e. deterministic) partial differential equation. The diffusion equation is a ubiquitous equation in the sciences, which is used to describe not only material diffusion, but also the distribution of heat in a region or of alleles in a genetic population, to name only a few applications. This equation can be derived using Brownian motion, provided one assumes that each individual pollen grain is performing an independent Brownian motion. However, it is quite evident that this assumption is wrong. For one thing, the motions of the individual grains are not independent, since they are interacting with one another through collisions. Moreover, they are not even performing Brownian motions. Grains near the center of the cloud will be constrained by their neighbors, and grains near the periphery will experience an outward drift as the cloud expands. What process or processes, then, if not Brownian motion, should be used to describe the trajectories of these colliding pollen grains?

The first-order approximation is fairly easy to obtain. The mean path of a grain in the cloud is a path whose velocity is proportional to −∇u(x,t). It is more difficult, however, to determine the fluctuations about this mean, which give the second-order approximation. This problem is already quite difficult in the simplified setting where space is reduced to one dimension, with work dating back to T. E. Harris's 1965 paper, “Diffusion with ‘collisions’ between particles.” In this talk, I will discuss in more detail the background of this problem, as well as some new results and work in progress.

A continuation of the previous talk.

A continuation of the Jan 23 talk.