Kay Kirkpatrick

Scaling Limits

Consider a drop of dye in water. If we could zoom in to watch it on the scale of a molecule, the dye particles would look like billiards colliding with each other and with the water molecules. Each one goes in a straight line unimpeded for a short time, and then hits another particle and bounces off to go in a new direction until the next collision, and so on. This motion is governed by Newton's laws, with a chaotic sensitivity to the initial position and velocity of the molecules. Zooming back out to the large scale (called the scaling limit), we see the dye diffusing throughout the water, spreading out uniformly. This macroscopic behavior (Brownian motion for an individual particle) is governed by the diffusion equation (or, heat equation: ∂_t f = Δ_x f ). As Einstein explained in 1905, this equation can be derived from the microscopic chaos.

With mathematical tools from both the theories of partial differential equations (PDEs) and probability, we can tackle more complex scaling limits. I work on rigorously deriving PDEs in scaling limits for some interesting phenomena--plasmas, condensates, and biopolymers--that would be wonderful to explain from the microscopic first principles. I also study the resulting PDEs, deepening our understanding of these physical and biological phenomena.

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(2D animation of the microscopic model of diffused particles. Courtesy of Greg L, Wikipedia. See also: Diffusion applet and Brownian motion applet.)

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