The Diffusion Applet

Project (adapted by E. Sontag for Rutgers Math 336)

Run the applet shown below, by clicking on "Remove lid".
(Whenever you need to restart, sometimes "reloading" the page will not work, depending on your browser. In that case, you should press the shift key and, while it is pressed, click on the "reload" [or "refresh"] button.)

The applet models perfume molecules trapped inside a bottle. Removing the lid, the perfume molecules diffuse throughout the room.
(The real model would have to be in three dimensions, but let us just think of it in the plane, for simplicity.)

Assignment:

0. Play with this for a couple of minutes. (No write-up required for this part. Honor system. :)

1. Write down an appropriate diffusion partial differential equation to model what happens to the concentration of particles in the room (suppose that the room is the unit square [0,1]x[0,1] and that the bottle is the square [.4,.6]x[0,.2]).

You must carefully state what the initial conditions and boundary conditions are. Fos instance, c(x,y,t)=what? if x is 0.4 and y is... what?, etc.

For the model, we assume that the lid has already been opened, so the "walls" to consider are the three closed sides of the bottle as well as the four sides of the room.

Suppose that the initial concentration in the bottle is "1" units/area, that the diffusion coefficient is D=1.

2. What is the steady state distribution, that is to say, what does the solution c(x,t) look like, for very large t? You do not need to do any complicated computations nor solve any PDE; use your intuition. But do give a precise answer (for example: "the density will be constantly equal to 0.3 in the region such and such").

Designed by Paul O. Lewis, U of Connecticut, who wrote many other neat applets as well, and who wrote this in relation to the present one:

Diffusion

Imagine the particles are perfume molecules trapped inside a bottle. Remove the lid of the bottle by pressing the button and watch the perfume molecules diffuse throughout the room. The perfume molecules that escape just happen to be moving in a (random) direction that takes them out of the opening. No extra input of energy is needed to produce the net effect, which is the dispersion of perfume from an area of higher concentration (inside the box) to an area of lower concentration (outside the box). To reverse this process of diffusion (i.e. climb up the concentration gradient) would require an input of energy.

While this is a rather simplistic model of diffusion (well, ok, it is extremely simplistic), it does I think get across the main ideas in a way that is much more fun than reading the above paragraph.