Diffusion to a disk-like adsorber

1.

• Read Berg Links to an external site. pages 27-28, and pages 38-39.
• hw3/3d_diff/plane_diff.py is a program that starts a particle at some point on a sphere and lets it random walk inside it until it's adsorbed by a disk below, or it hits the outer sphere again. Add comments to the code to explain what it is doing in detail.
• The current when the radius is infinite is given by Berg. When the radius is sufficiently large, this is expected to be a good approximation. Compare the results of this simulation with the formula for the current given by Berg. The crucial question to answer, is how the current depends on the radius of the disk. Naively one might expect it to be proportional to the surface area, but it is instead proportional to the disk's radius. What do you find numerically?

Note: You can do a more accurate analysis of this by using the concept of diffusion resistance and considering this as analogous to the following electrical problem: A weakly conducting medium has a metal conductor at the radius where particles are released (starting_radius) at a voltage V, and there is another grounded conductor at the outer radius (outer_radius). There is an insulating plane going through the center of the spheres, with a metal disk of some radius (absorb_radius) also held at ground. You can calculate the ratio of the current flowing between the release radius and the disk, compared to the total current.

2. Give three biological examples of where you have diffusion to a roughly disk-like absorber. What idealizations are made in the physical model that might affect the results seen in real biological systems?