Workout: Diffusion and random walks (Plinko)

Read

Read the web page Diffusion and random walks.

Launch

To see how this works in practice, go to the PhET simulation Plinko to perform a few tasks. 

Set up

Select the tab "Lab". This show a ball being dropped down into an array of pegs. At each peg, the ball has a 50-50 chance of going left or right. Press the green "go" button a few times to see how this works. When the ball gets to the bottom it falls into one bin or another and the display cumulates the number of balls that have fallen into each bin.

Answer these questions

1. Predict where you expect it is most likely for a ball dropped through this array to end up. Then choose the multiple ball button next to the green "go" arrow, and run the simulation until about 1000 balls have fallen. Was your prediction correct? If you can, explain why your prediction was right or wrong. The picture below shows a screen after 33 balls have fallen and more are on their way.
2. This model is sometimes used to analyze random walk on a line. Consider an example where space and time are both discretized. A set of molecules are released at point 0 on a line and at every time step move to the left (-1) or right (+1) with equal probability. Do you think the plinko board is a good model of this or not? Explain why you think so. What would the number of rows in the plinko board correspond to?
3. Now switch the button in the go-arrow box back to the single ball and switch the view control button (next to the top center of the board on the right) to "Path". Run a few balls through the board. How does the path through the board correspond to the left-right motion of a molecule in the random-walk-on-a-line model? In terms of paths, can you explain why the balls are most likely to end up in the middle?

Joe Redish 10/21/17