4.3.3.P16

Helpful readings:

• Random Walks and Counting Paths
• Expectation Values Via Counting Paths

The equation 

describes diffusion of a molecule in one dimension. The goals of this problem are to understand where this equation comes from using the model of a random walk and to practice working with distributions and calculations on distributions.

Instead of treating space and time as continuous variables, we'll describe motion in one dimension as on a discrete grid with discrete time steps.

Suppose a molecule begins at x=0 and can take one step of 1 nm to either the left or right.

Then after one step, it could be at -1 nm or +1 nm. There is one path leading to -1 nm and one path leading to +1 nm, so two paths over all. Each past is equally probably, so if we did many trials, the average position, <x>, is zero, which we can calculate via

On the other hand, the average square of position, <x²>, is equal to 1 nm². The quick way to see this is that (-1 nm)² = 1 nm² and (1nm)² = 1 nm², so whether x=-1 nm or x=1 nm, x²=1 nm², so <x²> must be 1 nm² on average. We can do the longer calculation as:

A. Suppose the molecule takes two steps and moves left or right 1 nm at random each time. The possibilities for where it can end up are -2 nm, 0, and 2 nm. Make sure you understand why it can't end up at -1 nm or 1 nm in this scenario. Make a table for each of the positions where the molecule can end up, and fill in the number of paths that lead to that position. Each distinct path is equally probable.

B. Find <x> and <x²> after two steps. To make sure you don't go off the rails in the rest of the problem, you should find 0 and 2 nm², respectively.

C. Make a conjecture about what will happen for three steps. Then calculate the possibilities and the the numbers of paths after three steps, and find <x> and <x²> after three steps. Was your conjecture correct?

Now suppose the molecule starts at x = 6 nm. It takes one random step, ending up at 5 nm or 7 nm with equal probability. Then <x> = 6 nm and 

D. In the scenario just described, how much did <x> increase due to taking the one step? How much did <x²> increase due to taking that step?

E. Now imagine that the molecule starts at a position a, which could be any number of nm. If it takes one step on 1 nm randomly left or right with equal probability, how much do <x> and <x²> increase? Show your work via algebra, not just a guess based on what you've done above (although a guess will also help).

Let's ramp the complexity up a little. Imagine you have, as a starting position, that a molecule had one path to get to  -2 nm and one path to get to 3 nm (and no path to get anywhere else). This starting position might not be likely by diffusion, but other scenarios could lead us to it. Then before taking any steps,

Then imagine taking one step of length 1nm, left or right with equal probability. Then there are 4 possible ending points: -3nm, -1nm, 2nm, 4nm, all with one path leading to them (and there are four paths total).

F. Find <x> and <x²> for the situation in the last sentence. How much did these increase due to taking the one step?

In the previous part, you should have found that  <x²> increases by 1 nm² when we take one step. (That's the answer; you'll be graded on the quality and clarity of your work as you demonstrate it.) Here is another way to see this, besides the algebraic calculation. If the molecule starts at position -2 nm then <x²> will increase by 1 nm² after one step, which we know from work earlier in the problem. Also, if the molecule starts at 3 nm, <x²> will increase by 1 nm². So, regardless of where we start, <x²> will increase by 1 nm². Therefore, the answer to our calculation in part G must be 1 nm².

G. Extend the argument in the previous problem to explain why if you start with any distribution of starting positions at all, then take one steps, <x²> increases by 1 nm². Also explain why if you take n steps, <x²> increases by n nm².

H. We started with taking discrete steps of a specified distance. Now we have to make the connection between our "steps" and an actual time. The diffusion constant is defined by <x²> = 2Dt. In our model, suppose the molecule takes one random 1 nm step each microsecond. Then what would its diffusion constant be?

I. Suppose we used a different model where the molecule could take one step 1nm to the right, stay put, or take one step 1 nm to the left each microsecond. Would the diffusion constant be higher or lower? Why? (You don't need to calculate it.)

J. The random walk we've been discussing is not a complete model of diffusion of molecules! Brainstorm some aspects of random motion of molecules that you think might not be captured by the random walk, and suggest one part of the random walk model you might modify if you wanted to create a more-detailed model.

Mark Eichenlaub 10/20/17