Chapter 4 of Nelson introduces the idea of a random walk as a way of describing (amongst other things) the motion of colloidal particles. Reading this section reminded me of a discussion I had on the topic with a chemistry lecturer last semester. The question I had was, “Given that I know the position of a colloidal particle at time t, what is the expectation value of the particle’s position at time t+Δt?” Or perhaps more precisely, “...what are the points of equal maximum expectation at time t+Δt?”
I had expected the lecturer to respond that the expectation value for a single particle will remain the position at time t; the only thing that changes is the uncertainty. However the answer I got was along the lines of, “The expectation value evolves according to the formula: r=sqrt(kDt)” [where k is equal to 2 times the dimensionality]. Thus in 2D, the points of equal maximum expectation lie on a circle which has radius proportional to sqrt(Δt).
I’m fairly sure this is incorrect, and I also have an idea as to why [r should be mean(r)] but it’s still difficult to get my head around.
What do you think? Is there something obvious I’m missing?
Hi Martin,
ReplyDeleteYour lecturer was correct, with a few caveats.
Firstly though, I am wary of your usage of the word expectation: surely what you mean is the most likely value?
The expectation value of the position of a particle undergoing a random walk is equal to its initial position because the motions away from that point are not biased in any particular direction.
What your lecturer answered with was the RMS (root mean square) displacement. The most likely magnitude of the displacement grows as the square root of the time, which you will recall from BIPH2000 as an example of a sub-ballistic trajectory.
The result for the RMS displacement is a classic in physics and is definitely true (I even have data sets from my own experiment which show that this is true)! It's also quite a pretty result. In full,
= 2Dt',
where the average <> is taken over all starting times t at constant temperature.
My mathematics has been interpreted as HTML and magicked away from the last post...
ReplyDeleteif we let brackets [] denote an average as explained above, then
[(r(t+t')-r(t))^2] = 2Dt'.
I think we're in agreement; but I also think that makes the lecturer incorrect...
ReplyDeleteThe lecturer was arguing that for a single particle the "most likely value" (what I called the "expectation value of the particle's position") behaves like the RMS displacement. Which I don't think is correct.