While in Hong Kong, I have been working with Jonathan Wylie on the theory of diffusion with “traps”. This is related to some of my experiments, and what we are aiming for is a way to extract some new physical information from our experimental measurements. The experiments involve measuring the diffusion of little tracer particles suspended in a complex fluids. For simple Brownian motion, the mean squared displacement increases linearly with time. In many complex fluids, we see something rather different: At very short times, the mean squared displacement increases linearly. Then over some intermediate time range, the mean squared displacement flattens out into a plateau. At longer times, it grows linearly again, but with a lower diffusion constant than at short times. The plateau occurs because the tracers get trapped by the microstructure of the complex fluid. However over longer times, the microstructure evolves and the particles can escape from the traps and diffuse freely again. The calculations we are doing – actually it’s mostly Jonathan so far – involve using the formal theory of random walks to figure out the distribution of distances traveled by the tracers in the presence of traps. The particles diffuse freely until they hit a trap, then they are held in the trap for a time drawn from a particular distribution, then they diffuse again. Parts of the calculation are straightforward, if you know about random walk theory. But parts are what you might call interesting. I have had to refresh my memory about Green functions, which caused me nightmares when I was a student and played a significant role in my decision to become an experimentalist. We have also run across one of those pathological cases that mathematicians warn us about, but which never happen in real life – we get an expression involving an integral over an infinite sum, and it turns out that you cannot interchange the order of the sum and the integral. All good stuff. The idea is that this will model our experimental data, and the ratio of the diffusion constants we find at short and long times will depend on the time the particles spend in the traps.