Intersegment transfer
In a free polymer coil, contacts between different parts of the polymer are much more likely for positions close by on the chain. The compression of the polymer into a small volume increases the change that very distant parts touch each other. Hence confinement should increase the speed of exploration by reducing redundancy in the search.
The distribution of return times in a closed box
The probability of a diffusing particle to return to the origin in free 3D space is given by
$$P(0,t) = \frac{dx^3}{(4\pi Dt)^{3/2}}$$
where \(dx^3\) is a small infinitesimal volume around the origin. When considering the same problem in an enclosed volume, we encounter the same discrete spectrum of eigenmodes that we derived in a previous assignment.
In the middle of our box, the only the even modes make a contribution. And all these contributions are equal. Hence we should have
$$P(0,t) = \sum_{n=0}^\infty e^{-\frac{n^2\pi^2 D}{L^2}t} $$