## 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

$$P(0,t) = \sum_{n=0}^\infty e^{-\frac{n^2\pi^2 D}{L^2}t} $$