The end-to-end distribution function of a polymer chain with a pairwise additive self-interaction

We analyse the end-to-end distribution function of a polymer chain with a pair-wise interaction between its repeat units in a continuous space. By means of a cluster expansion we classify its contributions according to the number and arrangement of the contacts between the repeat units. We then collect together diagrams corresponding to the same “underlying graph”, defined by a set of contact clusters connected by a pattern of oriented free polymer sections. The number of distinct diagrams belonging to a given underlying graph can be expressed in terms of the so-called Kirchhoff matrix. The analysis results in an expression for the end-to-end distribution function, which is analogous to the pair correlation function of a “gas of contact clusters” with the free polymer propagator acting as a “pair-interaction potential” between them.