Legendre transformation of a self-interacting polymer chain

The end-to-end distribution function for a polymer chain with self-interaction is derived from a functional having the significance of a generator for connected diagrams which consist of vertices representing clusters of monomers in contact, and lines representing free polymer sections. In spite of the dependence of the vertex weights on the multiplicities of the contacts and the presence of an extra weight-factor peculiar to the polymer chain, the generator for connected diagrams is shown to be expressible in terms of a similar generator for irreducible diagrams through a reduction procedure that is analogous to the transformation from connected to irreducible cluster diagrams in classical gas theory. The relation between the generators for connected and irreducible diagrams can be cast into the form of a functional Legendre transformation by considering them as characteristic functionals of new activity and density functions, that embody the multiplicity dependent cluster activity and densities figuring as vertex weights in the connected and irreducible diagrams respectively. This Legendre transformation can be reduced to a much simpler form by imposing an upper bound to the allowed contact multiplicity of the interacting chain.