The statistical mechanics of a concatenated polymer chain

The elastic and configurational properties of mesoscopic structures, which have been formed by linking a finite number of polymer loops to each other, are considered. For simplicity it is assumed that the linking numbers between pairs of loops provides a sufficient description of the topological structure. To conserve the topology the statistical mechanical calculation of the partition sum is restricted to the configuration space labelled by the specified linking numbers. An expansion of the partition sum is proposed where the linking numbers, corresponding to a particular structure, select a sub-set of terms from the expansion. The case of a linear concatenated chain of loops is studied and it is shown that the first non-zero contribution from this expansion is already able to describe the chain as a re-scaled Gaussian chain capable of supporting stress. This confirms that the approximation maintains the structural integrity of the chain. The effective step length and step length distribution function are also identified together with the elastic properties. The relationship of linking numbers to the Abelian Chem-Simons gauge field theory is used to illustrate essential parts of the calculation. The relevant term in the partition sum is represented in terms of Feynman diagrams describing the two-particle scattering amplitude for vector gauge bosons. It is shown that the expected re-scaled Gaussian chain result comes from the dominance in the s-channel of a massive scalar boson representing the structure of an individual loop in the chain.