TOPOLOGY, GAUGE-FIELDS AND THE STATISTICAL-MECHANICS OF A MELT OF POLYMER RINGS

The topological properties of a melt of polymer molecules in the form of closed loops is considered. The use of topological invariants withi n a statistical mechanical framework is explained as well as their con nection to gauge fields and gauge field theory. Attention is focused i n this paper on the configurational properties of a single loop in a m elt of loops where the winding number between each pair is zero. The c onfigurations of all the chains, except one, are averaged out in the p artition sum. The effect of the topological constraint on the remainin g chain is to produce two configurational weighting factors which invo lve unusual geometrical properties. One is a phase factor depending on the writhe of the configuration and the other is a Boltzmann-like fac tor with the self-inductance of the configuration in the role of the i nteraction energy. The term based on the writhe has the remarkable pro perty of transforming random-walk-like configurations into those of st iff rods, while the inductance term promotes a transition to a complet ely collapsed state. The actual configuration of the loop is shown to be a non-trivial balance between these opposing tendencies with the si ze R of the loop scaling with the length as R(2) similar to L(0.89).