The statistical mechanics of a melt of polymer rings

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Abstract
Topological restrictions are introduced into a melt of flexible polymer rings by specifying the linking number between each pair of rings. Attention is focused on the configurational properties of a single ring in the melt, where the winding number between each pair of rings is zero. The configurations of all the other chains are averaged out in the partition sum. The effect of the topological constraint on the remaining chain is expressed as a product of two configurational weighting factors which involve unusual geometrical properties of the configuration. One is a phase factor depending on the torsion of the configuration and the other is a Boltzmann-like factor with the self-inductance of the configuration in the role of the interaction energy. The term based on the torsion has the remarkable property of transforming random walk-like configurations into those of stiff rods, while the inductance term promotes a transition to a completely collapsed state. Our preliminary results suggest that the actual configuration of the loop is a non-trivial balance between these opposing tendencies with the size R of the loop scaling with the length as R2 approximately L1-1(3 pi /) approximately L0.89.