Statistics of the entanglement of polymers: Unentangled loops and primitive paths

Abstract

Attempts to apply the reptation and tube model to describe polymer disentanglement and associated relaxations have had wide success. However, questions have arisen in various applications as to the statistics of entanglement of a random walk with an obstacle net. Calculations are presented of a number of probabilities which characterize the degree to which a polymer, represented by a lattice walk, entangles with an array of obstacles. In particular, we have calculated the number of ways that such a walk can form unentangled closed loops of various types. If one reels in a general random walk from its ends, pulling out unentangled loops, one is left with the so‐called primitive path, which is taken to represent the path of the tube. The probability that an N step random walk has a K step primitive path has been determined. Asymptotic formulas for this probability are presented.