Mean-Square Intrachain Distances in a Self-Avoiding Walk

Abstract

An investigation is undertaken of the second‐order moments associated with the distribution of intrachain elements in a self‐avoiding‐walk model of chain polymer. The mean‐square distance of an element of the chain from an end point of the chain, 〈Q n 2 〉 , and the mean‐square distance of an element of the chain from the centre of mass, 〈S n 2 〉 , are enumerated exactly on various two‐ and three‐dimensional lattices for short chains of n links. It is conjectured that as n → ∞ , the limiting values of 〈Q n 2 〉 / 〈R n 2 〉 and 〈S n 2 〉 / 〈R n 2 〉 , 〈R n 2 〉 being the mean‐square end‐to‐end distance of the chain, depend on dimensionality rather than on lattice structure. Estimates for these limiting values in two and three dimensions are given.