The Excluded Volume Effect in Polymer Chains and the Analogous Random Walk Problem

Abstract

An excluded volume random walk is studied in an m‐dimensional space. A general expression is obtained for the mean square length of a walk of N steps, 〈R_N²〉. It is shown that the increase in 〈R_N²〉, δ_s, resulting from the interaction of only the ith and i+sth steps in the walk is a constant independent of s in two dimensions. In three and four dimensions δ_s is of the order s^−½ and s⁻¹, respectively. Thus, the interaction of the steps is surprisingly large. An estimate is made of the upper bound on 〈R_N²〉 caused by the simultaneous interaction of all steps. The result in three dimensions is that in the limit of large N, 〈R_N²〉/N is at most of order N^½.