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

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, 〈RN2〉. It is shown that the increase in 〈RN2〉, δs, 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 δs 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 〈RN2〉 caused by the simultaneous interaction of all steps. The result in three dimensions is that in the limit of large N, 〈RN2〉/N is at most of order N½.