The statistical mechanics of polymers with excluded volume

The probability distribution of the configurations of a polymer consisting of freely hinged links of length l and excluded volume v is studied. It is shown that the interaction of the polymer with itself can be represented by considering the polymer under the influence of a self-consistent field which reduces the problem to an equation like the Hartree equation for an atom. This can be solved asymptotically, giving the probability of the nth link of the polymer passing through the point r to be (L)exp[-27{r-(5/3)^3/5(v/3πl)^1/5L^3/5}²(1/20Ll)] where L = nl is the length along the polymer and (L) the normalization. Thus the mean square of r, r², is (5/3)^6/5(v/3πl)^2/5L^6/5.

The theory is extended to polymers of finite length, to the excluded random walk problem and to n dimensions.