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 n th link of the polymer passing through the point r to be ( L )exp[-27 r -(5/3) 3/5 ( v /3π l ) 1/5 L 3/5 2 (1/20 Ll )] where L = nl is the length along the polymer and ( L ) the normalization. Thus the mean square of r , r 2 , is (5/3) 6/5 ( v /3π l ) 2/5 L 6/5 . The theory is extended to polymers of finite length, to the excluded random walk problem and to n dimensions.