Two-dimensional polymer configuration via mean-field theory

We consider determining the configurational properties of a neutral polymer in two dimensions (2D) via self-consistent mean-field methods. By suitably scaling the problem we recover the Flory result for polymers under the excluded volume interaction, i.e. R(N) similar to N-3/4, where R(N) is the mean scaling length of a polymer which consists of (N + 1) monomers. If we let x denote the scaled distance from one end of the polymer to a point in space we find that there exists a point y*, where the scaled polymer density f(N)(x), decays rapidly to zero. Physically the existence of such a point is expected since the polymer has a finite length. For y* - x > O(N--1/3) we find f(N)(x) similar to 1/2x[f(N)(x)-f(N)(y*)](1/2) while for x - y* > O(N--1/3) we obtain f(N)(x) similar to o(1). We discuss the consequence of these results on the validity of the asymptotic methods used.