The statistics of long chains with non-Markovian repulsive interactions and the minimal gaussian approximation

Abstract

We study long chains (or rings) which occupy a space of s dimensions and which have repulsive interactions between all the points of the chain (N being the number of links) ; the method consists in introducing trial probabilities which are determined by minimization of the free energy FN ; these probabilities definite the mean size of the chain. Current theories are examined critically and their inconsistencies are revealed. The Minimal Gaussian approximation, which seems the simplest consistent approach, is described in detail for a ring of N links whose end points are assigned coordinates rj (j = 1, ..., N). The calculation shows that the mean square distance between two such points rj and rj+n (n >> 1, n/N << 1) is of the form : < (rj+ n — rj)2 > = bn2α(log n)βwith the following values : α=1,β=—1 for s = 2;α = 2/3, β = 0 for s = 3 ; α =1/2, β = 1/2 for s = 4 ; α= 1/2, β = 0 for s > 4. The structure of a large ring is investigated and the term ΔFN = FN— NlimN'→∞ (FN'/N') is calculated for s = 3 and N >> 1 (ΔFN oc log N). It is also shown that a large class of trial probabilities leads to the same qualitative results as the Gaussian approximation