Short range correlation between elements of a long polymer in a good solvent

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Abstract
Correlation properties of a long polymer in a good solvent are studied with the help of renormalization techniques. The probability distribution of the vector r joining, in a space of dimension d, the end points of a polymer segment made of N links, is a function Ps,N(r), where s is an index referring to various situations. Case s = 0 : the segment coincides with the polymer itself. Case s = 1 : the segment is at one extremity of an infinite polymer. Case s = 2 : the segment is located in the central part of an infinite polymer. It is shown that the probability distributions obey scaling laws of the form Ps,N(r) = N-νd fs (r/N ν), that, for small x, fs(x) ∝ x θs and that the indices θ0, θ1, θ 2 are given by the renormalization of simple vertices. The second order expansions of these indices with respect to ε = 4 — d have been calculated. The results are : Estimations of these indices for d = 3 give θ0 = 0.273, θ1 = 0.46, θ 2 = 0.71. These results show that the probability of contact of the end points of a segment made of N links, belonging to the central part of an infinite polymer is proportional to N-ν(d+θ2 )≃ N-2.18. This conclusion agrees with the fact, predicted by the author, that for N>> 1, the dominant terms of the energy E of a polymer made of N monomers, are of the form E = aN + b, in a good solvent