Moments and distribution functions for polymer chains of finite length. I. Theory

Abstract

The persistence vector a ≡ 〈r〉, representing the configurational average of the end‐to‐end vector r for a chain molecule, and 〈g〉, the configurational average of the vector g extending from the beginning of the chain to its center of mass, are discussed. These and other vectors and related tensors are expressed in a reference frame defined by the first two bonds of the chain. Algebraic relations between the various vectors and tensors characterizing the spatial distribution of the chain are derived, and matrix generators for their evaluation are presented. The spatial distribution of the remote end of the chain in the reference frame embedded in its first two bonds is discussed in terms of the vector ρ = r − a representing the displacement from the mean position of the terminus of r. The density distribution functionWa (ρ) is developed in terms of the moments of vector ρ with the aid of three‐dimensional tensorpolynomials which are analogs of the Hermite polynomials.