Shape of a Random-Flight Chain

Abstract

An analysis is made of the distribution function and its moments for a linear combination of three randomly chosen orthogonal components of the so‐called radius of gyration of an unrestricted random‐flight chain, and certain averages of moments are obtained of the three‐dimensional distribution ${\mathrm{W\; (L}}_1^2{L}_2^2{L}_3^2)$ , where $L_1{\mathrm{\hspace{0.17em}\le \hspace{0.17em}L}}_2{\mathrm{\hspace{0.17em}\le \hspace{0.17em}L}}_3$ are the orthogonal components of the radius of gyration along the principal axes of inertia of the chain. The strong departures of the chain shape from spherical symmetry indicated by these results are confirmed and complemented by Monte Carlo studies of unrestricted random walks on a simple cubic lattice. A surprisingly high ratio of principal components is found for chains with 50 and 100 bonds, ${\mathrm{〈L}}_3^2{\mathrm{〉:\; 〈L}}_2^2{\mathrm{〉:\; 〈L}}_1^2\text{〉\hspace{0.17em}≃\hspace{0.17em}11.7:2.7:1}$ .