Computed Probability Density Functions of the Radii of Gyration of Long Random-Flight Chains

Probability density functions for the orthogonal components of the radius of gyration, Si, of a random-flight chain were computed using an indirect numerical integration algorithm which was based upon a multiple convolution proposed by Forsman and Hughes. Computations were performed for relatively long chains of 100–1000 statistical segments t. By using a reduced radius of gyration ξi = (π / 23) Si〈Si2〉−1 / 2 the distributions converged to a limiting function at about t = 500, except for extremely small radii, ξi ≤ [10(2t)1/2]−1. Additional computations using the one-dimensional distributions yielded the corresponding probability densities for both the two- and three-dimensional (polar) radii. The computed functions were in excellent agreement with all their limiting properties previously determined by analytical methods.