The Shapes of Random Walks: Development of 1/D Expansion.

A newly developed analytical method for the study of the shapes of random walks is presented. Three kinds of randomly generated objects have been studied and their shapes are characterized in terms of the invariants of a tensor, analogous to the moment-of-inertia tensor, whose eigenvalues are the squares of the principal radii of gyration. The method is used to derive exact analytical expressions and valid in an arbitrary spatial dimensionality, d , for the asphericities of unrestricted open walks or linear chain polymers; closed walks or polymer rings; and walks with fixed end-to-end distance which correspond to long polymer chain molecules whose heads and tails are fixed in space. A graphical procedure is developed to systematize a l/d series expansion for the individual principal radii of gyration and their respective probability distribution functions P(R_sp {i}{2})(1 <=q i <=q d) . The average principal radii of gyration are calculated using the l/d formalism to second order in l/d for open walks; and to first order in l/d for both closed walks and walks with fixed end-to-end distance. Selected terms in the l/d expansion are summed to all order in l/d in the determination of P(R _sp{i}{2}), leading to an explicit analytical form for the probability distribution function for open walks. The distributions of the eigenvalues is compared with the distributions obtained from numerical simulations of walks in two to five dimensions. The agreement between the theoretical and numerical probabilities is extremely good. Other predictions for various parameters that characterize the average shape of open and closed walks in three dimensions are also found to agree remarkably well with the results of simulations, the difference being of order 5%. The method discussed here can, in principle, be used to study the shapes of other random fractal objects.