Stiff Chains and Levy Flight: Two Self Avoiding Walk Models and the Uses of Their Statistical Mechanical Representations

Abstract

Two self-avoiding walk models are described: In the stiff chain model, useful in the description of some polymers, the walk “prefers” to continue in the direction of the last step with probability 1-p. The limit of small p and a large number of steps N is of particular interest. We present numerical results indicating the nature of the crossover from “ballistic” to random walk behavior. In two dimensions, the walk is asymptotically behaving like a self avoiding walk but in three dimensions, the numerical results suggest a random (not self-avoiding) behavior for a wide range of N. We describe this walk in terms of a vector spin model in the limit that the number of components n -> 0 and use this formulation to account for the difference between two and three dimensions observed numerically. Secondly, we consider self-avoiding Levy flight: the step length distribution is of the form P(x) = C/x ^μ+1. We study a type of Levy flight with a self-avoiding constraint called node-avoiding Levy flight here. This node-avoiding case is shown to be obtained from the n -> 0 limit of the statistical mechanics of another kind of vector spin model. The critical properties of this model were previously studied by Fisher, Nickel and Saks using the ε expansion. By comparing numerical simulations of node-avoiding Levy flight with their results, we can obtain information about the ε expansion in statistical mechanics at very small values of ε.