Statistical mechanics with topological constraints: I

Abstract

The entropy of very long flexible molecules in the presence of topological constraints is studied, and a formula deduced which needs the probability that a random walk will have a particular topological specification. Examples are solved, including a plane random walk sweeping out a given angle around a point in the plane which is generalized to three dimensions including the passage of a random walk past many lines in space, and the probability that a random walk will penetrate through or become multiply entangled with a closed ring.