Gauge Theory of Topological Entanglements. II: -- A Pair of Fluctuating Chains --

Abstract

A statistical theory is developed for the study of interactions between two long flexible polymers which are of topological origin. The probability of finding a pair of closed Brownian chains in entangled states of fixed topological numbers are given as a function of their separation. The Gauss linking number is studied for the topological number by introducing fictitious "magnetic fields" produced by the chains. Comparisons are made with the results of Monte Carlo study on random walks. § 1. Introduction For chain-like excitations embedded in three dimensional continuum, such as polymer melts, rubbers, vorticies in a fluid and dislocation lines in a solid, there exist many situations in which topological relationships among such linear chains can be considered permanent during the observation and play an essential role for reali'Zation of macroscopic phenomena. In· a preceding paper l) (hereafter referred to as I), we developed a field theory for studying two mutually entangled Brownian chains by taking the Gauss linking number as a constraint of the topological relation. In relatively simple situations in which one of the chains is fixed in various geometrical forms, the average degree of entanglements was found exactly in the limit of infinite chain length. The underlying characteristics of the Brownian chains were assumed to be of Wiener process but can easily be extended to other stochastic processes depending on the properties of the excita-tions mentioned above. In this paper we study two fluctuating closed Wiener paths placed in a certain distance r and with fixed Gauss number m. The formation probabilities Pm(r) of such paths are of our main interest. The average degree of entanglements can be expressed by the moments < m 2k >r = ~ m 2k Pm(r) which were called topological moments. 2) Let ri(s) (i=l, 2) be the two Wiener paths of length L.-The formation probability of such paths can be described by the Green's function