Second topological moment 〈m2〉 of two closed entangled polymers

We calculate exactly by field theoretical techniques the second topological moment hm 2 i of entanglement of two closed polymersP1 andP2. This result is used to estimate approximately the mean square average of the linking number of a polymer P1 in solution with other polymers. © 2000 Elsevier Science B.V. All rights reserved. 1. Consider two closed polymersP1 and P2 which statistically can be linked with each other any number of timesmD 0; 1; 2;:::. The situation is illustrated in Fig. 1 formD 2. An important physical quantity is the probability distribution of the linking number m as a function of the lengths ofP1 andP2. As a first step towards finding it we calculate, in this Letter, an exact expression for the second moment of the distribution,hm 2 i. An approximate result for this quantity was obtained before in Ref. [1] on the basis of a a mean-field method, considering the density of bond vectors ofP2 as Gaussian random variables. Such methods are usually quite accurate when a large number of polymers is involved [2,3]. As an unpleasant feature, however, they introduce a dependence on the source of Gaussian noise, and modify the critical behavior of the system, whereas topological interactions are not expected to do