Exact Evaluation of the Mean Square Radius of Gyration for Gaussian Topological Polymer Chains

Abstract

Various polymers with nontrivial molecular structures expressed by graphs have recently been synthesized in experiments. We call them topological polymers for the graphs. We consider a set of Gaussian chains describing an ideal topological polymer or ideal polymer network for a given graph and call it topologically constrained random walks (TCRW) of the graph. In this chapter we show an exact evaluation of the mean square radius of gyration for the TCRW of complete graphs. We first review fundamental aspects of the novel method for constructing TCRW through the boundary operator of homology theory, as given in Ref. (Cantarella et al., Gaussian Random Embeddings of Multigraph, [5]). Then we rigorously derive the average size of the TCRW for complete graphs. By making use of the asymptotic formula (Cantarella et al., Radius of Gyration, Contraction Factors, and Subdivisions of Topological Polymers, [6]) we exactly derive the mean square radius of gyration for the subdivided topological polymers consisting of Gaussian chains or the subdivided TCRW for a given graph in the large subdivision degree limit. That is, in the limit of sending the number of subdivided segments in each branch to infinity. Throughout the chapter we emphasize the key point of the TCRW method that the probability distribution function of the edge displacements or bond vectors of the TCRW for a connected graph is directly derived from the normal distribution with unit variance. For instance, thanks to it we can rapidly generate conformations of the Gaussian network of a given graph where external forces are applied at the surfaces so that it has a nonzero finite volume in equilibrium.