Topological theory of entanglement (two body topological interaction approximation). I. Friction coefficients of the center of polymer chains

The topological theory of entanglement presented in the previous papers [J. Phys. Soc. Jpn. 37, 1413, 1423, 1429 (1974)] is generalized to a system composed of N three dimensional (D3) linear polymer chains and a system composed of a two dimensional (D2) linear polymer chain and N fixed barriers. The fundamental assumption of the present theory is that entanglement states of the systems are described by a set of (slightly modified) Gauss linking coefficients, ea, each of which represents topological states of two D3 polymer chains or of a D2 polymer chain and a point. These ea’s are called ’’(two body) entanglement coordinates.’’ The Verdier–Stockmayer model is used for representing the Brownian motion of the polymer chains, and diffusion equations are derived for simultaneous distribution functions in regard to the entanglement coordinates {e} and the Rouse coordinates {q}. It is found that the previous theory leads two contradictions: (i) When it is applied to the so-called phantom chains, nonzero interactions, which must be essentially zero, appear among the D3 chains or between the D2 chain and the barriers. (ii) The friction coefficient of the center of the polymer chains derived before diverges in the present systems, when N is infinity. These contradictions are removed by considering the non-Fokker–Planck terms explicitly in the diffusion equations, and nondivergent expressions of the friction coefficient are derived.