Reptation theory: geometrical and topological aspects

This paper discusses topological and geometrical aspects of reptation theory which are common to all versions of reptation theory. These are: the postulated existence of the tube, the functional relationship between the tube diameter a and the polymer/monomer density p, the crossover from the Rouse to reptation regime. Statistical mechanics of the geometrically confined polymer chain is reanalyzed by careful separation of the diffusive motion of the chain into the longitudinal and transversal parts. Connection between old results and the new formalism is established. It is shown that the longitudinal motion resembles that known for directed polymers. This provides a source of the effective rigidification of the reptating chain's backbone thus facilitating the viscosity exponent to be larger than 3. The transversal motion is also reanalyzed. It is shown that the diffusion on the Bethe lattice used before to describe the transversal (planar) motion (conformational statistics) of the trapped chain is actually the diffusion on the universal covering of the corresponding Riemannian surface. This fact allows to reanalyze the tube stability using topological arguments. Detailed numerical comparison of the obtained new theoretical results with available experimental and Monte Carlo data is provided. Very good agreement between theory and experiment is found. It is also shown that the emerging physical picture of the tube destruction is isomorphic to that which was developed earlier with the help of the quantum Hall effect analogy (J. Phys. I 4 , 843 (1994)). Remarkable connections between the reptation theory and the theory of quantum chaotic/mesoscopic systems are established thus making the reptation theory part of the more general theory of quantum chaotic systems.