Equilibrium statistical distributions for subchains in an entangled polymer melt

n de Gennes-Doi-Edwards theory for entangled polymeric melts, a length scale r(0) is introduced, giving the equilibrium mesh size of the physical network of chains. Each polymer molecule is then represented as a random walk, with a step size r(0) (a subchain, made up of n(0) Kuhn segments) dictated by the existence of entanglements. Progressing from this simple picture, an issue that has been constantly overlooked so far, despite its potential relevance, is that of finite-size effects at the de Gennes-Doi-Edwards characteristic length scale. Actually, since a subchain in a melt is a small, nonmacroscopic system, fluctuations of both its length and its number of Kuhn segments are certainly nonnegligible. An ad hoc theoretical treatment from nonstandard (nano) statistical mechanics and thermodynamics seems then required, to find the anticipated equilibrium statistical distributions of the subchain population. In this contribution, we carefully discuss this topic. Some predictions from the nonstandard fluctuation-inclusive approach on the statistics of subchains are here obtained, and compared with existing simulations, even down to the atomistic level.