(Nano)granules Involving Aggregation at a Passage to the Nanoscale as Viewed in Terms of Diffusive Heisenberg Relation

Abstract

This feature study attempts on setting a granules involving mesoscopic matter-aggregation context as a problem of passing locally by the matter-aggregating system from classical stochastic (mesoscopic limit) to a quantum description (nanoscale quantum-size effect limit). A d-dimensional entropy-production aggregation of the granules-involving matter is considered in terms of a (sub)diffusive realization. It turns out that when taking a full d-dimensional pathway of the aggregation toward the nanoscale, one is capable of disclosing a Heisenberg type (diffusional) relation, setting up an upper uncertainty bound for the (sub)diffusive very slow granules-including environment that matches the quantum limit of h/2pm (m– average mass of a granule; h – the Planck’s constant) for the diffusion coefficient of the aggregation, likely first proposed by Fürth in 1933, and qualitatively foreseen by Schrödinger some years before, both in the context of a diffusing particle cluster. The classical-quantum passage uncovered here, also termed insightfully the quantum-size effect (as borrowed from the quantum dots’ parlance), works properly for the three-dimensional (d=3) case, making use of a substantial physical fact that the (nano)granules interact readily via their surfaces with the also granular surroundings in which they are immersed. This natural observation is embodied in the basic averaging construction of the diffusion coefficient of the aggregation of interest. Certain, possibly biomedical, applications of the model can also be viewed by prospective inspection of tiny inhomogeneities in nanofibrillar and/or lamellar matter for which certain numerical realizations are plausible to prepare based on thorough nanoscale experiments.