We are looking at an aggregation of matter into granules. Diffusion plays a pivotal role here. When going down to the nanometer scale (the so-called nanoscale quantum-size effect limit), quantum mechanics, and the Heisenberg uncertainty relation, may take over the role of classical diffusion, as viewed typically in the mesoscopic/stochastic limit. A d-dimensional entropy-production aggregation of the granules-involving matter in the granule-size space 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, within the granule-size analogy invoked, matches the quantum limit of h/2πμ (μ-average mass of a granule; h-the Planck's constant) for the diffusion coefficient of the aggregation, first proposed by Fürth in 1933 and qualitatively foreseen by Schrödinger some years before, with both in the context of a diffusing particle. The classical quantum passage uncovered here, also termed insightfully as 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 entropy-productive (nano)aggregation of interest.
Keywords: Fokker–Planck and diffusion-type equation; Heisenberg-type relation for the granule evolution; granule evolution; matter aggregation; mesoscale; nanoscale; nanostructure formation; quantum-size effect; stochastic quantization.