Mean Values, Moments, Moment Ratios and a Generalized Mean Value Theorem for Size Distributions

Abstract

Generalized mean values of size distributions are defined via the general power mean, using Kronecker's delta to allow for the geometric mean. Special cases of these generalized mean values are the superarithmetic, arithmetic, geometric, harmonic and subharmonic means of number-, length-, surface-, volume-and intensity-weighted distributions. In addition to these special cases, however, our generalized r-weighted k-mean allows for non-integer values of k, which can be an advantage for describing material responses or effective properties of heterogeneous materials or disperse systems that are determined in a different way by different parts of a size distribution. For these generalized mean values a theorem is proved, which contains Herdan's theorem as a special case and turns out to be identical to Alderliesten's symmetry relation for moment ratios. In contrast to the moment-ratio notation, however, the interpretation of our notation is simple, intuitive and self-evident.

Universal Origin of Boson Peak Vibrational Anomalies in Ordered Crystals and in Amorphous Materials

“…Our model might be able to successfully describe also the anomalies in the thermal transport and heat capacity related to the boson peak [3]. In the future, it would be interesting to extend our formalism to liquids and in particular to the recently discovered gapped dispersion The elastic moduli are taken to obey the law ∼ α1 ρ + α2 ρ 2 in accordance with the experimental fits of [47,48]. For the damping Γ we assumed a dependence ∼ ρ as derived in [46] for isotropic solids.…”

“…In this case, importantly, the main contribution to the damping Γ is expected to come mainly from structural disorder: hence, this example illustrates the generality of the proposed framework. We take the phonon damping Γ to be proportional to the density ρ, as derived for isotropic solids in [46], and the elastic moduli to be described by ∼ α 1 ρ + α 2 ρ 2 as observed experimentally for densified silica in [47,48]. Taking into account that the Debye wavenumber q D is, by definition, proportional to the cubic root of the density ρ, we plot our results in Fig.4.…”