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.

Keywords

• size distributions (number-, length-, surface-, volume-weighted)
• mean size (superarithmetic, arithmetic, geometric, harmonic and subharmonic)
• moments
• moment ratios
• herdan's theorem