Logarithmic-normal law of distribution of matter and its properties
Abstract
Considering various distributions of natural qualities and properties, mainly from the field of geological sciences, the author came to the idea of replacing one argument with another functionally related to it. With such a replacement, the type of distribution function changes; for example, symmetrical forms of distribution become asymmetrical and vice versa. The functional transformation of variables leads to the identification of families of distribution functions that reflect the same natural process in different forms. Among these forms, there may also be simpler ones, for instance symmetrical forms, and finding them is advisable. The author poses a general problem: what should be the type of relationship between mutually replaceable arguments so that an asymmetric monotonic form is reduced to a Gaussian function, and in such a way that certain constants remain unchanged? The solution to this problem establishes that the form of the function is logarithmic. A particular form of this relationship will have the expression y = lg(x - a) + c. The author deduces some properties of the logarithmic distribution and points out numerous cases (in geochemistry, petrology, geology, mineral processing, mineral reserve estimation, distribution of matter during sedimentation, hydraulic classification, etc.) where asymmetric distributions can be transformed into Gaussian form by a logarithmic transformation.
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