Concentration distribution of trace elements: from normal distribution to Lévy flights

Abstract

The paper discusses a nature of concentration distributions of trace elements in biomedical samples, which were measured by using the X-ray fluorescence techniques (XRF, TXRF). Our earlier observation, that the lognormal distribution well describes the measured concentration distribution is explained here on a more general ground. Particularly, the role of random multiplicative process, which models the concentration distributions of trace elements in biomedical samples, is discussed in detail. It is demonstrated that the lognormal distribution, appearing when the multiplicative process is driven by normal distribution, can be generalized to the so-called log-stable distribution. Such distribution describes the random multiplicative process, which is driven, instead of normal distribution, by more general stable distribution, being known as the Lévy flights. The presented ideas are exemplified by the results of the study of trace element concentration distributions in selected biomedical samples, obtained by using the conventional (XRF) and (TXRF) X-ray fluorescence methods. Particularly, the first observation of log-stable concentration distribution of trace elements is reported and discussed here in detail.