On the origin of logarithmic-normal distributions: An analytical derivation, and its application to nucleation and growth processes

Abstract

The logarithmic-normal (lognormal) distribution is one of the most frequently observed distributions in nature and describes a large number of physical, biological and even sociological phenomena. The origin of this distribution is therefore of broad interest but a general derivation from basic principles is still lacking. Using random nucleation and growth to describe crystallization processes we derive the time development of grain size distributions. Our derivation provides, for the first time, an analytical expression of the size distribution in the form of a lognormal type distribution. We apply our results to the grain size distribution of solid phase crystallized Si-films. The logarithmic-normal (lognormal) size distribution is one of the most frequently encountered distributions in nature [1,2]. It has been shown early on that a large number of technical processes and even sociological phenomena such as income distributions [2] or the productivity of researchers [3] follow

Flexible bacterial cellulose / permalloy nanocomposite xerogel sheets – Size scalable magnetic actuator-cum-electrical conductor

“…Such a particle size distribution is commonly observed in growth kinetics controlled nanoparticles formation. 18 An energy dispersive x-ray spectra obtained from the xerogel shows peaks corresponding to Fe, Ni, O and C indicating absence of other elemental impurities. The ratio of the elements Fe to Ni in several different areas of the nanocomposite is found to be ∼ 1:3, confirming the formation of FeNi 3 phase.…”

Estimation on Reliability Models of Bearing Failure Data

“…In particular, the Weibull distribution and log-normal distribution are widely used in reliability theoretical analysis. Though this has achieved certain results, they show large error and low precision in the process of product reliability estimation [10][11][12][13][14][15]. The accuracy of reliability models increasingly got the attention of scholars and experts.…”

The self-preserving size distribution of fractal aggregates coagulating by Brownian motion and simultaneous fluid shear at low Peclet numbers: Numerical solutions

“…It is interesting that whereas the average size and the width of an aerosol size distribution strongly depend upon the chemical composition, the generation and the age of each aerosol, the basic shape of an aerosol size distribution is usually very close to a log-normal distribution or to a linear combination of different lognormal distributions (Seinfeld & Pandis, 2006). However, there is hardly any theoretically and statistically detailed explanation for this practice although there are some investigations on this problem for aerosol particles undergoing Brownian coagulation (Friedlander & Wang, 1966;Otto, Fissan, Parkt, & Lee 1999) and for the nucleation and growth processes (Bergmann & Bill, 2008;Kiss, Söderlund, Niklasson, & Granqvist, 1999).…”