Lognormal distribution function

Coalescence Growth Mechanism. Following the very early step of the growth represented by Eq. (1), many nuclei exist in the growth zone. Hence Eq. (2) would be a major step for the crystal growth. Since there are many nuclei and embryos with various sizes in the zone, Uy in Eq. (2) can be assumed to be a random variable. Due to mathematical statistics, the fraction of volume approaches a Gaussian after many coalescence steps (3). A lognormal distribution function is defined by... 

Figure 4.5. Modeling of volume-averaging effect with the lognormal distribution function on the real parts of the linear (a) and cubic (b) susceptibilities.

Aerosol size distributions often have large standard deviations caused by a long upper "tail for particles larger than the peak in the distribution. Such distributions can be represented approximately by the lognormal distribution function ... 

The lognormal distribution function can be interpreted physically as the result of a process of breakup of larger particles at rates that are normally distributed with respect to particle size (Aitchison and Brown, 1957). Approximately lognormal distributions also result when the aerosol size distribution is controlled by coagulation (Chapter 7). In this case the value of the standard deviation is determined by the form of the particle collision frequency function. Multimodal aerosols may result when particles from several different types of sources are mixed. Such distributions are often approximated by adding lognormal distributions, each of which corresponds to a mode in the observed distribution and to a particular type of source. 

Verify that the nonnal and lognormal distribution functions defined in the text in equations (1.26) and (1,27) are dimensionally correct. 

Aitchison, J., and Brown, J. A. C. (1957) Tire Lognormal Distribution Function, Cambridge University Press, Cambridge, UK. 

Growth of a Lognormal Aerosol Size Distribution Aerosol size distribution data are sometimes represented by the lognormal distribution function [see also (8.33)]... 

They were able to fit experimental modulus data for a calcium-potassium nitrate melt and a lithium aluminosilicate glass using a double lognormal distribution function. 

The result of fragility analysis is the fragility function, which represents the probability that an engineering demand parameter (EDP) will exceed a certain limit-state value edp given an intensity IM = im. In the simplest case, it is assumed that the seismic fragility function is based on a lognormal distribution function ... 

If, for example, F(s) is taken as a cumulate lognormal distribution function, dF(s)lds is the lognormal probability density function, i.e.,... 

The fragility curve is defined by a median value of the engineering demand parameter (EDP), which is, for example, spectral displacement in HAZUS and maximum interstory drift ratio (IDR) in Jayaram et al. 2012, and by the variability associated with the estimation of that damage state for that EDP. The median EDP corresponds to the threshold of the damage state. The curve is assumed to follow the cumulative lognormal distribution function as shown below ... 

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