Logarithm Normal Distribution

The normal distribution function extends to both the positive and negative sides. To avoid the negative molecular weights (which do not exist), an assumption is made that the logarithm of the molecular weight is normally distributed. Thus, we replace x by In X and m by In m. Then the weight distribution becomes 

---

It is demonstrated in Figure 22.11 that the quasi-static stress-strain cycles at different prestrains of silica-filled rubbers can be well described in the scope of the above-mentioned dynamic flocculation model of stress softening and filler-induced hysteresis up to large strain. Thereby, the size distribution < ( ) has been chosen as an isotropic logarithmic normal distribution (< ( i) = 4> ) = A( 3)) ... 

Of interest for analytical chemistry are at least two further distributions, the logarithmic normal distribution for analytical results at the trace- and ultra-trace level, and the Poisson distribution for discrete results (e.g., counts of impulse summator in XRF). 

In the first case it is not the measured values themselves that are normally distributed but their logarithms. Consequently, the parameters of logarithmic-normal-distributed results are estimated as geometric mean... 

Two approaches attempt to solve this problem. The one is a transformation of the data to logarithms prior to the statistical calculations corresponding to a logarithmic normal distribution. The other is a modification of the z-scores with correction factors. This method was introduced first in a German standard for proficiency testing (DIN 38402 - 45), which in the meantime partially was transferred into ISO/TS 20612. 

Special distribution functions are specified in some standards (e.g., power distribution, logarithmic normal distribution, and RRSB distribution). Methods of determination for pigments are rated in Section 1.2.2. 

For anisometric particles (e.g., needle- or platelet-shaped particles) mathematical statistics may likewise be applied [1.10]. The two-dimensional logarithmic normal distribution of the length L and breadth B of the particles also allows the representation and calculation of the characteristic parameters and mean values. The eccentricity of the calculated standard deviation ellipse (Fig. 2) is a measure of the correlation between the length and breadth of the particle. By using more than two... 

Figure 2. Standard deviation ellipses of a logarithmic normal distribution (yellow iron oxide pigment)...

Fractionation Data and Distribution Analysis of the HEC Before Hydrolysis. The results of the fractionation of HEC not subjected to cellulase attack are given in Table I. It appeared that the distribution of these fractionation data could be described by the Lansing-Kraemer distribution (39), also known as the logarithmic normal distribution, i.e. ... 

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... 

Fig. 5 Pore size distribution function of microporous membranes, parameterization with logarithmic normal distribution, for the parameter sets specified in Table 4. (a) Differential psd,...

The developed approach will be applied to model membranes whose psds closely resemble those of Nafion and similar PFSI membranes [60]. A common parameterization of experimental data for ultrafiltration membranes is given by the so-called logarithmic normal distribution [59],... 

A Empirically determined constant — — < (z) Logarithmic normal distribution —... 

Model Distribution While a PSD with n intervals is represented by 2n + 1 numbers, further data reduction can be performed by fitting the size distribution to a specific mathematical model. The logarithmic normal distribution or the logarithmic normal probability function is one common model distribution used for the distribution density, and it is given by... 

Fig. 6.3 Standardized cumulative volume distribution g3(dp/d5o) as a function of in the form of logarithmic normal distribution for the test conditions given in Fig. 6.2 from [166]... 

The above-discussed particle size distributions are shown, for the sake of comparison, as logarithmic normal distributions in Fig. 6.8 for the 6-blade turbine stirrer and different material systems. It emerges, that, with the exception of car-nauba wax, all organic liquids in the range investigated behaved similarly in this... 

Size and shape of the embedded particles can be varied during film deposition. They mainly depend on the amount of the evaporated metal, the pressure during the deposition, and the distance of the sample from the metal evaporation source. The particles have a broad size distribution that can be described in principle with a logarithmic-normal distribution [6]. 

Now, consider the parametric method. As was already mentioned, the underlying assumption is that the distribution belongs to a certain class. The choice of the distribution class can be made on the basis of general considerations about the kind of distribution that is likely to be formed in a specific physical process. For example, for a turbulent flow of emulsion in a pipe, the distribution can be described with sufficient accuracy by a logarithmic normal distribution or a gamma-distribution. Consider these two cases successively. 

In the considered case, the basic mechanisms of formation of droplets in the turbulent gas flow are processes of coagulation and breakage of drops. These two processes proceed simultaneously. As a result, the size distribution of the drops is established. Assuming homogeneity and isotropy of the turbulent flow, this distribution looks like a logarithmic normal distribution [1] ... 

To close the system of equations represented by Eq. (15.41) it is necessary to express the right-hand part in terms of moments. To this end, the coagulation kernel should have a special form (for example, the form of a homogeneous polynomial of degrees V and co), or it is necessary to accept that the distribution conforms to a certain class (for example, a logarithmic normal distribution or a gamma distribution). The first method is called the method of fractional moments, and the second one the parametric method. 

---