Particle size, statistics Gaussian distribution

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

Particle size, like other variables in nature, tends to follow well-defined mathematical laws in its distribution. This is not only of theoretical interest since data manipulation is made much easier if the distribution can be described by a mathematical law. Experimental data tends to follow the Normal law or Gaussian frequency distribution in many areas of statistics and statistical physics. However, the log-normal law is more frequently found with particulate systems. These laws suffer the disadvantage that they do not permit a maximum or minimum size and so, whilst fitting real distributions in the middle of the distribution, fail at each of the tails. 

An aerosol rarely consists of particles that are the same size, and usually a distribution of sizes around a mean is observed. The observed data may be fitted by statistical approximation to a distribution. The number of particles in a size range when plotted against the logarithm of the particle diameters frequently exhibit a normal (Gaussian) distribution. This is known as a log-normal distribution and is described by a parameter known as the geometric standard deviation. Theoretically, a monodisperse aerosol will exhibit a geometric standard deviation of 1 in practice, however, an accepted limit is 1.2 [6]. 

Log normal distribution (logarithmic normal distribution). A statistical probabiUty-density function, characterized by two parameters, that can sometimes provide a faithful representation of a polymer s molecular-weight distribution or the distribution of particle sizes in ground, brittle materials. It is a variant of the familiar normal or Gaussian distribution in which the logarithm of the measured quantity replaces the quantity itself. Its mathematical for is... 

The normal distribution is a symmetrical bell-shaped curve referred to in statistics as a Gaussian curve. It is a two-parameter function, one parameter is the mean, Xa which due to the symmetry of the curve coincides with the mode and median, and the other is the standard deviation a, which is a measure of the width of the distribution. The normal distribution of particle size is given by... 

Because of its mathematical properties, the standard deviation a is almost exclusively used to measure the dispersion of the partiele size distribution. When the skewed particle size distribution shown in Fig. 9 is replotted using the logarithm of the particle size, the skewed curve is transformed into a symmetrical bellshaped curve as shown in Fig. 10. This transformation is of great significance and importance in that a symmetrical bell-shaped distribution is amenable to all the statistical procedures developed for the normal or gaussian distribution. 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

Two other statistical diameters are often encountered, viz. the modal and median diameters both are determined from frequency plots (size interval versus number of particles in each interval). The modal diameter is the diameter at the peak of the frequency curve, whereas the median diameter defines a midpoint in the distribution - half the total number of particles are smaller than the median, half are larger. If the distribution curve obeys the Gaussian or Normal Error law, the median and modal diameters coincide. 

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