Distributions and the Problem of Small Numbers

If a large number of repeat observations on one and the same sample are plotted, most fall within a narrow interval around the mean, and a decreasing number is found further out. The familiar term bell curve is appropriate. (See Fig. 1.9.) 

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When a particle is illuminated by a beam of light with specified characteristics, the amount and angular distribution of the light scattered by the particle, as well as the amount absorbed, depends in a detailed way on the nature of the particle, that is, its shape, size, and the materials of which it is composed. This presents us with an almost unlimited number of distinct possibilities. Nevertheless, there are some features common to the phenomena of scattering and absorption by small particles. In this chapter, therefore, our goal is to say as much as possible about such phenomena without invoking any specific particle. This will establish the mathematical and physical framework underlying all the specific problems encountered in later chapters. 

Here, cp is a random number for the calculation step i . It is given by a standard procedure for the normal distribution values with a mean value of zero where Dj-Aj and F) are the corresponding Dj-a and F values for the j bin and the particle position i . The only limitation of the numerical method is concentrated in the fact that Ax must have very small values in order to eliminate all the problems of non-convergence caused by the second term on the right half of the equation (4.119). 

As mentioned above, for the first time this problem was pointed out in the work [14]. The authors related the generation of bound states to the ion collisions. Recently, this idea was used to take into consideration the presence of trapped ions in both analytical [21, 22] and numerical [23] studies. These papers give answers to a number of important questions, but many aspects of the problem still remain open. In particular, the bound ion distributions found in [21, 22] in the approximation of small collision frequency on the basis of calculations of free and bound ion balance, do not exhaust the variety of many other distributions which could exist in the absence of collisions. 

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