Statistical Description of Mixing

As mixing progresses, the black and white particles intermingle to a greater extent, and, after sufficient time passes, the mixture obtains its random mixing status. At this point the probability of finding a white particle at any point is constant and equal to the overall fraction of the white particles. If a number of samples of equal size are extracted from the mixture, the fraction of the white particles will vary from sample to sample. The mean value over all samples studied should equal the overall fraction of white particles and the distribution of values should follow the binomial or an equivalent distribution. 

Consider a sample, randomly extracted from the mixture, which contains n particles, where n is large enough for statistical treatment and small enough compared to the total number of particles in the mixture itself. Let p equal the fraction of black particles in the entire mixture. Then the probability P that this randomly selected sample has exactly b black particles (x = b/n) is given by the binomial (or Bernoulli) distribution (Spiegel, 1991)  

Similarly, if there are N samples tested, where A is a very large number, then the variance of the binomial distribution, for these samples, is given by the formula 

This variance is a measure of how much the concentration differs from the mean value. The procedure to determine whether or not the mixture is grossly uniform is obvious from the above discussion samples are extracted from the mixture, their average concentration is calculated, and finally the concentration distribution is checked against the binomial distribution. If there is a match between these two distributions, the mixture is considered to be grossly uniform or a random mixture. In the limit of zero variance the mixture attains the uniform state. 

The binomial distribution is considered to approach the normal (or Gaussian) distribution if the number of 

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The statistical description of mixing given in this section is restricted to a two-component system for simplicity. Of course, a four-component system can be considered as a... 

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