The Kinetic Method for Fermi-Dirac and Einstein-Bose Statistics

The Kinetic Method for Fermi-Dirac and Einstein-Bose Statistics. The arguments of the preceding sections must be modified in only two ways to change from the Boltzmann statistics to the Fermi-Dirac or Einstein-Bose statistics. In the first place, the law giving the number of collisions per unit time, Eq. (1.1), must be changed. Secondly, as 

In the Einstein-Bose statistics, there is no such clear physical way to find the revised law of collisions as in the Fermi-Dirac statistics. The law can be derived from the quantum theory but not in a simple enough way to describe here. In contrast to the Fermi-Dirac statistics, in which the presence of one molecule in a cell prevents another from entering the same cell, the situation with the Einstein-Bose statistics is that the presence of a molecule in a cell increases the probability that another one should enter the same cell. In fact, the number of molecules going into the fcth cell per second turns out to have a factor (1 + Nk), increasing linearly with the mean number Nk of molecules in that cell. Thus, the law of collisions for the Einstein-Bose statistics is just like Eq. (4.1), only with + signs replacing the — signs. In fact, we may write the law of collisions for both forms of statistics in the form 

we must consider the change of the mean number of molecules in the ith state, with time. Using the law of collisions (4.2) and proceeding as in the derivation of Eq. (1.6), we have at once 

Having found the number of collisions, we can find the change in entropy per unit time. Using the formulas (2.7) and (2.11) of Chap. V for the entropy in the case of Fermi-Dirac and Einstein-Bose statistics, we find at once that 

As in Sec. 2, we can write four expressions equivalent to Eq. (4.5), by interchanging the various indices i, j, k, l. Adding these four and dividing by four, we obtain 

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