Bose statistic

N is very large since the fluctuations around the average behave as 2. A quantum ideal gas with either Fenni or Bose statistics is treated in subsection A2.2.5.4. subsection A2.2.5.5. subsection A2.2.5.6 and subsection A2.2.5.7. 

The first temi is the classical ideal gas temi and the next temi is the first-order quantum correction due to Femii or Bose statistics, so that one can write... 

MSN.83. 1. Prigogine and A. P. Grecos, Quantum theory and dissipativity, in Proceedings, International Research Symposium on Statistical Physics with special sessions on Topics Related to Bose Statistics, University of Calcutta, 12, Suppl. 1, 177-184 (1975). 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

Assemblies in the Molecular Phase Space.—When we describe a system by giving the Ni s, the numbers of molecules in each cell of the molecular phase space, we automatically avoid the difficulties described in the last section relating to the identity of molecules. We now meet immediately the distinction between the Fcrmi-Dirac, the Einstein-Bose, and the classical or Boltzmann statistics. In the Einstein-Bose statistics, the simplest form in theory wa art. up a complexion bv giving a set of Nj s, and we say that any possible set of Ni s, subject only to the obvious restriction... 

In Eqs. (2.7) and (2.11), we have found the general expressions for the entropy in the Fermi-Dirac and Einstein-Bose statistics. From either one, we can find the entropy in the Boltzmann statistics by passing to the limit in which all N% s are very small compared to unity. For small N%, In (1 Nt) approaches iVt-, and (1 Nt) can be replaced by unity. Thus either Eq. (2.7) or (2.11) approaches... 

Fig. V-2.—Distribution functions for Fermi-Dirac statistics (a) Maxwell-Boltzmann statistics (b) and Finstoin-Bose statistics (c).

We now ask, how many collisions per second are there in which molecules in the ith and jth colls disappear and reappear in the fcth and Zth cells We can be sure that this number of collisions will be proportional both to the number of molecules in the ith and to the number of molecules in the jth cell. This is plain, since doubling the number of either type of molecule will give twice as many of the desired sort that can collide, and so will double the number of collisions per unit time. In the case of the Boltzmann statistics, which we first consider, the number of collisions will be independent of the number of molecules in the kth and Zth cells, though we shall find later that this is not the case with the Fermi-Dirac and Einstein-Bose statistics. We can then write the number of collisions of... 

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... 

The interesting feature of Eq. (2.11) is that the induced and spontaneous emission combine into a factor as simple as Nv + 1. This is strongly suggestive of the factors N3- + 1, which we met in the probability of transition in the Einstein-Bose statistics, Eq. (4.2), of Chap. VI. As a matter of fact, the Einstein-Bose statistics, in a slightly modified form, applies to photons. Since it does not really contribute further to our understanding of radiation, however, we shall not carry through a discussion of this relation, but merely mention its existence. 

---