Assemblies in the Molecular Phase Space

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 

We shall now rewrite Eq. (2.3), using Stirling s theorem. This states that, for a large value of N, 

Stirling s formula is fairly accurate for values of N greater than 10 for still larger N% where JV and (N/c)N are very large numbers, the factor /27rN is so near unity in proportion that it can be omitted for most purposes, so that we can write Nl simply as (N/e)N. Adopting this approximation, we can rewrite Eq. (2.3) as 

Equation (2.5) is of an interesting form being a quantity independent of G, raised to the G power, we may interpret it as a product of terms, one for each cell of the molecular phase space. Now to get the whole number of complexions for the system, we should multiply quantities like (2.5), for each group of G cells in the whole molecular phase space. Plainly this will give us something independent of the exact way we divide up the cells into groups, or independent of G, and we find 

For the Einstein-Bosc statistics, we wish the number of ways of arranging NXG molecules in G cells, allowing as many molecules as we please in a cell. This number can be shown to be 

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