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Distribution functions are used in statistical physics to estimate the mean number of particles occupying an energy level (hence also called occupation numbers). These distributions are mostly derived as those numbers for which the system under consideration is in its state of maximum probability. But one really requires average numbers. These average numbers can be obtained by the Darwin-Fowler method. Of course, for systems in the thermodynamic limit (large number of particles), as in statistical mechanics, the results are the same as with maximization.
In most texts on statistical mechanics the statistical distribution functions in Maxwell-Boltzmann statistics, Bose-Einstein statistics, Fermi-Dirac statistics) are derived by determining those for which the system is in its state of maximum probability. But one really requires those with average or mean probability, although - of course - the results are usually the same for systems with a huge number of elements, as is the case in statistical mechanics. The method for deriving the distribution functions with mean probability has been developed by C. G. Darwin and Fowler and is therefore known as the Darwin-Fowler method. This method is the most reliable general procedure for deriving statistical distribution functions. Since the method employs a selector variable (a factor introduced for each element to permit a counting procedure) the method is also known as the Darwin-Fowler method of selector variables. Note that a distribution function is not the same as the probability - cf. Maxwell-Boltzmann distribution, Bose-Einstein distribution, Fermi-Dirac distribution. Also note that the distribution function which is a measure of the fraction of those states which are actually occupied by elements, is given by or , where is the degeneracy of energy level of energy and is the number of elements occupying this level (e.g. in Fermi-Dirac statistics 0 or 1). Total energy and total number of elements are then given by and .
For independent elements with on level with energy and for a canonical system in a heat bath with temperature we set
The average over all arrangements is the mean occupation number
Insert a selector variable by setting
In classical statistics the elements are (a) distinguishable and can be arranged with packets of elements on level whose number is
so that in this case
Allowing for (b) the degeneracy of level this expression becomes
The selector variable allows to pick out the coefficient of which is . Thus
This result which agrees with the most probable value obtained by maximization does not involve a single approximation and is therefore exact, and thus demonstrates the power of this Darwin-Fowler method.
We have as above
where is the number of elements in energy level . Since in quantum statistics elements are indistinguishable no preliminary calculation of the number of ways of dividing elements into packets is required. Therefore the sum refers only to the sum over possible values of .
We note that similarly the coefficient in the above can be obtained as
Differentiating one obtains
One now evaluates the first and second derivatives of at the stationary point at which . This method of evaluation of around the saddle pointis known as the method of steepest descent. One then obtains
We have and hence
(the +1 being negligible since is large). We shall see in a moment that this last relation is simply the formula
We obtain the mean occupation number by evaluating
This expression gives the mean number of elements of the total of in the volume which occupy at temperature the 1-particle level with degeneracy (see e.g. a priori probability). For the relation to be reliable one should check that higher order contributions are initially decreasing in magnitude so that the expansion around the saddle point does indeed yield an asymptotic expansion.