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 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. Total energy and total number of elements are then given by and. The Darwin–Fowler method has been treated in the texts of E. Schrödinger, Fowler and Fowler and E. A. Guggenheim, of K. Huang, and of H. J. W. Müller–Kirsten. The method is also discussed and used for the derivation of Bose–Einstein condensation in the book of.
Classical statistics
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 statisticsthe elements are distinguishable and can be arranged with packets of elements on level whose number is so that in this case Allowing for the degeneracy of level this expression becomes The selector variable allows to pick out the coefficient of which is. Thus and hence 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.
Quantum statistics
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. In the case of Fermi–Dirac statistics we have per state. There are states for energy level. Hence we have In the case of Bose–Einstein statistics we have By the same procedure as before we obtain in the present case But Therefore Summarizing both cases and recalling the definition of, we have that is the coefficient of in where the upper signs apply to Fermi–Dirac statistics, and the lower signs to Bose–Einstein statistics. Next we have to evaluate the coefficient of in In the case of a function which can be expanded as the coefficient of is, with the help of the residue theorem of Cauchy, We note that similarly the coefficient in the above can be obtained as where Differentiating one obtains and One now evaluates the first and second derivatives of at the stationary point at which. This method of evaluation of around the saddle point is known as the method of steepest descent. One then obtains We have and hence . 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 . 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.
