Darwin–Fowler method

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In statistical mechanics, the Darwin–Fowler method is used for deriving the distribution functions with mean probability.

Distribution functions 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 with a large number of elements, as in statistical mechanics, the results are the same as with maximization.

Darwin–Fowler method

In most texts on statistical mechanics the statistical distribution functions (average number of particles 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 R. H. Fowler[1] 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.

The Darwin–Fowler method has been treated in the texts of Schrödinger,[2] Fowler[3] and Fowler and Guggenheim,[4] by Huang,[5] and Müller–Kirsten.[6] The method is also discussed and used for the derivation of Bose–Einstein condensation in the book of R. B. Dingle.[7]

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 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

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

or

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

(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 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.

External links

References

  1. ^ C.G. Darwin and R.H. Fowler, Phil. Mag. 44(1922) 450–479, 823–842.
  2. ^ E. Schrödinger, Statistical Thermodynamics, Cambridge University Press (1952).
  3. ^ R.H. Fowler, Statistical Mechanics, Cambridge University Press (1952).
  4. ^ R.H. Fowler and E. Guggenheim, Statistical Thermodynamics, Cambridge University Press (1960).
  5. ^ K. Huang, Statistical Mechanics, Wiley (1963).
  6. ^ H.J.W. Müller–Kirsten, Basics of Statistical Physics, 2nd ed., World Scientific (2013), ISBN 978-981-4449-53-3.
  7. ^ R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press (1973); pp. 267–271.
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