Darwin–Fowler method

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In statistical mechanics, the Darwin–Fowler method is used for deriving the distribution functions with mean probability. It was developed by Charles Galton Darwin and Ralph H. Fowler in 1922-1923.[1][2]

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.

Darwin–Fowler methodEdit

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[2] 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  .

The Darwin–Fowler method has been treated in the texts of E. Schrödinger,[3] Fowler[4] and Fowler and E.A. Guggenheim,[5] of K. Huang,[6] and of H.J.W. Müller–Kirsten.[7] The method is also discussed and used for the derivation of Bose–Einstein condensation in the book of R.B. Dingle [de].[8]

Classical statisticsEdit

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 statisticsEdit

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   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.

Further readingEdit

  • Mehra, Jagdish; Schrödinger, Erwin; Rechenberg, Helmut (2000-12-28). The Historical Development of Quantum Theory. Springer Science & Business Media. ISBN 9780387951805.

ReferencesEdit

  1. ^ "Darwin-Fowler method". Encyclopedia of Mathematics. Retrieved 2018-09-27.
  2. ^ a b C.G. Darwin and R.H. Fowler, Phil. Mag. 44(1922) 450–479, 823–842.
  3. ^ E. Schrödinger, Statistical Thermodynamics, Cambridge University Press (1952).
  4. ^ R.H. Fowler, Statistical Mechanics, Cambridge University Press (1952).
  5. ^ R.H. Fowler and E. Guggenheim, Statistical Thermodynamics, Cambridge University Press (1960).
  6. ^ K. Huang, Statistical Mechanics, Wiley (1963).
  7. ^ H.J.W. Müller–Kirsten, Basics of Statistical Physics, 2nd ed., World Scientific (2013), ISBN 978-981-4449-53-3.
  8. ^ R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press (1973); pp. 267–271.