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 method

In most texts on statistical mechanics the statistical distribution functions ${\displaystyle f}$ 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 ${\displaystyle f_{i}}$ which is a measure of the fraction of those states which are actually occupied by elements, is given by ${\displaystyle f_{i}=n_{i}/g_{i}}$ or ${\displaystyle n_{i}=f_{i}g_{i}}$, where ${\displaystyle g_{i}}$ is the degeneracy of energy level ${\displaystyle i}$ of energy ${\displaystyle \varepsilon _{i}}$ and ${\displaystyle n_{i}}$ is the number of elements occupying this level (e.g. in Fermi–Dirac statistics 0 or 1). Total energy ${\displaystyle E}$ and total number of elements ${\displaystyle N}$ are then given by ${\displaystyle E=\sum _{i}n_{i}\varepsilon _{i}}$ and ${\displaystyle N=\sum n_{i}}$.

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.^[8]

Classical statistics

For ${\displaystyle N=\sum _{i}n_{i}}$ independent elements with ${\displaystyle n_{i}}$ on level with energy ${\displaystyle \varepsilon _{i}}$ and ${\displaystyle E=\sum _{i}n_{i}\varepsilon _{i}}$ for a canonical system in a heat bath with temperature ${\displaystyle T}$ we set

${\displaystyle Z=\sum _{\text{arrangements}}e^{-E/kT}=\sum _{\text{arrangements}}\prod _{i}z_{i}^{n_{i}},\;\;\;z_{i}=e^{-\varepsilon _{i}/kT}.}$

The average over all arrangements is the mean occupation number

${\displaystyle (n_{i})_{\text{av}}={\frac {\sum _{j}n_{j}Z}{Z}}=z_{j}{\frac {\partial }{\partial z_{j}}}\ln Z.}$

Insert a selector variable ${\displaystyle \omega }$ by setting

${\displaystyle Z_{\omega }=\sum \prod _{i}(\omega z_{i})^{n_{i}}.}$

In classical statistics the ${\displaystyle N}$ elements are (a) distinguishable and can be arranged with packets of ${\displaystyle n_{i}}$ elements on level ${\displaystyle \varepsilon _{i}}$ whose number is

${\displaystyle {\frac {N!}{\prod _{i}n_{i}!}},}$

so that in this case

${\displaystyle Z_{\omega }=N!\sum _{n_{i}}\prod _{i}{\frac {(\omega z_{i})^{n_{i}}}{n_{i}!}}.}$

Allowing for (b) the degeneracy ${\displaystyle g_{i}}$ of level ${\displaystyle \varepsilon _{i}}$ this expression becomes

${\displaystyle Z_{\omega }=N!\prod _{i=1}^{\infty }\left(\sum _{n_{i}=0,1,2,\ldots }{\frac {(\omega z_{i})^{n_{i}}}{n_{i}!}}\right)^{g_{i}}=N!e^{\omega \sum _{i}g_{i}z_{i}}.}$

The selector variable ${\displaystyle \omega }$ allows one to pick out the coefficient of ${\displaystyle \omega ^{N}}$ which is ${\displaystyle Z}$. Thus

${\displaystyle Z={(\underset{}{\sum <!--\; \sum \; -->}}^{}}$Zdroj:https://en.wikipedia.org?pojem=Darwin-Fowler_method
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