Darwin–Fowler method
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]
External links
- [1]
- [2]
- [3]
- [4]]J.Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Springer, New York (1987), ISBN 0-387-95180-6]
References
- ^ C.G. Darwin and R.H. Fowler, Phil. Mag. 44(1922) 450–479, 823–842.
- ^ E. Schrödinger, Statistical Thermodynamics, Cambridge University Press (1952).
- ^ R.H. Fowler, Statistical Mechanics, Cambridge University Press (1952).
- ^ R.H. Fowler and E. Guggenheim, Statistical Thermodynamics, Cambridge University Press (1960).
- ^ K. Huang, Statistical Mechanics, Wiley (1963).
- ^ H.J.W. Müller–Kirsten, Introduction to Statistical Physics, 2nd ed., World Scientific (2013).
- ^ R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press (1973); pp. 267–271.