The Stirling approximation, , is used in the literature to derive the exponential Boltzmann distribution. We generalize the latter for a finite number of particles by applying the more exact Stirling formula and the exact function . A more accurate and analytical formulation of Boltzmann statistics is found in terms of the Lambert W-function. The Lambert-Boltzmann distribution is shown to be a very good approximation to the exact result calculated by numerical inversion of the Digamma-function. For a finite number of particles the exact distribution yields results that differ from the usual exponential Boltzmann distribution. As an example, the exact Digamma-Boltzmann distribution predicts that the constant-volume heat capacity of an Einstein solid decreases with decreasing . The exact Digamma-Boltzmann distribution imposes a constraint on the maximum energy of the highest populated state, consistent with the finite total energy of the microcanonical ensemble.
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January 2009
PAPERS|
January 01 2009
Revision of Boltzmann statistics for a finite number of particles
S. Kakorin
S. Kakorin
a)
Biophysical Chemistry, Department of Chemistry,
University of Bielefeld
, P.O. Box 100 131, D-33501 Bielefeld, Germany
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Electronic mail: sergej.kakorin@uni-bielefeld.de
Am. J. Phys. 77, 48–53 (2009)
Article history
Received:
January 19 2008
Accepted:
July 11 2008
Citation
S. Kakorin; Revision of Boltzmann statistics for a finite number of particles. Am. J. Phys. 1 January 2009; 77 (1): 48–53. https://doi.org/10.1119/1.2967703
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