Revision of Boltzmann statistics for a finite number of particles

Kakorin S (2009)
AMERICAN JOURNAL OF PHYSICS 77(1): 48-53.

Journal Article | Published | English

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Abstract
The Stirling approximation, ln(N!)approximate to N ln(N)-N, 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 ln(N!). 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 N 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 N. 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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Kakorin S. Revision of Boltzmann statistics for a finite number of particles. AMERICAN JOURNAL OF PHYSICS. 2009;77(1):48-53.
Kakorin, S. (2009). Revision of Boltzmann statistics for a finite number of particles. AMERICAN JOURNAL OF PHYSICS, 77(1), 48-53.
Kakorin, S. (2009). Revision of Boltzmann statistics for a finite number of particles. AMERICAN JOURNAL OF PHYSICS 77, 48-53.
Kakorin, S., 2009. Revision of Boltzmann statistics for a finite number of particles. AMERICAN JOURNAL OF PHYSICS, 77(1), p 48-53.
S. Kakorin, “Revision of Boltzmann statistics for a finite number of particles”, AMERICAN JOURNAL OF PHYSICS, vol. 77, 2009, pp. 48-53.
Kakorin, S.: Revision of Boltzmann statistics for a finite number of particles. AMERICAN JOURNAL OF PHYSICS. 77, 48-53 (2009).
Kakorin, Sergej. “Revision of Boltzmann statistics for a finite number of particles”. AMERICAN JOURNAL OF PHYSICS 77.1 (2009): 48-53.
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