The Stirling approximation, ln(N!)Nln(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.

1.
Sean A. C.
McDowell
, “
A simple derivation of the Boltzmann distribution
,”
J. Chem. Educ.
76
,
1393
1394
(
1999
).
2.
D. K.
Russel
, “
The Boltzmann distribution
,”
J. Chem. Educ.
73
,
299
300
(
1996
).
3.
A. S.
Wallner
and
K. A.
Brandt
, “
The validity of Stirling’s approximation
,”
J. Chem. Educ.
76
,
1395
1397
(
1999
).
4.
G.
Marsaglia
, and
J. C. W.
Marsaglia
, “
A new derivation of Stirling’s approximation to n!
,”
Am. Math. Monthly
9
,
826
829
(
1990
).
5.
H.
Margenau
and
G. M.
Murphy
,
The Mathematics of Physics and Chemistry
, 2nd ed. (
Van Nostrand
,
New York
,
1965
), pp.
438
441
.
6.
D. A.
Barry
,
J. Y.
Parange
,
L.
Li
,
H.
Prommer
,
C. J.
Cunningham
, and
F.
Stagnitti
, “
Analytical approximations for real values of the Lambert W-function
,”
Math. Comput. Simul.
53
,
95
103
(
2000
).
7.
P.
Atkins
and
J.
De Paula
,
Physical Chemistry
, 8th ed. (
Oxford University Press
, Oxford,
2006
), pp.
560
576
.
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