The Boltzmann entropy as traditionally presented in statistical mechanics textbooks is only a special case and not Boltzmann's fundamental definition. The difference becomes important when the traditional expression for the entropy is applied to colloids, for which it makes incorrect predictions. Boltzmann's original definition of the entropy in terms of the probabilities of states of composite systems leads to consistent and correct statistical mechanics and thermodynamics.
REFERENCES
1.
D. H.
Everett
, Basic Principles of Colloid Science
(Royal Society of Chemistry
, London
, 1988
).2.
See, for example,
L. D.
Landau
and E. M.
Lifshitz
, Statistical Physics
(Pergamon Press
, New York
, 1980
), pp. 24
–25
.3.
F.
Reif
, Fundamentals of Statistical and Thermal Physics
(McGraw-Hill
, New York
, 1965
), pp. 243
–245
.4.
R. K.
Pathria
, Statistical Mechanics
(Butterworth-Heinemann
, Boston
, 1996
), pp. 22
–26
. See especially Pathria's Eq. (2a) on page 23, which agrees with my Eq. (1) if the temperature is replaced by the energy as a variable.5.
J. W.
Gibbs
, Elementary Principles of Statistical Mechanics
(Yale University Press
, New Haven
, 1902
). (Reprinted by Dover
, New York
, 1960
), pp. 206
–207
.6.
R. H.
Swendsen
, “Statistical mechanics of classical systems of distinguishable particles
,” J. Stat. Phys.
107
, 1143
–1165
(2002
).7.
L.
Boltzmann
, Vorlesungen über Gastheorie
[J. A. Barth, Leipzig, Germany, 1896 (Part I) and 1898 (Part II)], translated into English by Stephen G. Brush in L. Boltzmann, Lectures on Gas Theory (Dover
, New York
, 1995
), p. 72
.8.
L.
Boltzmann
, “Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht
,” Wien. Ber.
76
, 373
–435
(1877
),reprinted in
Wissenschaftliche Abhand-lungen von Ludwig Boltzmann
(Chelsea
, New York
, 1968
), Vol. II
, pp. 164
–223
. The quotations are to be found on pp. 165 and 166.9.
Because I was not able to find English translations of Boltzmann’s papers, the translations in this paper are my own. I have tried to be faithful to Boltzmann’s meaning, but not to his sentence structure.
10.
N. G.
van Kampen
, “The Gibbs paradox
,” in Essays in Theoretical Physics
, edited by W. E.
Parry
(Pergamon
, Oxford
, 1984
). The importance of considering composite systems was also pointed out by van Kampen in this article.11.
C.
Cercignani
, Ludwig Boltzmann: The Man Who Trusted Atoms
(Oxford University Press
, New York
, 1998
), p. xi
.12.
M.
Planck
, “Über das Gesetz der Energieverteilung im Normalspektrum
,” Drudes Annalen
, 553
–562
(1901
), reprinted in Ostwalds Klassiker der exakten Wissenschaften, Band 206, “Die Ableitung der Strahlungsgesteze,”
pp. 65
–74
. The equation is given with an arbitrary additive constant on p. 68
of the reprinted text.13.
translated into English by
Morton
Masius
, in M. Planck, The Theory of Heat Radiation
(Dover
, New York
, 1991
), p. 119
.© 2006 American Association of Physics Teachers.
2006
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