ENTROPY AND PROBABILITY 67Maxwell's work on kinetic theory only to the extent that it was discussed by Boltz-
mann in those same volumes. Thus Einstein did not know the true gaps in the
arguments of Maxwell and, especially, of Boltzmann; nor did he accidentally fill
them. The reading of Einstein's paper [E10] is not facilitated by the absence of
an explicit statement as to what, in his opinion, the gaps actually were. This paper
is devoted exclusively to thermal equilibrium. The statistical interpretation of tem-
perature, entropy, and the equipartition theorem are discussed. The tool used is
essentially (in modern terms) the canonical ensemble. The paper is competent and
neither very interesting nor, by Einstein's own admission [E2], very well written.
Einstein believed that in his next paper, completed in 1903 [Ell], he gave a
proof of the second law for irreversible processes. At this stage, he of course needed
some definition for the thermodynamic probability W, and it is here that he inde-
pendently introduced Boltzmann's first definition in terms of the time spent in the
appropriate interval in F space. His proof is logically correct but rests on an erro-
neous assumption: 'We will have to assume that more probable distributions will
always follow less probable ones, that is, that W always [my italics] increases until
the distribution becomes constant and Whas reached a maximum' [E49]. Three
days after he sent this paper to the Annalen der Physik, he wrote to Besso, 'Now
[this work] is completely clear and simple so that I am completely satisfied with
it' [E50]. He had been studying Boltzmann's book since 1901 [E51]. The book
does refer to the Loschmidt objection, but, in typical Boltzmann fashion, in a
somewhat tucked-away place [B16]. Einstein must have missed it; at any rate, it
is obvious that in 1903 he was unaware of the main subtlety in the proof of the
second law: the overwhelming probability, rather than the certainty, of entropy
increase.
It was not until 1910 that, for the first time, Einstein's 'derivation' was criticized
in the literature. At that time, Paul Hertz pointed out that 'if one assumes, as
Einstein did, that more probable distributions follow less probable ones, then one
introduces thereby a special assumption which is not evident and which is thor-
oughly in need of proof [HI]. This is a remarkable comment. Hertz does not say,
'Your assumption is wrong.' Rather, he asks for its proof. Here we have but one
example of the fact that, at the end of the first decade of the twentieth century,
Boltzmann's ideas had not yet been assimilated by many of those who were active
at the frontiers of statistical physics. A larger audience acquired some degree of
familiarity with Boltzmann's work only after its exegesis by the Ehrenfests, pub-
lished in 1911 [E3].
Einstein's reply to Hertz, also written in 1910 [E2] is remarkable as well. He
agrees with Hertz's objection and adds, 'Already then [i.e., in 1903] my derivation
did not satisfy me, so that shortly thereafter I gave a second derivation.' The latter
is contained in the only paper Einstein completed in 1904 [El2].* It is indeed a
*For other discussions of Einstein's 1902-4 papers, see [K4] and [K5].