ENTROPY AND PROBABILITY 65Einstein's precursors have now been sufficiently introduced. I conclude this sec-
tion with three final comments.
The first definition of probability, in terms of time spent, is the natural one,
directly linked to observation. For example, the most probable state is the state in
which the system persists for the longest time. The second definition (either for
w or for W) is not directly linked to observation; it is more like a declaration. It
has the advantage, however, that one can more readily compute with it. Logic
demands, of course, that these two definitions be equivalent, that 'time spent' be
proportional to 'volume in F space.' This is the profound and not yet fully solved
problem of ergodic theory.* Boltzmann was well aware of the need to show this
equivalence. Einstein's physical intuition made him comfortable with the first but
not with the second definition.
Second, why did Boltzmann himself not introduce the symbol k?** After all,
his 1877 paper [B6] contains a section entitled 'The Relation of the Entropy to
the Quantity Which I Have Called Partition Probability,' that quantity being
essentially In W. Moreover, in that section he noted that In W 'is identical with the
entropy up to a constant factor and an additive constant.' He was also quite
familiar with Eq. 4.9, with its two Lagrange multipliers [B14]. I can imagine that
he did not write down Eq. 4.3 because he was more concerned with understanding
the second law of thermodynamics than with the applications of an equation such
as Eq. 4.3 to practical calculations. I hope that this question will be discussed
some day by someone more at home with Boltzmann's work than I am.
Finally, Eq. 4.3 is evidently more general than Eq. 4.10. Boltzmann was aware
of this: '[InH^7 ] also has a meaning for an irreversible bodyf and also steadily
increases during [such a process]' [B6]. The first one to make use of Eq. 4.3 in its
broader sense was Einstein. It was also Einstein who, in 1905, in his paper on the
light-quantum hypothesis [E13], gave that equation its only fitting name: Boltz-
mann's principle.4c. Preludes to 1905
Boltzmann's qualities as an outstanding lecturer are not reflected in his scientific
papers, which are sometimes unduly long, occasionally obscure, and often dense.
Their main conclusions are sometimes tucked away among lengthy calculations.
Also (and especially in regard to the theoretical interpretation of the second law),
Boltzmann would change his point of view from one paper to the next without
*For introductions to this problem, see, e.g., [Ul] and [VI].
**As to what might have been, in 1860 Maxwell could have been the first to introduce k when he
derived his velocity distribution, in which the Boltzmann factor makes its first appearance. Maxwell
wrote this factor as exp( — v^2 /a^2 ), where v — velocity, showed that a^2 is proportional to the average
of v^2 , and knew full well that this average is proportional to T.
f Obviously, he must have meant process instead of body.