ENTROPY AND PROBABILITY 63
self in 1903 [Ell], evidently unaware of Boltzmann's paper of 1868. (Lorentz
later called this definition the time ensemble of Einstein [L3], perhaps not the
most felicitous of names.) Rather, Einstein had reservations about the second def-
inition of probability, which Boltzmann gave in the paper of 1877 [B6]. In that
paper, Boltzmann introduced for the first time a new tool, the so-called statistical
method, in which there is no need to deal explicitly with collision mechanisms and
collision frequencies (as there is in the kinetic method). His new reasoning only
holds close to equilibrium [BIO]. He applied the method only to an ideal gas
[Bll]. For that case, he not only gave his second definition of probability but also
showed how that probability can be computed explicitly by means of counting
'complexions.'
In preparation for some comments on Einstein's objections (Section 4d) as well
as for a later discussion of the differences between classical and quantum statistics
(Chapter 23), it is necessary to recall some elementary facts about this counting
procedure.*
Suppose I show someone two identical balls lying on a table and then ask this
person to close his eyes and a few moments later to open them again. I then ask
whether or not I have meanwhile switched the two balls around. He cannot tell,
since the balls are identical. Yet I know the answer. If I have switched the balls,
then I have been able to follow the continuous motion which brought the balls
from the initial to the final configuration. This simple example illustrates Boltz-
mann's first axiom of classical mechanics, which says, in essence, that identical
particles which cannot come infinitely close to each other can be distinguished by
their initial conditions and by the continuity of their motion. This assumption,
Boltzmann stressed, 'gives us the sole possibility of recognizing the same material
point at different times' [B13]. As Erwin Schroedinger emphasized, 'Nobody
before Boltzmann held it necessary to define what one means by [the term] the
same material point' [S5]. Thus we may speak classically of a gas with energy E
consisting of N identical, distinguishable molecules.
Consider next (following Boltzmann) the specific case of an ideal gas model in
which the energies of the individual particles can take on only discrete values
e,,e2).... Let there be rc, particles with energy e, so that
*See Lorentz [L3] for the equivalence of this method with the microcanonical ensemble of Gibbs.
Also, the notion of ensemble has its roots in Boltzmann's work [B12], as was stressed by Gibbs in
the preface of his book on statistical mechanics [Gl].
Since the gas is ideal, the particles are uncorrelated and therefore have no a priori
preference for any particular region in one-particle phase space (n space), i.e.,
they are statistically independent. Moreover, they are distinguishable in the sense