Equations 4.17 and 4.20 again give Eq. 4.15.
Equation 4.20 can of course also be derived from Boltzmann's formula Eq. 4.8,
since each factor WA can be chosen proportional to V (for all A). Therefore Eq.
4.8 can be written W = VN times a complexion-counting factor which is the
same for states a and b. Einstein was therefore quite right in saying that Eq. 4.15
(and, therefore, the ideal gas law which follows from Eqs. 4.15 and 4.16) can be
derived without counting complexions. 'I shall show in a separate paper [he
announced] that, in considerations about thermal properties, the so-called statis-
tical probability is completely adequate' [El3]. This statement was too optimistic.
Equation 4.8 yields much stronger results than Eq. 4.15. No physicist will deny
that the probability for finding n statistically independent particles in the subvol-
ume V of FQ is 'obviously' equal to (V/V 0 )". The counting of complexions gives
more information, however, to wit, the Maxwell-Boltzmann distribution. No
wonder that the promised paper never appeared.
Einstein did not cease criticizing the notion of complexion, however. Here he
is in 1910: 'Usually W is put equal to the number of complexions.... In order
to calculate W, one needs a complete (molecular-mechanical) theory of the system
under consideration. Therefore it is dubious whether the Boltzmann principle has
any meaning without a complete molecular-mechanical theory or some other the-
ory which describes the elementary processes. [Eq. 4.3] seems without content,
from a phenomenological point of view, without giving in addition such an Ele-
mentartheorie' [E29].
My best understanding of this statement is that, in 1910, it was not clear to him
how the complexion method was to be extended from an ideal to a real gas. It is
true that there are no simple and explicit counting formulas like Eqs. 4.5 and 4.8
if intermolecular forces are present. However, as a matter of principle the case of
a real gas can be dealt with by using Gibbs's coarse-grained microcanonical
ensemble, a procedure with which Einstein apparently was not yet familiar.
After 1910, critical remarks on the statistical method are no longer found in
Einstein's papers. His subsequent views on this subject are best illustrated by his
comments on Boltzmann and Gibbs in later years. Of Boltzmann he wrote in
72 STATISTICAL PHYSICSFor the case of an ideal gas, the subsystems may be taken to be the individual
molecules. Let the gas in the states a and b have volume and temperature (V,T)
and (F 0 ,T), respectively. Einstein next unveils his own definition of probability:
'For this probability [ Wa/Wb], which is a "statistical probability," one obviously
[my italics] finds the value