and asked what happens if T drops below T 0 (for given v 0 ). His answer:
I maintain that, in this case, a number of molecules steadily growing with
increasing density goes over in the first quantum state (which has zero kinetic
energy) while the remaining molecules distribute themselves according to the
parameter value A = 1. ... A separation is effected; one part condenses, the
rest remains a 'saturated ideal gas.' [E7]He had come upon the first purely statistically derived example of a phase tran-
sition, which is now called Bose-Einstein condensation. I defer a few comments
on this phenomenon to the next section and turn to other important facets of the
three Einstein papers.
- Einstein on Statistical Dependence. After the papers by Bose [B3] and the
first one by Einstein [E8] came out, Ehrenfest and others objected (so we read in
Einstein's second paper [E7]) that 'the quanta and molecules, respectively, are not
treated as statistically independent, a fact that is not particularly emphasized in
our papers' (i.e., [B3] and [E8]). Einstein replied, 'This [objection] is entirely
correct' [E7]. He went on to stress that the differences between the Boltzmann
and the BE counting 'express indirectly a certain hypothesis on a mutual influence
of the molecules which for the time being is of a quite mysterious nature.' With
this remark, Einstein came to the very threshold of the quantum mechanics of
identical particle systems. The mysterious influence is, of course, the correlation
induced by the requirement of totally symmetric wave functions. - Einstein on Indistinguishability. In order to illustrate further the differ-
ences between the new and the old counting of macrostates, Einstein cast W in a
with v = V/N. He then discussed the region A < 1, where the equation of state
(obtained by eliminating A between Eqs. 23.19) shows perturbative deviations
from the classical ideal gas. All this is good physics, though unusually straightfor-
ward for a man like Einstein.
In his second paper [E7], the most important one of the three, Einstein began
with the v — T relation at A = 1:
(23.20)430 THE QUANTUM THEORYthe critical value unity. He proceeded to the continuous limit, in which the sum
in Eq. 23.17 is replaced by an integral over phase space, and found(23.19)