432 THE QUANTUM THEORYcorresponds to the state of things. The reason is that I have not been able to
participate as intensively in the modern development of the quantum theory as
would be necessary for this purpose. This is in part because I have on the whole
too little receptive talent for fully following the stormy developments, in part
also because I do not approve of the purely statistical way of thinking on which
the new theories are founded.... Up until now, I kept hoping to be able to
contribute something of value in Brussels; I have now given up that hope. I beg
you not to be angry with me because of that; I did not take this lightly but tried
with all my strength.... Perhaps Herr Fermi in Bologna ... or Langevin...
could do a good job. [E12]23d. Postscript on Bose-Einstein Condensation
(1) In December 1924, Einstein wrote to Ehrenfest, 'From a certain tempera-
ture on, the molecules "condense" without attractive forces, that is, they accu-
mulate at zero velocity. The theory is pretty, but is there also some truth to it?'
[E13].
(2) In 1925, Einstein mentioned hydrogen, helium, and the electron gas as the
best possible candidates in which to observe his condensation phenomenon [E7].
In 1925, these were, of course, sensible proposals. Recall that the Fermi-Dirac
statistics was not discovered until 1926 [Fl, Dl], following Pauli's enunciation
of the exclusion principle in 1925 [P2]. Even then, it took some time until it was
sorted out when BE and FD statistics apply respectively: referring to Dirac's
paper [Dl], Pauli wrote in December 1926, 'We shall take the point of view also
advocated by Dirac, that the Fermi, and not the Einstein-Bose, statistics applies
to the material gas' [P3]. These matters were cleared up by 1927.
(3) In his 1925 paper, Einstein did not call the condensation phenomenon a
phase transition. According to Uhlenbeck (private communication), nobody real-
ized in 1925 that the existence of a phase transition was a 'deep' problem. In 1926,
Uhlenbeck himself raised an objection to Einstein's treatment of the condensation
problem [Ul]. This critique was to lead to a more precise theoretical formulation
of the conditions under which phase transitions can occur. Uhlenbeck noted that
the quantity N° in Eq. 23.17 -«• oo as A — 1 (for fixed T); hence also N -
oo. Thus, if A —» 1, it is impossible to implement the constraint that N is a fixed
finite number. Therefore A = 1 can be reached only asymptomatically and there
is no two-phase regime.
Uhlenbeck recently described the communications between Ehrenfest and Ein-
stein on this question [U2]. Uhlenbeck and Einstein were both right, however.
The point is that a sharp phase transition can occur only in the so-called ther-
modynamic limit N — oo, V — GO, v fixed. This view emerged in a morning-
long debate that took place during the van der Waals Centenary Conference in
November 1937. The issue was, Does the partition function contain the infor-
mation necessary to describe a sharp phase transition? The transition implies the
*Dirac has given a charming account of the time sequence of these discoveries [D2].