3 7 4 D O C . 3 8 5 Q U A N T U M T H E O R Y O F I D E A L G A S I I
(28)
By taking the product over all infinitesimal regions, one obtains the total number
of complexions of a state and from it, according to Boltzmann’s principle, the en-
tropy
(29a)
It is easily recognized that by this calculation approach the distribution of mole-
cules over the cells is not treated as statistically independent. This is because the
cases, here called “complexions,” would not be regarded of equal probability ac-
cording to the hypothesis of an independent distribution of the individual mole-
cules among the cells. For really statistically independent molecules, the counting
of these “complexions” of differing probabilities would not yield the entropy cor-
rectly. Consequently, the formula indirectly expresses a certain hypothesis about an
initially completely puzzling mutual influence of the molecules that determines
just the same statistical probability of the cases that are defined here as
“complexions.”
b) according to the hypothesis of statistical independence of the molecules:
A state is defined microscopically by the fact that for each molecule one indi-
cates in which cell it is sitting (complexion). How many complexions belong in one
macroscopically defined state? I can distribute n definite molecules in
different ways over the cells of the ν th elementary region. If the assignment of
the molecules to the elementary regions is already performed in a definite way, then
there are altogether
different distributions of the molecules over all the cells. In order to obtain the num-
ber of complexions in the defined meaning, this amount must now also be multi-
plied by the number
of possible assignments of all the molecules to the elementary regions for given .
Boltzmann’s principle then yields for the entropy the expression
(29b)
nν zν 1)! –+ (
nν!(zν 1)! –
--------------------------------
S
κ¦{
nν zν) + ( lg nν zν) + ( nν lg nν – zν – zν lg zν}. –
ν
=
[p. 6]
zννn
zν
zνν
n
)
∏(
n!
∏nν!
---------------
nν
S κ® n lg n nν lg zν nν lg nν) –
¦(
+
¯ ¿
¾.
­ ½
=