3 7 6 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

Consequently, a contradiction to the prediction of the Nernst theorem exists in cal-

culation approach (b). Calculation approach (a), by contrast, concurs with the

Nernst theorem, as is immediately seen when one considers that, at absolute zero

in the sense of calculation approach (a), only a single complexion exists (W = 1).

Approach (b) leads, according to this exposition, either to a violation of the Nernst

theorem or to a violation of the requirement that the entropy at a given inner state

be proportional to the number of

molecules.[9]

For these reasons I believe that cal-

culation approach (a) (i.e., Bose’s statistical approach) must be given preference,

although the preference for this calculation approach over other approaches cannot

be proven a priori. This result, for its part, constitutes a support for the notion of a

deep essential relationship between radiation and gas, in that the same statistical

approach that leads to Planck’s formula establishes agreement between gas theory

and the Nernst theorem in its application to ideal gases.

§8. The Fluctuation Properties of Ideal Gas

A gas of volume V communicates with one of the same kind of infinitely large

volume. Both volumes are separated by a wall that only allows molecules of the in-

finitesimal energy region ΔE to pass, but reflects molecules of other kinetic ener-

gies. This fictitious kind of wall is analogous to the quasi-monochromatically per-

meable wall in the domain of radiation theory. The issue to be examined is the

fluctuation Δ in the number of molecules n belonging to the energy region ΔE. It is

thereby assumed that an exchange of energy between molecules of different energy

regions within V does not take place, so that fluctuations in the number of mole-

cules with energies beyond ΔE are unable to

occur.[10]

Let be the mean of the molecules belonging to ΔE; the momentary

value. Then (29a) delivers the value for the entropy as a function of by plugging

into this equation instead of

.[11]

Going up to quadratic terms, one

obtains

A similar relation applies for the infinitely large remaining system, namely,

[p. 8]

nν nν Δ +

Δν

nν Δ + nν

S S

Δ∂

∂S

Δν

1

2∂Δν

----------- -

∂2S

2

. Δν 2. + + =

S° S°

∂Sx

∂Δν

---------Δν. –=