D O C . 4 2 7 Q U A N T U M T H E O R Y O F I D E A L G A S 4 2 3
(14)
where A and B are independent of L. Since, however, s and thus also is a func-
tion of ρ alone, one can solve this equation for ρ and obtain
(15)
where Ψ is an unknown function. A and B can, of course, be dependent on κT, ,
m, and h.
The quantity A can be determined by applying the entropy principle to an infin-
itesimally isopyknic heating of the gas. If one designates the energy of the gas E
and the changes that occur during this process D, then one initially has
Because, due to (14),
and due to the number of molecules that remain constant,
one obtains
or
Instead of (15), the result is thus
(15a)
§5. Gas in a Conservative Force Field
Let there be a gas in dynamic equilibrium under the influence of a conservative
force field. The potential energy Π of a molecule is a function of its location. ρ is
again the molecular density for the reduced six-dimensional phase space. We again
neglect collisions between the molecules and assume that the motion of the indi-
vidual molecule proceeds under the influence of the external force field according
to classical mechanics. The condition that the motion is a stationary one then yields
the condition
. (16)
ρ∂
∂s
AL B, +=
ρ∂
∂s
ρ Ψ( AL B,) + =
V
N
--- -
DE TdS
V
h3
----
-³LDρ
dΦ
V ----------³DsdΦ.-T3κh
= = =
[p. 23]
Ds Dρ( AL B), + =
dΦ
³Dρ
0, =
1 κTA) – dΦ(
³LDρ
0 =
A
1
κT
------. =
ρ
Ψ§
L
κT
------
B·
+
© ¹
. =
∂xi
∂
ρxi) (
∂pi
∂
ρpi)¹ ( +
§ ·
¦©
0 =