ln 1 þ
P
α
ð4:146Þ
could be expanded in powers of P/α, and Eq. (4.145) would yield a finite value for P/
α. There remains only the possibility that P/α is very large. Since
αn
Pn
ln
Pn
αn
ð4:146Þ
vanishes, Eq. (4.145) gives
α ¼
a
b
¼ pe
À
nP
L ,
ð4:147Þ
from which follows:
b ¼
P
L
,
a ¼
p
2
L
e
À
nP
L
ð4:148Þ
By the approach used in this section, using the mean kinetic energy of a molecule,
these equations show that in the limit of increasing p, W, L the probability of
dispersion in kinetic energy remains indeterminate. We now want to consider a
second, more realistic problem. We take the three velocity components u, v,
w parallel to the three coordinate axes as the independent variables, and find the
maximum of the expression
Q ¼
Z þ1
À1
Z þ1
À1
Z þ1
À1
ln f u, v, w
ð
Þdudvdw,
ð4:149Þ
while simultaneously the two expressions
n ¼
Z þ1
À1
Z þ1
À1
Z þ1
À1
f u, v, w
ð
Þdudvdw,
ð4:150Þ
L ¼
Z þ1
À1
Z þ1
À1
Z þ1
À1
u
2
þ v
2
þ w
2
À
Á
f u, v, w
ð
Þdudvdw
ð4:151Þ
remain constant, integrating over u, v, w. If the velocity magnitude is
ω ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
u 2 þ v 2 þ w 2
p
ð4:152Þ
4.3 Evolution of Thermodynamic State Index (Φ)
169