x ¼ 0:5088742 . . .
From this, one finds in accordance with Eq. (4.55)
w 0 ¼ 3:4535 w 4 ¼ 0:2316
w 1 ¼ 1:7574 w 5 ¼ 0:1178
w 2 ¼ 0:8943 w 6 ¼ 0:0599
w 3 ¼ 0:4551 w 7 ¼ 0:0304
These numbers satisfy the condition that
ffiffiffiffiffi
2π
p
w 0
e
w 0 Á
ffiffiffiffiffi
2π
p
w 1
e
w 1 . . . etc:
is minimized, while the minimized variables w obey the two constraints
w 0 þ w 1 þ w 2 þ w 3 þ w 4 þ w 5 þ w 6 þ w 7 ¼ 7,
ð4:79aÞ
w 1 þ 2w 2 þ 3w 3 þ 4w 4 þ 5w 5 þ 6w 6 þ 7w 7 ¼ 7,
ð4:79bÞ
which minimum, incidentally because of the first of Eqs. (4.79a) and (4.79b),
coincides with the minimum of w 0
ð Þ
w 0 w 1
ð Þ
w 1
. . . This provides only an approximate
solution to our problem, which asks for so many (w 0 ) zeros, so many (w 1 ) ones, etc.
with as many permutations as the resulting complexion permits, while the w
0 s
simultaneously satisfy the constraints Eqs. (4.79a, 4.79b). Since p and n here are
very small, one hardly expects any great accuracy, yet you already get the solution to
the permutation problem by taking the nearest integer for each w, with the exception
of w 3 , for which you have to assign the value of 1 instead of 0.4551. In this manner, it
is apparent
w 0 ¼ 3, w 1 ¼ 2, w 2 ¼ w 3 ¼ 1, w 4 ¼ w 5 ¼ w 6 ¼ w 7 ¼ 0
In addition, in fact we saw in the previous table that the complexion of 0001123
has the most permutations. We now consider the same special case with n ¼ λ ¼ 7,
but set p ¼ 1; that is, the molecules may have kinetic energies of 0, 1, 2, 3. . .1. We
know then that the values of the variables w will vary little from those of the former
case. In fact, we obtain
x ¼
1
2
; w 0 ¼
7
2
¼ 3:5, w 1 ¼
w 0
2
¼ 1:75, w 2 ¼
w 1
2
¼ 0:875 etc:
We consider a little more complicated example. Take n ¼ 13, λ ¼ 19, but we
only treat the simpler case where p ¼ 1. Then we have
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4 Unified Mechanics Theory
From this, one finds in accordance with Eq. (4.55)
w 0 ¼ 3:4535 w 4 ¼ 0:2316
w 1 ¼ 1:7574 w 5 ¼ 0:1178
w 2 ¼ 0:8943 w 6 ¼ 0:0599
w 3 ¼ 0:4551 w 7 ¼ 0:0304
These numbers satisfy the condition that
ffiffiffiffiffi
2π
p
w 0
e
w 0 Á
ffiffiffiffiffi
2π
p
w 1
e
w 1 . . . etc:
is minimized, while the minimized variables w obey the two constraints
w 0 þ w 1 þ w 2 þ w 3 þ w 4 þ w 5 þ w 6 þ w 7 ¼ 7,
ð4:79aÞ
w 1 þ 2w 2 þ 3w 3 þ 4w 4 þ 5w 5 þ 6w 6 þ 7w 7 ¼ 7,
ð4:79bÞ
which minimum, incidentally because of the first of Eqs. (4.79a) and (4.79b),
coincides with the minimum of w 0
ð Þ
w 0 w 1
ð Þ
w 1
. . . This provides only an approximate
solution to our problem, which asks for so many (w 0 ) zeros, so many (w 1 ) ones, etc.
with as many permutations as the resulting complexion permits, while the w
0 s
simultaneously satisfy the constraints Eqs. (4.79a, 4.79b). Since p and n here are
very small, one hardly expects any great accuracy, yet you already get the solution to
the permutation problem by taking the nearest integer for each w, with the exception
of w 3 , for which you have to assign the value of 1 instead of 0.4551. In this manner, it
is apparent
w 0 ¼ 3, w 1 ¼ 2, w 2 ¼ w 3 ¼ 1, w 4 ¼ w 5 ¼ w 6 ¼ w 7 ¼ 0
In addition, in fact we saw in the previous table that the complexion of 0001123
has the most permutations. We now consider the same special case with n ¼ λ ¼ 7,
but set p ¼ 1; that is, the molecules may have kinetic energies of 0, 1, 2, 3. . .1. We
know then that the values of the variables w will vary little from those of the former
case. In fact, we obtain
x ¼
1
2
; w 0 ¼
7
2
¼ 3:5, w 1 ¼
w 0
2
¼ 1:75, w 2 ¼
w 1
2
¼ 0:875 etc:
We consider a little more complicated example. Take n ¼ 13, λ ¼ 19, but we
only treat the simpler case where p ¼ 1. Then we have
150
4 Unified Mechanics Theory