w 0 ¼ 6,w 1 ¼ 1 w 2 ¼ w 3 ¼ w 4 ¼ w 5 ¼ w 6 ¼ 0 [w 1 is also zero. However Boltzmann
forgets to include it]. It is immaterial which molecule has kinetic energy 7E.
There are seven possible complexions, which represent this state distribution.
Denoting the sum of all possible complexions, 1716, by J, then the probability of the
first state distribution is 7/J; similarly, the probability of the second state distribution
is 42/J; the most probable state distribution is the tenth, as its elements permit the
greatest number of permutations. Hereon, we call the number of permutations the
relative likelihood of the state distribution; this can be defined in a different way,
which we next illustrate with a specific numerical example, since generalization is
straightforward. Suppose we have an urn containing an infinite number of paper
slips. On each slip is one of the numbers 0, 1, 2, 3, 4, 5, 6, 7; each number is on the
same amount of slips and has the same probability of being picked. We now draw the
first septet of slips, and note the numbers on them. This septet provides a sample
state distribution with a kinetic energy of E times the number written on the first slip
for molecule l and s forth. We return the slips to the urn and draw a second septet,
which gives us a second state distribution, etc. After we draw a very large number of
septets, we reject all those for which the total does not equal to seven. This still
leaves a large number of septets. Since each number [digit] has the same probability
of occurrence and the same elements in a different order from different complexions,
each possible complexion will occur equally often. By ordering the numbers within
each septet by size, we can classify each into one of the fifteen cases tabulated above.
It is not clear why Boltzmann does not work with probability of state distribution.
Since probability would not change with the sample size, urn with infinite slips
example would be unnecessary. Therefore, the number of septets which fall into the
class 0000007 relative to the 0000016 class will be 7:42. Similarly for all the other
septets. The most likely state distribution is the one which produces the most septets,
namely the 10th.
Boltzmann’s Footnote: If we divide the number of septets corresponding to a
particular state by the total number of septets, we obtain the probability of distribution. Instead of discarding all septets whose total is not 7, we could after drawing a
slip remove from the urn all those other slips for which a total of 7 is now impossible,
e.g., on drawing a slip with 6 on it, all other slips except those with 0 or l would be
removed. If the first six slips all had zero on them, only slips with 7 on them would
be left in the urn. One more thing should be noted at this point. We construct all
possible complexions. If we denote by W 0 the arithmetic mean of all values of w 0
which belong to different complexions and form analogous expressionsW 1 , W 2 ⋯,
in the limit these quantities would also form the same state distribution.
Translators Sharp and Matschinsky’s note: Boltzmann’s comments on the results
of Mr. Oskar Meyer beginning “Ich will hier einige worte € uber die von. Hrn. Oskar
Meyer” on p 172 (Wiss. A6) and ending with “Bearbeitung des allgemeinen
problems zur€ u ckkehren” on p 175 (Wiss. A6) are of historical interest only and
are omitted.
The first task is to determine the permutation number, previously designated byP,
for any state distribution. Denoting by J the sum of the permutations P for all
possible state distributions, the quotient P/J is the state distribution’s probability,
142
4 Unified Mechanics Theory
forgets to include it]. It is immaterial which molecule has kinetic energy 7E.
There are seven possible complexions, which represent this state distribution.
Denoting the sum of all possible complexions, 1716, by J, then the probability of the
first state distribution is 7/J; similarly, the probability of the second state distribution
is 42/J; the most probable state distribution is the tenth, as its elements permit the
greatest number of permutations. Hereon, we call the number of permutations the
relative likelihood of the state distribution; this can be defined in a different way,
which we next illustrate with a specific numerical example, since generalization is
straightforward. Suppose we have an urn containing an infinite number of paper
slips. On each slip is one of the numbers 0, 1, 2, 3, 4, 5, 6, 7; each number is on the
same amount of slips and has the same probability of being picked. We now draw the
first septet of slips, and note the numbers on them. This septet provides a sample
state distribution with a kinetic energy of E times the number written on the first slip
for molecule l and s forth. We return the slips to the urn and draw a second septet,
which gives us a second state distribution, etc. After we draw a very large number of
septets, we reject all those for which the total does not equal to seven. This still
leaves a large number of septets. Since each number [digit] has the same probability
of occurrence and the same elements in a different order from different complexions,
each possible complexion will occur equally often. By ordering the numbers within
each septet by size, we can classify each into one of the fifteen cases tabulated above.
It is not clear why Boltzmann does not work with probability of state distribution.
Since probability would not change with the sample size, urn with infinite slips
example would be unnecessary. Therefore, the number of septets which fall into the
class 0000007 relative to the 0000016 class will be 7:42. Similarly for all the other
septets. The most likely state distribution is the one which produces the most septets,
namely the 10th.
Boltzmann’s Footnote: If we divide the number of septets corresponding to a
particular state by the total number of septets, we obtain the probability of distribution. Instead of discarding all septets whose total is not 7, we could after drawing a
slip remove from the urn all those other slips for which a total of 7 is now impossible,
e.g., on drawing a slip with 6 on it, all other slips except those with 0 or l would be
removed. If the first six slips all had zero on them, only slips with 7 on them would
be left in the urn. One more thing should be noted at this point. We construct all
possible complexions. If we denote by W 0 the arithmetic mean of all values of w 0
which belong to different complexions and form analogous expressionsW 1 , W 2 ⋯,
in the limit these quantities would also form the same state distribution.
Translators Sharp and Matschinsky’s note: Boltzmann’s comments on the results
of Mr. Oskar Meyer beginning “Ich will hier einige worte € uber die von. Hrn. Oskar
Meyer” on p 172 (Wiss. A6) and ending with “Bearbeitung des allgemeinen
problems zur€ u ckkehren” on p 175 (Wiss. A6) are of historical interest only and
are omitted.
The first task is to determine the permutation number, previously designated byP,
for any state distribution. Denoting by J the sum of the permutations P for all
possible state distributions, the quotient P/J is the state distribution’s probability,
142
4 Unified Mechanics Theory