of all the bodies must necessarily increase, as is well known from the fact that
R
dQ/T
integrated over a nonreversible cyclic process is negative. According to Eq. (4.171)
the sum of the permutability measures of the bodies ΣΩ and the total permutability
measure must have the same increase. Therefore at thermal equilibrium the magnitude of the permutability measure times a constant is identical to the entropy, to
within an additive constant; but it also retains meaning during a nonreversible
process, continually increasing.
We can establish two principles: The first refers to a collection of bodies in which
various state changes have occurred, at least some of which are irreversible, e.g.,
where some of the bodies were not always in thermal equilibrium. If the system was
in a state of thermal equilibrium before and after all these changes, then the sum of
the entropies of all the bodies can be calculated before and after those state changes
without further ado, and it is always equal to 2/3 times the permutability measure of
all the bodies. The first principle is that the total entropy after the state changes is
always greater than before it. The same is of course true of the permutability
measure. The second principle refers to a gas that undergoes a change of state
without requiring that it begin and end in thermal equilibrium. Then the entropy of
the initial and final states is not defined, but one can still calculate the quantity, which
we have called the permutability measure, and that is to say that its value is
necessarily larger after the state change than before. We shall presently see that the
latter proposition can be applied without difficulty to a system of several gases, and it
can be extended as well to polyatomic gas molecules, and when external forces are
acting. For a system of several gases, the sum of the individual gas permutability
measures must be defined to be the permutability measure of the system; if one
introduces on the other hand the number of permutations itself, then the number of
permutations of the system would be the product of the number of permutations of
the constituents. If we assume the latter principle applies to anybody, then the two
propositions just discussed are special cases of a single general theorem, which reads
as follows:
We consider any system of bodies that undergoes some state changes, without
requiring the initial and final states to be in thermal equilibrium. Then the total
permutability measure for the bodies continually increases during the state
changes, and can remain constant only so long as all the bodies during the change
of state remain infinitely close to thermal equilibrium (reversible state changes.)
[C.B. note: it is important to point out that Boltzmann drops the term “gas,” in his
theorem]
To give an example, we consider a vessel divided into two halves by a very thin
partition. The remaining walls of the vessel should also be very thin, so that the heat
they absorb can be neglected, and surrounded by a substantial mass of other gas.
One-half of the vessel should be completely filled with gas, while the other is
initially completely empty. Suddenly pulling away the divider, which requires no
significant work, causes that gas to spread at once throughout the vessel. Calculating
the permutability measure for the gas, we find that this increases during this process,
without changes in any other body. Now the gas is compressed in a reversible
176
4 Unified Mechanics Theory
R
dQ/T
integrated over a nonreversible cyclic process is negative. According to Eq. (4.171)
the sum of the permutability measures of the bodies ΣΩ and the total permutability
measure must have the same increase. Therefore at thermal equilibrium the magnitude of the permutability measure times a constant is identical to the entropy, to
within an additive constant; but it also retains meaning during a nonreversible
process, continually increasing.
We can establish two principles: The first refers to a collection of bodies in which
various state changes have occurred, at least some of which are irreversible, e.g.,
where some of the bodies were not always in thermal equilibrium. If the system was
in a state of thermal equilibrium before and after all these changes, then the sum of
the entropies of all the bodies can be calculated before and after those state changes
without further ado, and it is always equal to 2/3 times the permutability measure of
all the bodies. The first principle is that the total entropy after the state changes is
always greater than before it. The same is of course true of the permutability
measure. The second principle refers to a gas that undergoes a change of state
without requiring that it begin and end in thermal equilibrium. Then the entropy of
the initial and final states is not defined, but one can still calculate the quantity, which
we have called the permutability measure, and that is to say that its value is
necessarily larger after the state change than before. We shall presently see that the
latter proposition can be applied without difficulty to a system of several gases, and it
can be extended as well to polyatomic gas molecules, and when external forces are
acting. For a system of several gases, the sum of the individual gas permutability
measures must be defined to be the permutability measure of the system; if one
introduces on the other hand the number of permutations itself, then the number of
permutations of the system would be the product of the number of permutations of
the constituents. If we assume the latter principle applies to anybody, then the two
propositions just discussed are special cases of a single general theorem, which reads
as follows:
We consider any system of bodies that undergoes some state changes, without
requiring the initial and final states to be in thermal equilibrium. Then the total
permutability measure for the bodies continually increases during the state
changes, and can remain constant only so long as all the bodies during the change
of state remain infinitely close to thermal equilibrium (reversible state changes.)
[C.B. note: it is important to point out that Boltzmann drops the term “gas,” in his
theorem]
To give an example, we consider a vessel divided into two halves by a very thin
partition. The remaining walls of the vessel should also be very thin, so that the heat
they absorb can be neglected, and surrounded by a substantial mass of other gas.
One-half of the vessel should be completely filled with gas, while the other is
initially completely empty. Suddenly pulling away the divider, which requires no
significant work, causes that gas to spread at once throughout the vessel. Calculating
the permutability measure for the gas, we find that this increases during this process,
without changes in any other body. Now the gas is compressed in a reversible
176
4 Unified Mechanics Theory