infinities are [nothing] but limiting cases one assumes each molecule can behave in
this fashion only in the limiting case where each molecule can assume more and
more values of the velocity.
To continue, however, we will consider the kinetic energy, rather than the
velocity of the molecules. Each molecule can have only a finite number of values
for its kinetic energy. As a further simplification, we assume that the kinetic energies
of each molecule form an arithmetic progression, such as the following:
0, E, 2E, 3E, . . . , pE
ð4:43Þ
We call p the largest possible value of the kinetic energy,pE. Before impact, each
of two colliding molecules shall have a kinetic energy of
0, or E, or 2E, etc: pE
ð4:44Þ
Which means that after the collision, each molecule still has one of the above
values of kinetic energy. The number of molecules in the vessel is n. If we know how
many of these n molecules have a kinetic energy of zero, how many have a kinetic
energy of E, and so on, then we know the kinetic energy distribution. If at the
beginning there is some state distribution among the gas molecules, this will in
general be changed by the collisions.
The laws governing this change have already been the subject of my previous
investigations. But right way, I note that this is not my intention here, instead I want
to establish the probability of a state distribution, regardless of how it is created or,
more specifically, I want to find all possible combinations of the P + 1 kinetic energy
values allowed to each of the n molecules and then establish how many of these
combinations correspond to each state distribution. [The term “state distribution” is
better translated/interpreted as the distribution of a state into English. However, I am
keeping the original translation, because the term refers to what Boltzmann refers to
“complexion”, which is the distribution of kinetic energies of molecules]. The latter
number ( p + 1) then determines the likelihood of the relevant state distribution, as I
have already stated in my published “Remarks about several problems in the
mechanical theory of heat” (Wiss. Abhand. Vol II, reprint 39, p 121).
As a preliminary, we will use a simpler schematic approach to the problem,
instead of the exact case. Suppose we have n molecules. Each of them is capable of
having kinetic energy
0, E, 2E, 3E, . . . , pE
ð4:45Þ
Moreover, suppose these energies are distributed in all possible ways among the
n molecules, such that the total energy is a constant, e.g., λE ¼ L. Any such
distribution, in which the first molecule may have a kinetic energy of, e.g., 2E, the
second may have 6E, and so on, upto the last molecule, we call a complexion and so
that each individual complexion can be easily enumerated. We write them in
sequence (for convenience we divide through by E), specifying the kinetic energy
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4 Unified Mechanics Theory
this fashion only in the limiting case where each molecule can assume more and
more values of the velocity.
To continue, however, we will consider the kinetic energy, rather than the
velocity of the molecules. Each molecule can have only a finite number of values
for its kinetic energy. As a further simplification, we assume that the kinetic energies
of each molecule form an arithmetic progression, such as the following:
0, E, 2E, 3E, . . . , pE
ð4:43Þ
We call p the largest possible value of the kinetic energy,pE. Before impact, each
of two colliding molecules shall have a kinetic energy of
0, or E, or 2E, etc: pE
ð4:44Þ
Which means that after the collision, each molecule still has one of the above
values of kinetic energy. The number of molecules in the vessel is n. If we know how
many of these n molecules have a kinetic energy of zero, how many have a kinetic
energy of E, and so on, then we know the kinetic energy distribution. If at the
beginning there is some state distribution among the gas molecules, this will in
general be changed by the collisions.
The laws governing this change have already been the subject of my previous
investigations. But right way, I note that this is not my intention here, instead I want
to establish the probability of a state distribution, regardless of how it is created or,
more specifically, I want to find all possible combinations of the P + 1 kinetic energy
values allowed to each of the n molecules and then establish how many of these
combinations correspond to each state distribution. [The term “state distribution” is
better translated/interpreted as the distribution of a state into English. However, I am
keeping the original translation, because the term refers to what Boltzmann refers to
“complexion”, which is the distribution of kinetic energies of molecules]. The latter
number ( p + 1) then determines the likelihood of the relevant state distribution, as I
have already stated in my published “Remarks about several problems in the
mechanical theory of heat” (Wiss. Abhand. Vol II, reprint 39, p 121).
As a preliminary, we will use a simpler schematic approach to the problem,
instead of the exact case. Suppose we have n molecules. Each of them is capable of
having kinetic energy
0, E, 2E, 3E, . . . , pE
ð4:45Þ
Moreover, suppose these energies are distributed in all possible ways among the
n molecules, such that the total energy is a constant, e.g., λE ¼ L. Any such
distribution, in which the first molecule may have a kinetic energy of, e.g., 2E, the
second may have 6E, and so on, upto the last molecule, we call a complexion and so
that each individual complexion can be easily enumerated. We write them in
sequence (for convenience we divide through by E), specifying the kinetic energy
140
4 Unified Mechanics Theory