Kinetische Theorie der Gase, Breslau 1877, Seite 262). He attempts to interpret, in
the described manner, the equations of my continued studies concerning the equilibrium of heat particles. However, the line of reasoning of Mr. Meyer remained
entirely unclear to me and I will return to my concerns with his approach on page
172 (Wiss. Abhand Vol. II).
We have to take a totally different approach because it is our main purpose not to
limit our discussion to thermal equilibrium, but to explore the relationship of this
probabilistic formulation to the second theorem of the mechanical theory of heat. We
want first to solve the problem which I referred to above and already defined in my
“Remarks on some problems of the mechanical theory of heat” [Wiss. Abhand,
reprint 39], namely to calculate the probability of state distributions from the number
of different distributions. We want first to treat as simple a case as possible, namely a
gas of rigid elastic spherical molecules trapped in a container with absolutely elastic
walls (which interact with central forces only within a certain small distance, but not
otherwise, the latter assumption, which includes the former as a special case, does
not change the calculations in the least). Even in this case, the application of
probability theory is not easy. The number of molecules is not infinite, in a
mathematical sense, yet the number of velocities each molecule is capable of is
effectively infinite. Given this last condition, the calculations are very difficult to
facilitate understanding. I will, as in earlier work, consider a limiting case.
4.3.2 Kinetic Energy Has Discrete Values
We assume initially each molecule is only capable of assuming a finite number of
velocities, such as
0,
1
q
,
2
q
,
3
q
, ⋯
P
q
ð4:41Þ
where P and q are arbitrary finite numbers. Upon colliding, two molecules may
exchange velocities.
The fact that Boltzmann allows exchange of energy between molecules automatically makes his formulation general, i.e., irrespective of gas, liquid, or solid state.
But after the collision, both molecules still have one of the above velocities, namely,
0, or
1
q
, or
2
q
, etc:till
P
q
ð4:42Þ
This assumption does not correspond to any realistic mechanical model, but it is
easier to handle mathematically and the actual problem to be solved is re-established
by letting P and q go to infinity.
Even if at first sight, this seems a very abstract way of treating the problem, it
rapidly leads to the desired objective, and when you consider that in nature all
4.3 Evolution of Thermodynamic State Index (Φ)
139
the described manner, the equations of my continued studies concerning the equilibrium of heat particles. However, the line of reasoning of Mr. Meyer remained
entirely unclear to me and I will return to my concerns with his approach on page
172 (Wiss. Abhand Vol. II).
We have to take a totally different approach because it is our main purpose not to
limit our discussion to thermal equilibrium, but to explore the relationship of this
probabilistic formulation to the second theorem of the mechanical theory of heat. We
want first to solve the problem which I referred to above and already defined in my
“Remarks on some problems of the mechanical theory of heat” [Wiss. Abhand,
reprint 39], namely to calculate the probability of state distributions from the number
of different distributions. We want first to treat as simple a case as possible, namely a
gas of rigid elastic spherical molecules trapped in a container with absolutely elastic
walls (which interact with central forces only within a certain small distance, but not
otherwise, the latter assumption, which includes the former as a special case, does
not change the calculations in the least). Even in this case, the application of
probability theory is not easy. The number of molecules is not infinite, in a
mathematical sense, yet the number of velocities each molecule is capable of is
effectively infinite. Given this last condition, the calculations are very difficult to
facilitate understanding. I will, as in earlier work, consider a limiting case.
4.3.2 Kinetic Energy Has Discrete Values
We assume initially each molecule is only capable of assuming a finite number of
velocities, such as
0,
1
q
,
2
q
,
3
q
, ⋯
P
q
ð4:41Þ
where P and q are arbitrary finite numbers. Upon colliding, two molecules may
exchange velocities.
The fact that Boltzmann allows exchange of energy between molecules automatically makes his formulation general, i.e., irrespective of gas, liquid, or solid state.
But after the collision, both molecules still have one of the above velocities, namely,
0, or
1
q
, or
2
q
, etc:till
P
q
ð4:42Þ
This assumption does not correspond to any realistic mechanical model, but it is
easier to handle mathematically and the actual problem to be solved is re-established
by letting P and q go to infinity.
Even if at first sight, this seems a very abstract way of treating the problem, it
rapidly leads to the desired objective, and when you consider that in nature all
4.3 Evolution of Thermodynamic State Index (Φ)
139