BOLTZMANN furthermore informs us that, as soon as in a gas, the mean length of path is great in comparison with the mean distance between two adjacent molecules, the neighboring molecules will quickly become different from what they formerly were. Therefore it is exceedingly probable that a "molekular-geordnete" (but molar-ungeordnete) distribution would shortly pass into a "molekular-ungeordnete" distribution.
Furthermore, from the constitution of a gas results that the place where a molecule collided is entirely independent of the spot where its preceding collision took place. Of course, this independence could be maintained for an indefinite time only by an infinite number of molecules.
The place of collision of a pair of molecules must in our Theory of Probabilities be independent of the locality from which either molecule started.
From all the preceding we must infer what measure of haphazard BOLTZMANN considers necessary for the legitimate use of the Theory of Probabilities.
BOLTZMANN in proving his H-Theorem,[5] which establishes the one-sidedness of all natural events, makes the explicit assumption that the motion at the start is both "molar- und molekular-ungeordnet" and remains so. Later on, he assumes the same things but adds that if they are not so at the start they will soon become so; therefore said assumption does not preclude the consideration by Probability methods of the general case or the passage from "ordnete" to "ungeordnete" conditions which characterizes all natural events.
In fact these very definitions show solicitude for securing the uninterrupted operation of the laws of probability. BOLTZMANN intimates his approval of S. H. BURBURY'S statement of the condition of independence underlying his work.
Here S. H. BURBURY[6] simplifies the matter by assuming that any unit of volume of space contains a uniform mixture of differently speeded molecules and then says:
"Let
be the velocity of the center of gravity of any pair of molecules and