is here fully satisfied.
At this point we again call attention to the fact, that in both the unsettled and settled states of a system all conceivable micro-states are not equally likely to obtain. On [p. 19] mention was made that the unsettled and the settled state each possessed a host of conceivable micro-states which agreed with the characteristic averages of their respective macro-states (the unsettled and the settled ones), and yet in each set some of these led subsequently to events which did not accord with experience. Therefore for both the unsettled and the settled state we must limit the manifold character of their micro-states by eliminating all those micro-states which lead to results contrary to experience. This is accomplished by assuming the hypothesis of "elementary-disorder" (elementar-ungeordnet) to obtain for the unsettled as well as the settled state. Now so far as the haphazard character of the remaining motions are concerned, we might stop right here, for the very nature of this hypothesis insures results in harmony with experience, i.e., with the undisturbed operation of the laws of probability.
But if we do not stop here, preferring to examine some of the special features of fortuitous motion, as detailed on pp. [10], [13], [14] and [17], we still see that by this hypothesis we have not removed the haphazard character of the remaining motions in either the unsettled or the settled state. For instance, we have not removed BURBURY'S condition
. We must remember, too, that in PLANCK'S briefest statement of "elementary disorder" (bot. of [p. 11]), two important features of haphazard are emphasized, viz.: the independence and great number of the constituents. BOLTZMANN in his Gas Theorie of course considers the special features which underlie the application of the Calculus of Probabilities; thus he says they are, the great number of molecules and the length of their paths, which together make the laws of the collision of a molecule in a gas independent of the place where it collided before. Neither has the introduction of the hypothesis of "elementary disorder" done away with these special features. There have simply been excluded from consideration such pre-computed and prearranged regularities in the paths and directions of molecules as purposely interfere with the operation of the laws of probability. We are still free to consider all the imaginable positions and velocities of the individual molecules which are compatible with the mean velocity, density, and temperature properly characteristic of each stage of the passage from the unsettled to the settled state. For adequate haphazard we only need the assumption that the molecules fly so irregularly as to permit the operation of the laws of probabilities. Such a presentation as this of course calls for complete trust that all the specified requirements have been adequately met and BOLTZMANN'S eminence as a mathematical physicist and the endorsement of his peers must be our guarantee for such confidence and trust.
Before closing this discussion of unsettled and settled states we will insert here two remarks, really at this stage, anticipatory in their nature. The first is, that under the limitation imposed by our supplementary hypothesis of "elementary chaos," the very sharpest definition of any macro-state is the number of its possible micro-states. This is evidently the number of permutations, possible with the given locus and velocity elements under the restriction imposed above. Later on we will find that this number of possible micro-states is smaller for the unsettled state than for the settled one. This gives us a clean-cut distinction between the two states contemplated. The second remark is that the inevitable change in the system as a whole is always from the less probable to the more probable, is a passage from an unsettled state of the system to its settled state and this is here synonymous with the growth of the number of possible micro-states. It is this difference between the initial and final states which constitutes the universal driving motive in all natural events.
[SECTION B]
THE APPLICATION OF CALCULUS OF PROBABILITIES
IN MOLECULAR PHYSICS.
(1) The Probability Concept, its Usefulness in the Past, its Present
Necessity, and its Universality.
An indication of its essential value in this physical discussion is evidenced by the fact that we have almost unwittingly been forced to constantly refer to it in all of our preliminaries. But when this concept is first broached to a student, he feels about it like the "man in the street"; it is by the latter regarded as a matter of chance and hence of uncertainty and unreliability; moreover, the latter knows in a vague way that the subject has to do with averages, that it is often of a statistical nature, and knows that statistics in general are widely distrusted. The student is at first likely to share these views with said man in the street, and at best feels that its introduction is of remote interest, far fetched, and tends to hide and dissipate the kernel of the matter. The student must disabuse himself of these false notions by reflecting how much there is in Nature that is spontaneous, in other words, how many events there are in which there is a passage from a less probable to a more probable condition and that he cannot afford to despise or ignore a Calculus which measures these changes as exactly as possible.