Physical significance of entropy or of the second law

by Joseph Frederic Klein

These may be briefly stated to be (a) atomic theory, (b) the likeness of particles (or elements), (c) very numerous particles, and (d) "elementary chaos."

The first prerequisite is that the body (here a gas) is made up of small, discrete particles. This atomic theory has long been the foundation stone of chemistry, and is again coming into deserved esteem in Physics pure and simple. (See simple and clear article in Harper's Monthly, June, 1910). But this minute subdivision must be accompanied by the particles being of the same kind, or at least belonging to comparatively few groups, each containing many particles of the same sort. This likeness is necessary; for only from this likeness results law and order in the whole from disorder in the parts. If the constituents were of many different kinds, the results in the aggregate would not be so simple as we actually find them to be. There is an example of this sort of complexity in chemistry. We have already intimated that the particles of each kind must be very numerous, but special emphasis must be laid on this prerequisite. If we ask how numerous these elements must be in order that the Theory of Probabilities may be applicable, the answer is, as many constituents as are necessary to determine the mean values which define the state in the macroscopic sense (i.e., in the aggregate condition). An idea of the extent to which Nature carries this subdivision is furnished by the fact that one grain (avoirdupois) of air, under standard conditions, contains,

i.e., millions of billions of particles!

The last one of said prerequisites is "elementary chaos," and needs further elucidation and limitation; we will therefore go into this feature at greater length.

BOLTZMANN has used the term "molekular-ungeordnet" (molecularly disordered) to designate this chaotic condition of the particles, and PLANCK has introduced a more general term still, "elementar-ungeordnet" (elementary disorder or chaos) in order to make the method applicable to phenomena like radiation, in which the elements are not atoms or particles but partial oscillations of different periods. The essence of the matter seems to consist in excluding from consideration all such regularities in the conditions of the elements as would lead to results at variance with the well-known laws of physical phenomena, justifying this exclusion by the assumption that no such elementary regularities obtain in Nature. This only means that not all of the many molecular arrangements, which are conceivable from the purely mechanical standpoint, are actually realized. For instance, in an isolated gaseous system we could conceive of a succession of elementary states at variance with the principle of conservation of energy; such a set would obviously not be realized. This exclusion or limitation leaves room for various hypotheses as to said elementary disarrangement, but to be admissible they must all permit of the legitimate application of the Theory of Probabilities, the best one being ultimately determined by its agreement in the whole with known facts or laws. Evidently by prearrangement and precomputation there could be obtained molecular arrangements which would establish long-continued regularities, which would furnish mean results in the aggregate, that would be at variance with the well-known behavior of Nature. All such cases are here excluded.

According to PLANCK the unregulated, confused and whirring intermingling of very many atoms (in the case of a monatomic gas) is the prerequisite for the validity of this hypothesis of "elementary chaos."

(2) Differences in the States of "Elementary Chaos"

When we consider the general state of a gas we need not think of the state of equilibrium, for this is still further characterized by the condition that its entropy is a maximum.[24]

Hence in the general or unsettled state of the gas an unequal distribution of density may prevail, any number of arbitrarily different streams (whirls and eddies) may be present, and we may in particular assume that there has taken place no sort of equalization between the different velocities of the molecules. To conceive of said differences we may assume beforehand, in perfectly arbitrary fashion, the velocities of the molecules as well as their co-ordinates of location. But there must exist (in order that we may know the state in the macroscopic sense), certain mean values of density and velocity, for it is through these very mean values that the state is characterized from the aggregate (macroscopic) standpoint. The differences that do exist in the successive stages of disorder of the unsettled state are mainly due to the molecular collisions that are constantly taking place, thus changing the velocity and locus of each molecule.