Physical significance of entropy or of the second law

by Joseph Frederic Klein

In this connection BOLTZMANN says: (W. S. B. d. Akad. d. Wiss., Vol. LXVI, B 1872, p. 275).

"The mechanical theory of heat assumes that the molecules of gases are in no way at rest but possess the liveliest sort of motion, therefore, even when a body does not change its state, every one of its molecules is constantly altering its condition of motion and the different molecules likewise simultaneously exist side by side in most different conditions. It is solely due to the fact that we always get the same average values, even when the most irregular occurrences take place under the same circumstances, that we can explain why we recognize perfectly definite laws in warm bodies. For the molecules of the body are so numerous and their motions so swift that indeed we do not perceive aught but these average values. We might compare the regularity of these average values with those furnished by general statistics which, to be sure, are likewise derived from occurrences which are also conditioned by the wholly incalculable co-operation of the most manifold external circumstances. The molecules are as it were like so many individuals having the most different kinds of motion, and it is only because the number of those which on the average possess the same sort of motion is a constant one that the properties of the gas remain unchanged. The determination of the average values is the task of the Calculus of Probabilities. The problems of the mechanical theory of heat are therefore problems in this calculus. It would, however, be a mistake to think any uncertainty is attached to the theory of heat because the theorems of probability are applied. One must not confuse an imperfectly proved proposition (whose truth is consequently doubtful) with a completely established theorem of the Calculus of Probabilities; the latter represents, like the result of every other calculus, a necessary consequence of certain premises, and if these are correct the result is confirmed by experience, provided a sufficient number of cases has been observed, which will always be the case with Heat because of the enormous number of molecules in a body."

To become more specific we will mention some of the problems to which the Theory of Probabilities has been profitably applied. In business to life and fire insurance; in engineering to reducing the inevitable errors of observations by the Method of Least Squares; and in physics to the determination of Maxwell's Law of the distribution of velocities. The results thus obtained are universally trusted and accepted by experts. Why then should this Calculus not be applicable to the more general natural events?

In this connection consider some of its good points: (a) It eliminates from a problem the accidental elements if the latter are sufficiently numerous; (b) it deals legitimately with averages; (c) it involves combination considerations other than averages; (d) it is available for non-mechanical as well as mechanical occurrences and thus (e) has a capacity for covering the whole range of natural events, giving it a character of universality which is now its most valuable asset.

As an example of this we may instance BOLTZMANN'S deservedly famous H-theorem, which establishes the one-sidedness of all natural events.[7] Concerning it, this master in mathematical physics says:

"It can only be deduced from the laws of probability that, if the initial state is not especially arranged for a certain purpose, the probability that

decreases is always greater than that it increases. In this connection we may add that BOLTZMANN looked forward to a time, "when the fundamental equations for the motion of individual molecules will prove to be merely approximate formulas, which give average values which, according to the Theory of Probabilities, result from the co-operation of very many independently moving individuals constituting the surrounding medium, for example, in meteorology the laws will refer only to average values deduced by the Theory of Probabilities from a long series of observations. These individuals must of course be so numerous and act so promptly that the correct average values will obtain in millionths of a second."

To further strengthen our faith we may point out that the probability method has been successfully used to determine unique results from complicated conditions and has been employed for the general treatment of problems. In the case before us it has solved the entropy puzzle which has exercised physicists, as well as engineers, for decades, and it has thereby emancipated the Second Law from all anthropomorphism, from all dependence on human experimental skill. When we take the broadest possible view of its character, this Calculus enables us to read the present riddle of our universe, namely, why it is in its present improbable state. We have therefore in this Calculus an engine for investigation which is of great power and is likely to play a large part in the future in the ascertainment of physical truth. Of course it must then be in the hands of masters. It is they and they alone who can properly and adequately interpret such a physical problem as the one before us. In scientific work our last court of appeal must be Nature, and we therefore say: The best justification for the use of the Theory of Probabilities in our problem is that its results are in such complete accord with the facts.

In dealing with this physical engine of investigation, we must again call attention to some of the features of haphazard necessary for its legitimate application. Of course the statement of these features will vary with the mechanical or non-mechanical character of the problem to which it is applied. As we are here dealing mainly with the former, we will limit ourselves to its features: (a) The elements dealt with must be very numerous, strictly speaking, infinite; (b) as a phase of (a) we may say also that when we speak of the probability of a state we express the thought that it can be realized in many different ways; (c) when we speak of the relative directions of a pair of molecules all possible directions must be considered; (d) we must so weight the elements say, in (a), (b), and (c) that they are equally likely; (e) every one of the entering elements must possess constituents of which each individual is independent of every other; for instance, (f) in a gas the place where a molecule collided must be independent of the place where it collided before. In our physical problem all of these features are not always realized; for instance, the number of particles of gas are only finite instead of being infinite; again, all relative velocities after collision of a pair of molecules are not equally likely; BOLTZMANN and BURBURY provide for these shortcomings by very truly asserting that in actual cases we are not dealing with isolated systems, that the surrounding walls are not impervious to external influences, and that the latter come at haphazard without regard to internal state of the system at the time, thus renewing and maintaining the desired state of haphazard.