Physical significance of entropy or of the second law

by Joseph Frederic Klein

BOLTZMANN says this "normal" state is permanent (stationary) for given external conditions because magnitude

does not vary; such a normal state has many configurations, but all agree in having same number of complexions.

Also, "MAXWELL'S Velocity Distribution is not a state which assigns to each molecule a particular place (locus) and a particular velocity, which are reached say by the locus and velocity of each molecule asymptotically approaching said assigned locus and velocity. With a finite number of molecules MAXWELL'S state will never be exactly but only approximately realized. MAXWELL'S velocity is not a singular one which is confronted by an immense number of non-Maxwellian velocity distributions. On the contrary, among the immense number of possible velocity distributions by far the greater number possess the characteristics of the MAXWELL velocity distribution."

MAX PLANCK (Festschrift, p. 113) lucidly dwells on thermal equilibrium, entropy and temperature, as follows:

"The mechanical significance of the temperature idea is most closely connected with the mechanical significance of entropy, for the two are connected by

. By answering one of these questions we at the same time settle the other."

In the earlier days interest was naturally centered in the directly measurable magnitude temperature and entropy appeared as a more complicated idea which was to be derived from the former. Nowadays this relation is rather reversed and the prime question is to first explain entropy mechanically and this will then define temperature. The reason for this change of attitude is that in all such explanatory efforts to present Thermodynamics mechanically and give temperature a complete mechanical definition it is necessary to come back to the peculiarities of "thermal equilibrium." But the full significance of this equilibrium conception is only to be reached from the standpoint of irreversibility. For thermal equilibrium can only be defined as the final state toward which all irreversible processes strive. In this way the question as to temperature leads necessarily to the nature of irreversibility and this in turn is solely founded on the existence of the entropy function. This magnitude is therefore the primary, general conception which is significant for all kinds of states and changes of state, while temperature emerges from this with the help of the special condition of thermal equilibrium, in which condition the entropy attains its maximum.

[19]In MAXWELL'S distribution the molecules are assumed to be uniformly scattered throughout the unit volume; it is the velocities only that are variously distributed in the different elementary regions. To realize the haphazard character (necessary in Calculus of Probabilities) of the motions of the molecules, we must bear in mind that each of the molecules in the unit volume has a different velocity and direction; here no direction has preference over another, i.e., one direction of a molecule is as likely as another. Here at first we write expression for the number of molecules whose velocities parallel to the co-ordinate axes are respectively confined between the velocity limits: