. If a body has as general a motion as possible, it may be resolved into translations parallel to the
axes and to rotations about these axes. Each of these two sets furnishes three components of motion or a total of six components; then we say that the perfectly unconstrained motion of the body has six degrees of freedom. If a body moves parallel to one of the co-ordinate planes, we say it has two degrees of freedom. When we come to consider molecular motion in general and the independence which characterizes the motion of each of the many molecules we see that altogether we have here an extraordinary number of degrees of freedom, and composed of such is the realm of our "elementary chaos."
If we go to the other extreme and think of only one atom, we see at once that we cannot properly speak of its disorder. But the case is different with a moderate number of atoms, say, a hundred or a thousand. Here we surely can speak of disorder if the co-ordinates of location and the velocity components are distributed by haphazard among the atoms. But as the process as a whole, the sequence of events in the aggregate, may not with this comparatively small number of atoms take place before a macroscopic observer in a unique (unambiguous) manner, we cannot say that we have here reached a true state of "elementary chaos." If we now ask as to the minimum number of atoms necessary to make the process an irreversible one, the answer is, as many as are necessary to form determinate mean values which will define the progress of the state in the macroscopic sense. Only for these mean values does the Second Law possess significance; for these, however, it is perfectly exact, just as exact as the theorem of probability, which says that the mean value of numerous throws with one cubical die is equal to 3½.
We may now properly infer from all these views that the state of "elementary chaos" (or "molekular ungeordnete" motion) is the necessary condition for adequate haphazard and makes the application of the Theory of Probabilities possible.
[4]On p. 133 of Wärmestrahlung PLANCK says, "only measurable mean values are kontrollierbar," and this may help us to get the meaning here.
[5]In BOLTZMANN'S H-Theorem we have a process (consisting of a number of separately reversible processes) which is irreversible in the aggregate.
[6]Nature, Vol. LI, p. 78, Nov. 22, 1894.
(3) Settled and Unsettled States; Distinction between Final Stage
of Elementary Chaos and its Preceding Stages