(1) The "State" of a Body and its "Change of State"
As we will make constant use of the terms contained in this heading and as they here represent fundamentally important conceptions, we will seek to make them clear by presenting them in the various forms into which they have been cast by the different investigators, even at the risk of being considered prolix.
In the Introduction to this article we called attention to the two distinct modes of attacking any physical problem. Now the conception "state of a body" varies with the chosen mode of attack. Of course as both modes are legitimate and lead to correct results, these differences in the conception of "state" can be reconciled and a broader definition reached. We can illustrate these different methods of approach, as PLANCK has done, by assuming two different observers of the state of the body, one called the microscopic-observer and the other the macroscopic-observer. The former possesses senses so acute and powers so great that he can recognize each individual atom and can measure its motion. For this observer each atom will move exactly according to the elementary laws prescribed for it by General Dynamics. These laws, so far as we know them, also at once permit of exactly the opposite course of each event. Consequently there can be here no question of probability, of entropy or of its growth. On the other hand, the "macro-observer," (who perceives the atomic host, say as a homogeneous gas, and consequently applies to its mechanical and thermal events the laws of thermodynamics) will regard the process as a whole to be an irreversible one in accordance with the Second Law.... Now a particular change of state cannot at the same time be both reversible and irreversible. But the one observer has a different idea of "change of state" from the other; the micro-observer's conception of "change of state" is different from that of the macro-observer. What then is "change of state?" The state of a physical system can probably not be rigorously defined, otherwise than the conception, as a whole, of all those physical magnitudes whose instantaneous values, under given external conditions, also uniquely determine the sequence of these changing values.
BOLTZMANN'S statement is much more clear, namely, "The state of a body is determined, (a) by the law of distribution of the particles in space and (b) by the law of distribution of the velocities of the particles; in other words, a body's condition is determined (a) by the number of particles which lie in each elementary realm of the space and (b) by a statement of the number of particles which belong to each elementary velocity group. These elementary realms are all equal and so are the elementary velocity groups equal among themselves. But it is furthermore assumed that each elementary realm and each elementary velocity group contains very many particles."
Now if we ask the aforesaid two observers what they understand by the state of the atomic host or gas under consideration, they will give entirely different answers. The micro-observer will mention those magnitudes which determine the location and the velocity condition of all the individual atoms. This would mean in the simplest case, in which the atoms are regarded as material points, that there would be six times as many magnitudes as atoms present, namely, for each atom there would be three co-ordinates of location and three of velocity components; in the case of composite molecules there would be many more such magnitudes. For the micro-observer, the state and the sequence of the event would not be determined until all these many magnitudes had been separately given. The state thus defined we will call the "micro-state." The macroscopic-observer on the other hand gets along with much fewer data; he will say that the state of the contemplated homogeneous gas is already determined by the density, the visible velocity and the temperature at each place of the gas and he will expect, when these magnitudes are given, that the course of the physical events will be completely determined, namely, will occur in obedience to the two laws of thermodynamics and therefore be bound to show an increase in entropy. The state thus defined we will call the "macro-state." The difference in the two observers is that one sees only the atomic events and the other the occurrences in the aggregate. The former would have the absolute mechanical idea of state and the latter the statistical idea. Before attempting to reconcile their apparently conflicting conclusions, we will here call attention to some necessary relations between the micro-state and the macro-state. In the first place we must remember that all a priori possible micro-states are not realized in nature; they are conceivable but never attain fruition. How shall we select what may be called these natural micro-states? The principles of general dynamics furnish no guide for such selection and so recourse may be had to any dynamic hypothesis whose selection will be fully justified by experience.
Now PLANCK says: "In order to traverse this path of investigation, we must evidently first of all keep in mind all the conceivable positions and velocities of the individual atoms, which are compatible with particular values of the density, the velocity and the temperature of the gas, or, in other words, we must consider all the micro-states which belong to a particular macro-state and must examine all the different events which follow from the different micro-states according to the fixed laws of dynamics. Now up to this time, the closer calculation and combination of these minute elements has always given the important result that the vast majority of these micro-states belong to one and the same macro-state or aggregate, and that only comparatively few of the said micro-states furnish an anomalous result, and these few are characterized by very special and far-reaching conditions existing between the locations and the velocities of adjacent atoms. And, furthermore, it has appeared that the almost invariably resulting macro-event is just the very one perceived by the macroscopic observer, the one in which all the measurable mean values have a unique sequence, and consequently and in particular satisfies the second law of thermodynamics."
"Herewith is revealed the bridge of reconciliation between the two observers. The micro-observer needs only to take up in his theory the physical hypothesis, that all such particular cases (which premise very special, far-reaching conditions between the states of adjacent and interacting atoms) do not occur in Nature; or in other words, the micro-states are in 'elementary disorder' (elementar ungeordnet). This secures the unique (unambiguous) character of the macroscopic event and makes sure that the Principle of the Growth of Entropy will be satisfied in every direction."
Before elaborating all that is implied in this hypothesis of "elementary disorder" we will again point out that for each macro-state (even with settled values of density and temperature) there may be many micro-states which satisfy it in the aggregate.
According to PLANCK, "it is easy to see that the macro-observer deals with mean values; for what he calls density, visible velocity, temperature of the gas, are for the micro-observer certain averages, statistical data, which have been suitably obtained from the spatial arrangement and the velocities of the atoms. But with these averages the micro-observer at first can do nothing even if they are known for a certain time, for thereby the sequence of events is by no means settled; on the contrary, he can easily with said given averages ascertain a whole host of different values for the location and velocities of the individual atoms, all of which correspond to said given averages, and yet some of these lead to wholly different sequences of events even in their mean values," events which do not at all accord with experience. It is evident, if any progress is to be made, that the micro-observer must in some suitable way limit the manifold character of the multifarious micro-states. This he accomplishes by the hypothesis of "elementary disorder" about to be more fully defined.
In passing we may here note for future use, that what has just been said concerning macro-states (aggregates) with "settled" mean velocity, density and temperature, applies also to states unsettled in the aggregate, so far as concerns the manifold character of the conceivable constituent micro-states and the differences in the mean character of their sequences. Even after the above limiting hypothesis removes all illegitimate micro-states, an enormously greater number of legitimate ones will be left to constitute the number of complexions properly belonging to the state contemplated. We may also add that it seems quite evident that the numbers representing these complexions will be different in the settled and unsettled states even if the latter should ultimately possess the mean velocity, density and temperature of the former.
On the other hand, we also point out that for one and the same set of external conditions the macro-state may itself vary very greatly. When it has a settled density and temperature, it is said to be in a stationary state, to be in thermal equilibrium and, anticipating, we may add that it is then has maximum entropy, in short we may say it is in a "normal" condition. But the external conditions remaining the same, before attaining to said "normal" ultimate state, it may pass through a whole series of so-called "abnormal" states after it leaves its initial condition. While it is in any one of these "abnormal" states, it may be said to be in a more or less turbulent condition; it may then possess whirls and eddies; it may have different densities and temperatures in its different parts and then it will be difficult or impossible to measure these external physical features of its state as a whole. All this implies ever-varying atomic locations and velocities, but does not indicate any such special far-reaching regularities between adjacent and interacting particles as would vitiate at any stage our hypothesis of "elementary disorder" (elementar ungeordnet) or "molecular chaos."
Before going into more detail concerning this particular chaotic condition of the particles we will give PLANCK'S somewhat fuller statement of what constitutes the "state" of a physical system at a particular time and under given external conditions. It is, "the conception as a whole of all those mutually independent magnitudes which determine the sequence of events occurring in the system so far as they are accessible to measurement; the knowledge of the state is therefore equivalent to a knowledge of the initial conditions. For example, in a gas composed of invariable molecules the state is determined by the law of their space and velocity distribution, i.e., by the statement of the number of molecules, of their co-ordinates and velocity components which lie within each single small region. The number of molecules in any one of these different regions is in general entirely independent of the number in any other region, for the state need not be a stationary one nor one of equilibrium; these numbers should therefore all be separately known if the state of the gas is to be considered as given in the absolute mechanical sense. On the other hand, for the characterization of the state in the statistical sense, it is not necessary to go into closer detail concerning the molecules present in each elementary space; for here the necessary supplement is supplied by the hypothesis of molecular chaos, "which in spite of its mechanically indeterminate character guarantees the unambiguous sequence of the physical events."
(2) Further Elucidation of this Essential Condition of "Elementary Chaos." Sundry Aspects of Haphazard
To gain as complete an understanding as possible of this fundamental idea we will now give the views of the several investigators as to the physical features of this chaotic state. We have seen how PLANCK, the chief expositor of BOLTZMANN, boldly excludes from consideration all cases leading to anomalous results, because of the very special conditions existing between the molecular data, by assuming that these cases do not occur in Nature. PLANCK reminds the physicists who object to the hypothesis of elementary disorder because they feel it is unnecessary or even unjustifiable, that the hypothesis is already much used in Physics, that tacitly or otherwise it underlies every computation of the constants attached to friction, diffusion and the conduction of heat. On the other hand he reminds others, those inclined to regard the hypothesis of "elementary disorder" as axiomatic, of the theorem of H. POINCARÉ, which excludes this hypothesis for all times from a space surrounded with absolutely smooth walls. PLANCK says that the only escape from the portentous sweep of this proposition is that absolutely smooth walls do not exist in Nature.
The foregoing thought PLANCK has also put in a slightly different way. Appreciating that all mechanically possible simultaneous arrangements and velocities of molecules are not realized in Nature, the concept of "elementary disorder" implies one limitation of the conceivable molecular states, namely that, between the numerous elements of a physical system there exist no other relations than those conditioned by the existing measurable mean values of the physical features of the system in question.
Another, briefer but equivalent, definition is that: "In Nature all states and processes which contain numerous independent (unkontrollierbar) constituents are in 'elementary disorder' (elementar ungeordnet)." The constituents are molecular elements in mechanics and in thermodynamics and the energy elements in radiation.
The German word "unkontrollierbar"[4] here used may also with some justice be translated as, unconditioned, undetermined, unmeasurable, unregulated, uncorrelated, ungovernable or haphazard. But whichever term is best, PLANCK, mechanically speaking, meant by it, the confused, unregulated and whirring intermingling of very many atoms.
Either of these two equivalent definitions implies that such elementary disorder or chaos is a condition of sufficiently complete haphazard to warrant the application of the Theory of Probabilities to the unique (unambiguous) determination of the measurable physical features of the process viewed as a whole.
The foregoing ideas more or less tacitly underlie the whole of BOLTZMANN'S great pioneer work in this vast field. He it was who clearly showed that the Second Law could be derived from mechanical principles: that entropy was a property of every state, turbulent or otherwise; that the entropy idea would be emancipated from all thought of human, experimental, skill, and who thereby raised the Second Law to the position of a real principle. He did all this by a general basing of the idea of entropy on the idea of probability. Consequently we find much attention paid in all his work to haphazard molecular conditions. He first used the terms "molekular-geordnet" (molecularly ordered, or arranged), and "molekular ungeordnet" (molecularly disordered or disarranged), which latter phrase we must regard as synonymous with the term "elementar ungeordnet" (elementary disorder or chaos) with which we have already become acquainted in PLANCK'S presentation. We will, therefore, confine ourselves here to BOLTZMANN'S illustrations of these terms, for his work does not, in these particulars, contain any sharp definitions. Indeed he may have feared over-precision and may have trusted to the use he made of the terms at different times to convey their meaning.
Concerning some of the characteristics of BOLTZMANN'S haphazard motion we take the following from Vol. I of his "Vorlesungen über Gas Theorie."
If in a finite part of a gas the variables determining the motion of the molecules have different mean values from those in another finite part of the gas (for example if the mean density or mean velocity of a gas in one-half of a vessel is different from those in the other half), or more generally, if any finite part of a gas behaves differently from another finite part of a gas, then such a distribution is said to be "molar-geordnet" (in molar order). But when the total number of molecules in every unit of volume exists under the same conditions and possesses the same number of each kind of molecules throughout the changes contemplated, then the same number of molecules will leave a unit volume and will enter it so that the total number ever present remains the same; under such conditions we call the distribution "molar-ungeordnet" (in molar disorder) and that finite distribution is one of the characteristics of the haphazard state to which the Theory of Probabilities is applicable. [As another illustration of the excluded molar-geordnet states we may instance the case when all motions are parallel to one plane.]
But although in passing from one finite part to another of a gas no regularities (of average character) can be discerned, yet infinitesimal parts (say of two or more molecules) may exhibit certain regularities, and then the distribution would be "molekular-geordnet" (molecularly-ordered) although as a whole the gas is "molar-ungeordnet." For example (to take one of the infinite number of possible cases) suppose that the two nearest molecules always approached each other along their line of centers, or if a molecule moving with a particularly slow speed always had ten (10) slow neighbors, then the distribution would be "molekular-geordnet." But then the locality of one molecule would have some influence on the locality of another molecule and then in the Theory of Probabilities the presence of one molecule in one place would not be independent of the presence of some other molecule in some other place. Such dependence is not permissible by the Theory of Probabilities. Before, however, we can further describe what is here perhaps the most important term (molekular-ungeordnet), we must point out that BOLTZMANN considers the number of molecules of one kind whose component velocities along the co-ordinate axes are confined between the limits, and also the number of molecules of another kind whose component velocities similarly lie between the limit then, considering the chances that a molecule shall have velocities between the limits specified in (1) and molecule have velocities between limits (2), BOLTZMANN intimates that these chances are independent of the relative position of the molecules. Where there is such complete independence, or absence of all minute regularities, the distribution, according to BOLTZMANN, is "molekular-ungeordnet" (molecularly-disordered).
BOLTZMANN furthermore informs us that, as soon as in a gas, the mean length of path is great in comparison with the mean distance between two adjacent molecules, the neighboring molecules will quickly become different from what they formerly were. Therefore it is exceedingly probable that a "molekular-geordnete" (but molar-ungeordnete) distribution would shortly pass into a "molekular-ungeordnete" distribution.
Furthermore, from the constitution of a gas results that the place where a molecule collided is entirely independent of the spot where its preceding collision took place. Of course, this independence could be maintained for an indefinite time only by an infinite number of molecules.
The place of collision of a pair of molecules must in our Theory of Probabilities be independent of the locality from which either molecule started.
From all the preceding we must infer what measure of haphazard BOLTZMANN considers necessary for the legitimate use of the Theory of Probabilities.
BOLTZMANN in proving his H-Theorem,[5] which establishes the one-sidedness of all natural events, makes the explicit assumption that the motion at the start is both "molar- und molekular-ungeordnet" and remains so. Later on, he assumes the same things but adds that if they are not so at the start they will soon become so; therefore said assumption does not preclude the consideration by Probability methods of the general case or the passage from "ordnete" to "ungeordnete" conditions which characterizes all natural events.
In fact these very definitions show solicitude for securing the uninterrupted operation of the laws of probability. BOLTZMANN intimates his approval of S. H. BURBURY'S statement of the condition of independence underlying his work.
Here S. H. BURBURY[6] simplifies the matter by assuming that any unit of volume of space contains a uniform mixture of differently speeded molecules and then says:
"Let be the velocity of the center of gravity of any pair of molecules and their relative velocity. Then the following condition (here called ) holds: For any given direction of before collision, all directions of after collision are equally probable. Then BOLTZMANN'S H-theorem proves that if condition be satisfied, then if all directions of the relative velocity for given are not equally likely, the effect of collisions is to make diminish." [In essence BURBURY'S condition says no more than that Theory of Probabilities is applicable for finding number of collisions.] Furthermore, "any actual material system receives disturbances from without, the effect of which coming at haphazard without regard to state of system for the time being is, pro tanto, to renew or maintain the independence of the molecular motions, that very distribution of co-ordinates (of collision) which is required to make diminish. So there is a general tendency for to diminish, though it may conceivably increase in particular cases. Just as in matters political, change for the better is possible, but the tendency is for all change to be from bad to worse." Here BURBURY states what is practically true in all actual cases and thus furnishes an additional reason, if that were needed, for the legitimacy of the Probability method pursued by Boltzmann, and, another explanation of why the results obtained are in such perfect accord with experience.
As BURBURY'S remarks with respect to the nature of "elementary chaos" under consideration are always illuminating, we will, at the risk of repeating something already said, quote the following:
"The chance that the spheres approaching collision shall have velocities within assigned limits is independent of their relative position, and of the positions and velocities of all other spheres, and also independent of the past history of the system except so far as this has altered the distribution of the velocities inter se. In the following example this independence is satisfied for the initial state and, for the assumed method of distribution, has no past history.
"Example. A great number of equal elastic spheres, each of unit mass and diameter , are at an initial instant set in motion within a field of no force and bounded by elastic walls. The initial motion is formed as follows: (1) One person assigns component velocities to each sphere according to any law subject to the conditions that and that given constant. (2) Another person, in complete ignorance of the velocities so assigned, scatters the spheres at haphazard throughout . And they start from the initial positions so assigned by (2) with the velocities assigned to them respectively by (1)."
The system thus synthetically constructed would without doubt, at the start be "molekular-ungeordnet"—in fact, it is as near an approach to chaos as is possible in an imperfect world. But there is reason to doubt if it would continue to be thus "molekular-ungeordnet." For the distribution of velocities is according to any law consistent with the above-mentioned conditions and some such laws would lead to results hostile to the Second Law, and then we may safely say such laws of velocity distribution would never occur in Nature and would therefore belong to the cases which have been specially excepted.
Now there are mechanical systems which possess the entropy property and it has been truly said that the Second Law and irreversibility do not depend on any special peculiarity of heat motion, but only on the statistical property of a system possessing an extraordinary number of degrees of freedom. In this sense Professor J. W. GIBBS treated Mechanics statistically and showed that then the properties of temperature and entropy resulted. This matter has already been touched upon, but as numerous degrees of freedom is a feature of the "elementary chaos" under consideration it deserves repetition here and more than a passing mention.
Illustration of Degrees of Freedom. Refer a body's motion to three axes, . 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
The immediate purpose in the next few pages is to establish the (a) distinction between the successive stages of "elementary disorder" (chaos) as they develop in their inevitable passage from "abnormal" conditions to the final and so-called "normal" condition of thermal equilibrium and, furthermore, (b) to show that each of these stages is "elementar-ungeordnet" and (c) that in each one sufficient haphazard prevails to permit the legitimate application of the Theory of Probabilities.
We will first describe the unsettled (abnormal) and settled (normal) states, respectively. 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. 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. 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 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 and which thus change the locus and velocity of each molecule.
We may now easily describe the settled state as a special case of the unsettled one. In the settled state there is an equal distribution of density throughout all the elementary spaces, there are no different streams (whirls or eddies) present, and an equal partition of energy exists for all the elementary spaces. For it thermal equilibrium exists, the entropy is a maximum, and temperature of the state has now a definite meaning, because temperature is the mean energy of the molecules for this state of equilibrium. The condition is said to be a "stationary" or permanent one, for the mean values of the density, velocity, and temperature of this particular aggregate no longer change, although molecular collisions are still constantly occurring.
Well-known examples of the unsettled state of a system are: The turbulent state with its different streams, whirls, and eddies, the state in which the potential and kinetic energy is unequally distributed; for instance, when one part is at a high pressure and another part at a lower pressure, when one part is hotter than another part, and when unmixed gases are present in a communicating system.
A more specific feature of the unsettled state may be found in the accompaniment to BURBURY'S condition (already mentioned at bottom of p. 15) where it is intimated that (at the start and after collision) all directions of the relative velocity may not be equally likely.
When such differences have all disappeared to the extent that equal elementary spaces possess their equal shares of the different particles, velocities, and energies, the system will be a settled one, be in thermal equilibrium, and will possess a maximum entropy and a definite temperature. Moreover, BURBURY'S condition 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.