On the Logical Justification of Combinatorial Methods for Determining the Entropy of an Ideal Gas

B. Kedrov, Leningrad

Introduction. 1. The concept of randomness. 2. The concept of thermodynamic probability \(W\). 3. The two basic premises of the calculation of \(W\). 4. Classification of methods for calculating \(W\). 5. The first method for calculating \(W\) from the formal-mathematical side. 6. Classification of combinations according to the features of permutations. 7. The sequence in which permutations arise in the process of gas formation. 8. Examination of the methods for calculating \(W\) from the logical side. 9. The third premise of the calculation of \(W\). 10. Critique of the principle of the absolute identity of molecules. 11. Correction of the theory in accordance with objective reality. 12. Conclusion.

The pages of this journal have more than once carried articles devoted to the creation and development of the new quantum statistics in connection with concrete physical problems. In the articles of I. E. Tamm (1926), Sommerfeld (1928), Jordan (1930), Darrow (1930), Fowler (1931), and others, the enormous role of the new statistics as a method of investigation in a whole series of fields of contemporary physics and, above all, in the field of the theory of the structure of matter, found its reflection. The new content of physical theories required a new mathematical form corresponding to it, a new statistical apparatus by means of which it would be possible to solve those concrete physical problems that again and again confront human knowledge as it penetrates into the very depths of the nature of matter. At the same time, the old, seemingly insoluble contradictions and paradoxes to which classical statistics led proved to be comparatively easily and simply resolvable with the aid of the new, more perfect and correct approach to the study of reality.

As is well known, the new statistics is developing in two principal directions, and the peculiarity of each of these directions depends entirely on the character of the physical object under investigation (the physical aggregate or collective) to the study of which the given form of statistics is applied. On the one hand, the development of the field of ideal gases and, in particular, of the theory of radiation led to the creation of Bose–Einstein statistics; on the other hand, the development mainly of the electron theory of metals conditioned the development of Fermi–Dirac statistics, whose basis is the well-known Pauli principle.

On these theories one can historically trace the fact that both the birth of quantum statistics, and its branching into special forms, and the paths of the further development of each of these forms entirely follow from the character of the physical problems themselves, in whose solution quantum statistics is called upon to play, though a very active one, nevertheless only an auxiliary role. We are entitled to say that in modern physics statistics is not a self-sufficient factor and, on the whole, occupies a subordinate position. Therefore any exhaustive and, especially, historical investigation of physical statistics must above all proceed from an analysis of the concrete physical problems with which the given form of statistics is connected.

However, we may also approach the consideration of physical statistics from the purely combinatorial side, since combinatorics makes it possible to connect with one another such two quantities as the entropy of a system \(S\) and the thermodynamic probability of its state \(W\)*. Analyzing, from the combinatorial point of view, the connection between these fundamental quantities, we can reveal certain logically interesting points common to the most diverse methods of statistically determining \(S\), which usually remain in the background when particular cases of the application of statistics to concrete physical problems are investigated. Such a consideration is of interest also because, up to now, comparatively little attention has been paid to the general logical foundation of the combinatorial methods for calculating \(W\) and to their selection for one or another purpose of physical research. At the same time it is necessary once more to emphasize that, in confining ourselves primarily to the field of physical combinatorics, we by no means wish thereby to diminish the leading significance of those properly physical problems without connection with which physical statistics in itself would have no meaning and would not have received its development at all.

1

Combinatorial determination of \(S\), at first glance, seems to be a simple computational operation, a simple counting of a certain number of combinations of molecules, or of other physical individuals of some kind, composing the given ensemble whose entropy we wish to calculate. The aim of the present article is to show that this operation has a deep logical basis in the doctrine of chance and necessity, the role of the chance and necessary aspects being played by different sides in the motion and state of the physical ensemble under investigation.

* In what follows we shall denote the entropy of a system by the letter \(S\), and its thermodynamic probability by the letter \(W\).

Let us briefly outline the general path of our investigation. First of all, we reveal the opposing moments that underlie the combinatorial determination of $S$. Then, considering them in their unity, we arrive at an understanding of the essence of the calculus $W$, which in its most general form is the unity of chance and necessity. Tracing further the mutual connection of the revealed opposites and their transition into one another, we arrive at an understanding of the mutual connection of the combinatorial methods of determining $S$.

Finally, considering the accidental and necessary moments in the calculation of $W$, their unity and their transition into one another as a reflection of the accidental and necessary moments of reality itself, contained in the relation between the mechanical behavior of individual molecules and the thermodynamic behavior of the gas as a whole, i.e. in the relation between the individual and the collective, we make an attempt to eliminate certain purely formal assumptions accepted in contemporary statistics, and to give logical grounding to the most correct method of the combinatorial determination of $S$ for an ideal gas.

To solve the problem posed, we shall have to involve a number of logical categories; therefore we shall give a brief definition of at least two of them, as the most important for our investigation: chance and necessity.

A preliminary definition of these categories may be constructed by way of a simple opposition of the accidental to the necessary. Considering some event in relation to another event, we shall call the first accidental with respect to the second if it does not follow necessarily from that lawful connection in which the second event develops. Thus the motion of an individual molecule (its momentum and position inside the gas) is accidental with respect to the state of the entire gas, for it does not follow from the law-governed character of the whole collective of molecules as a whole; likewise, the collision of two molecules is accidental, for the displacement of one molecule does not follow from the displacement of another molecule. Conversely, if one event follows from the nature of another in such a way that the first proves to have been prepared by the entire course of development of the second, then the connection between the two events we shall define as necessary. Thus a change in the total volume of a gas necessarily leads to a change in the properties of the collective of molecules, above all in their internal interaction, consisting of their continuous collisions, which is then manifested in the form of a change in the spatial distribution of the molecules and in the pressure of the gas. Hence the connection between the volume of the gas, the spatial distribution of its molecules, and its pressure is established as necessary; this is also expressed by Boyle–Mariotte’s law.

In short, we may preliminarily define the accidental as that which could have occurred in this way, but could also have occurred in another way.

to another, while the necessary is that which must necessarily occur only in this way, and not in any other way.

A complete and developed definition of the two opposite categories can be given only when they are considered in unity and in their mutual connection with one another; this definition consists in the fact that necessity contains within itself its own negation—contingency; that necessity consists of contingencies, and that behind contingencies necessity is concealed; in short, in the fact that contingency is a form of manifestation of necessity in nature and society.

In this definition of contingency given by Engels, two basic aspects must be emphasized: first, the theoretical-cognitive, gnoseological aspect, consisting in the fact that contingency is defined as a form inherent in objective reality itself and reflected in our consciousness as a logical category; second, the methodological aspect, consisting in the fact that contingency is defined in unity with its opposite—with necessity.

These two aspects are not separated from one another and constitute a single whole.

The definition of contingency as a form of manifestation of necessity stands in sharp opposition to all other definitions of this category, which are given: first, by objective idealists: contingency is complete causelessness, absolute indeterminism, utter arbitrariness; second, by subjective idealists: contingency, like every logical category in general, is only a form of ordering our experience and, consequently, is devoid of objective significance; third, by idealist-rationalists: contingency is an a priori form of our thinking; fourth, by metaphysicians of the Wolffian type: contingency is absolutely torn away from necessity; the two opposites absolutely exclude one another; fifth, by mechanists: contingency is unrecognized necessity, is that whose causes are at present unknown to us, in other words—contingency has an exclusively subjective character; and, finally, sixth, by Plekhanov–Kurno: contingency is the point of intersection of necessary processes. The first five definitions prove unsuitable for the purposes of our investigation; the application of the last is restricted to the domain of internally unconnected and, in this sense, mechanical phenomena of nature.

2

Relying on the developed definition of contingency given by Engels, let us now analyze the concept of thermodynamic probability \(W\).

As the object of our investigation will serve

the simplest physical collective—namely: a highly rarefied, monatomic, chemically homogeneous gas in equilibrium, to whose behavior, with sufficient approximation, the laws of the so-called ideal gases are applicable. At the same time, in order to obtain the derivation in its purest form, reflecting only the essential point in the relation we are studying between the behavior of the individual and that of the collective, we shall abstract from the infinite complexity of the structure of molecules and shall consider them abstractly—as the simplest systems, completely subject in their motion to the laws of ordinary mechanics.

The general problem, both of calculating \(W\) and of the combinatorial determination of \(S\) for an ideal gas, consists in establishing, by means of the method of probability theory, from the quantitative side, the form of the dependence between the random mechanical motion of individual molecules and the necessary thermodynamic state of the gas as a whole.

In general form the dependence between \(S\) and \(W\) was established by Boltzmann:

\[ S = k \ln W. \tag{1} \]

Equation (1) is the starting point for all methods in general of the statistical determination of \(S\), which differ among themselves according to the way in which the value \(W\) is established.

By the thermodynamic probability \(W\), according to Planck’s definition, is understood the number of equally possible combinations corresponding to the microstates of systems, by means of which the given general state of the system (macrostate) can be realized.

Passing to the logical analysis of this basic concept, we state that Planck’s definition of \(W\) is constructed on the basis of two features not only of different, but also of directly opposite character: first, of a single combination corresponding to the macrostate of the system, by whose change we judge the change in the general state of the gas; and second, of a certain number of combinations of another kind, corresponding to all possible microstates of the system, by counting which we determine the probability of the given definite macrostate of the system.

Taking into account the general problem of calculating \(W\), we conclude that the combination of the first kind must be connected in an essentially necessary way with the thermodynamic state of the system and, consequently, with its \(S\); the combination of the second kind must be connected with the mechanical motion of individual molecules, must arise owing to their motion and, consequently, in relation to the whole gas as a whole must be random.

Thus the opposition of both combinations is reduced to the general opposition of the random and the necessary.

moments in the motion of a gaseous collective. This opposition is not of an absolutely polar character; the two combinations are not only not torn away from one another, but, on the contrary, are connected with one another in such a way that one cannot exist without the other, receiving its full determination only in unity with its opposite. As a concrete example we may take classical statistics, which held that the combination corresponding to the necessary behavior of the whole gas is the distribution of molecules, while the combinations by means of which distributions are realized are combinations of molecules. In other words, the random combination of molecules is the form behind which there is concealed, and in which there is manifested, their necessary distribution.

Developing this proposition further, we establish the different aspects of the interrelation of chance and necessity in their application to the concrete problem of physical combinatorics. We state that the necessary distribution cannot exist by itself as such, independently and outside the random combinations of individual molecules. It is precisely through these chances, and not in any other way, that the necessity of the behavior of the whole gas is manifested. For their part, random combinations do not represent purely external events, devoid of internal connection and appearing only on the surface of phenomena; on the contrary, behind individual combinations there is always concealed a distribution as the necessary combination of molecules.

All this means that the determinacy of the behavior of the gas, the determinacy of its necessity, consists precisely in the fact that within this necessity is contained its opposite, its negation—chance.

The proposition that “necessity consists of chances” is concretized in our case by the purely quantitative equality \(W = Z^N\). Here the necessity of the behavior of the gas (the distribution of its molecules) is expressed as equal to the number \(Z^N\) of counted chances (combinations).

Since the necessity inherent in the gas can exist only by being embodied in these chances, we are entitled to say that the random is necessary, for it must necessarily give form to the behavior of the whole gaseous collective as a whole, and that, at the same time, the necessary is random, for only in the form of the random does it assume a definite aspect, and it does not exist outside this form.

But if an individual combination of molecules is random when it is considered from the standpoint of the necessary thermodynamic behavior of the whole gas, then, on the other hand, from the standpoint of the necessary mechanical behavior of all the molecules composing the gas, this random combination reveals itself rather as an absolute necessity, as the result of the realization of a real possibility for each molecule to find itself in the given

moment in time at a given point in space and possess the given velocity.*

Thus the logical basis of the concept \(W\), hidden beneath the formal-mathematical shell of Planck’s definition, is the general proposition—formulated long ago by Engels—that chance is the form in which necessity manifests itself.

Here we encounter the indisputable fact that physics, in the course of its development, retrospectively not only fully confirms the correctness of general propositions that had long before been developed by philosophy, but, moreover, spontaneously, in addition to, and often contrary to, the views of many physicists, begins to rely on these general propositions. To remove any possible doubt about this, let us compare the following statement by Engels with our analysis of the definition of \(W\). Examining the views of the metaphysicians, some of whom regarded “chance and necessity as categories that absolutely exclude one another,” while others—the mechanists—denied chance altogether, Engels wrote in his Dialectics of Nature as early as 1881–1882: “In opposition to both these views Hegel comes forward with propositions unheard of until then, namely that the accidental has a ground because it is accidental, but just as much has no ground because it is accidental; that the accidental is necessary, that necessity determines itself as chance, and that, on the other hand, this chance is rather absolute necessity (Logik, Book II, section ‘Actuality’). Natural science preferred to ignore these propositions as self-contradictory nonsense, as a paradoxical play on words, theoretically remaining stuck, on the one hand, in the vacuity of Wolffian metaphysics, according to which something is either accidental or necessary, but in no case both at the same time; or, on the other hand, in an equally vacuous mechanical determinism, which in words denies chance in general, only to recognize it in practice in every individual case.”

Let us note further that the treatment of a combination of molecules precisely as a chance form in which the necessary tendency in the change of a gas manifests itself is the only correct treatment on the basis of which one can logically construct the concept, and hence the calculation, of the thermodynamic probability \(W\). Any other definition of chance, such as is given by idealists and metaphysicians, reveals its complete inadequacy as soon as we try to apply it to the analysis of the concept \(W\). Thus

* In what follows we shall show that in reality the absolutely necessary, from the point of view of the mechanical behavior of individual molecules, is only one arrangement; in actual fact, from the point of view of an individual molecule (the displacement of which we consider in the general case, abstracting from the displacement of other molecules), what is absolutely necessary is its transition from one cell to another, i.e., a single uncompensated rearrangement, which precisely underlies the formation of an arrangement of molecules.

Thus physics confirms the correctness of the proposition of dialectical materialism, as Lenin pointed out, and refutes idealistic attitudes, as well as the erroneous propositions of inconsistent metaphysical materialism.

Finally, let us note that we shall arrive at exactly the same conclusion if we subject to logical analysis any other combinatorial method of determining \(S\).

3

Analysis of the concept \(W\) leads directly to the conclusion that any method based on the calculation of \(W\) must have as its logical basis the establishment of two opposite combinations—one as necessary, the other as accidental, serving as the form of manifestation for the former. At the same time, all accidental combinations must be equiprobable among themselves, owing to which they can be taken as a scale, i.e., as the unit of enumeration of \(W\), and with their aid the value of \(W\) can be measured (computed) numerically. Hence follow two basic premises on the basis of which one or another combinatorial method of determining \(S\) may be constructed: first, the choice of a combination essentially and necessarily connected with \(S\), and, second, the selection of the unit of enumeration of \(W\). But in order to establish these basic premises correctly, it is necessary first to establish an exact characterization of all the various combinations.

First of all, let us note that any combination of molecules changes only as a result of the displacement of individual molecules and changes in their velocities. We shall call a permutation of a molecule any change in the values of its coordinates and its momenta, when this change is considered by us only from the point of view of the initial and final mechanical state of the molecule, independently of the path by which the molecule passed from one state to another. Hence any change and emergence of any combination of molecules may be represented as the result of permutations of individual molecules.

Let us now characterize the individual combinations. The total number of different kinds (orders) of combinations is six:

1) Arrangement of molecules (Permutation)*, whose feature is the position of the molecules inside elementary cells; it changes with the simultaneous permutation (exchange of places) of two molecules located in one and the same cell.

2) Placement of molecules (Elargissure); its feature is the presence of a group of individual molecules in a definite cell; it changes with the permutation of two identical groups located in different cells.

3) Combination of molecules (Combinaison); its feature is

* Usually this combination is inaccurately called a permutation; for convenience we retain the former French term.

finding individual molecules in definite cells; it changes when two molecules located in different cells are interchanged.

4) Distribution of molecules; its characteristic is the numbers \(n_i\) of molecules located in each \(i\)-th cell; it changes when a single molecule is transferred from one cell to another (i.e., when the system of numbers \(n_i\) is changed).

5) Grouping of molecules; its characteristic is the numbers \(\pi_n\) of groups (cells) containing \(n\) molecules each; it changes when the system of numbers \(\pi_n\) is changed.*

6) Répartition of molecules; its characteristic is the total number of cells \(Z\) and the total number of molecules \(N\); it is the aggregate of all possible, under the given conditions, arrangements of \(N\) molecules among \(Z\) cells; it changes when the numbers \(Z\) or \(N\) are changed.

Let us now consider how these combinations are connected with one another.

The universal, law-governed connection of all natural phenomena is not a uniform, abstractly necessary connection manifesting itself in exactly the same way at all stages in the development of matter; on the contrary, it is bound up with the qualitative specificity of one or another form of the motion of matter, and at each stage in the development of matter it appears in its own particular form; if, at the lower stages, it revealed itself as necessary, then at the higher stages, in this same part of it, it may already appear as accidental, revealing its necessity with respect to other, more complex phenomena.

This proposition can be demonstrated by the relation of two forms of the motion of matter, one of which represents the form of motion of the individual, and the other the form of motion of the collective—in particular, by the relation between the mechanical motion of molecules and the thermodynamic state of a gas. We have already seen how combinations of molecules that are accidental with respect to the behavior of the gas as a whole, in another connection—namely, with respect to the mechanical motion of the molecules—proved to be no longer accidental, but rather absolutely necessary.

Arranging in a sequence only the combinations we have just analyzed, we can establish various intermediate stages which, from the combinatorial point of view, will characterize the successive transition from the necessary behavior of individual molecules within the gas, through the necessary behavior of their individual groups, to the equally necessary behavior of their collective as a whole. Each such stage we shall call an order of combinations.

Each order reveals itself as accidental in relation to the higher order and as necessary in relation to the lower order.

* Arrangement, distribution, and grouping of molecules from the formal side may be characterized respectively as arrangement, combination, and distribution of cells, whose characteristic is the \(n_i\) of molecules contained in them.

As a result, we obtain, as it were, a certain chain or ladder of reciprocal transitions from the necessary to the accidental and back, where each pair of steps, i.e. of orders of combinations, is connected with one another by the relation of the accidental and the necessary. Therefore, when considered in this chain of successive transitions from one opposite into another, each order of combinations appears as a link in the common chain, having its own definiteness in that in one connection it reveals itself as accidentality, and in another connection—as necessity.

For brevity we shall denote the various combinations by the symbols \(R, G, D, C, E\), and \(P\), taking the initial letters of the corresponding French terms. Let us further agree, when composing formulas and relations, to enclose a given symbol in square brackets \([\,]\) when the corresponding combination is considered in the connection in which it appears as accidental; in braces \(\{\,\}\), when it appears as necessary; and in mixed brackets \([\,\}\), when it appears, on the one hand, as accidental and, on the other—as necessary. Let us also agree to arrange the combinations from right to left in accordance with the transition from higher orders to lower ones.* Then the general ladder of all combinations of molecules may be represented by the following relation:

\[ \{R\}\ldots [G]\ldots [D]\ldots [C]\ldots [E]\ldots [P]. \]

We shall call this sequential series of combinations the combinational series.

Each combination of a definite order, on the one hand, is an accidental form through which definite combinations of higher orders are realized, i.e. those located in the scheme to its left; and at the same time, on the other hand, it is itself realized as necessary through one or another combina-

* Comparing the orders \(C\) and \(E\), we see that \(E\) is at the same time a rearrangement of all \(\eta_i\)-molecules contained in the \(i\)-th cell in comparison with the same number of \(\eta_i\)-molecules constituting the contents of any other cell within the given combination \(C\), without causing it to change; only a rearrangement of empty cells can occur, i.e. of groups formed from 0-molecules. Therefore, approaching the question from the purely formal side, we may say that \(C\) appears as necessary in relation to \(E\), for the same \(C\) formally (but not really) corresponds to one and the same \(\tau_0!\) arrangements. The very same, in essence, also applies to the relation between each \(E\) and other combinations. But, obviously, since we consider combinations as ultimately determined by the interaction of molecules, the rearrangement of empty cells is devoid of any physical meaning, since it has no relation whatever to the real interaction between molecules. Therefore \(E\) must be considered only as an auxiliary combination, by means of which we eliminate from the total number \(Z!\) of rearrangements of groups the number of rearrangements \(\Pi_i \pi_i!\), immaterial for \(G\), which occur between identical groups. In this connection we do not include the order \(E\) among the number of combinatorial premises that are possible in principle for calculating \(W\), and those expressions into which this order enters we shall hereafter omit when considering the quantitative relations of various combinations, preserving it, however, for the sake of completeness of the picture when establishing the general classification of combinations.

of each of the lower orders, i.e., those located in the scheme to the right of it.

Here we have come right up to the resolution of the question of the degree of randomness in a number of different physical phenomena in their relation to one another. Within the framework of idealistic and metaphysical definitions of randomness this problem proves insoluble. Only the definition of randomness as a form of manifestation of necessity makes it possible to construct an exactly quantitatively determined relation, in which the degree of randomness appears as the number of intermediate forms in which the given necessary phenomenon is successively realized.

With the help of the developed definition of randomness as the logical basis of the calculation of \(W\), we can analyze the mutual connection of all combinations with one another and precisely establish the degree of their randomness in relation to one another.

Now we can definitively formulate what the basic premises of the calculation of \(W\) consist in. Since the random behavior of a molecule and the necessary behavior of a gas are connected with different combinations of molecules, before proceeding to the calculation of \(W\) we must correctly establish, in the series \(RGDCEP\), two combinations, of which one, taken as necessary, must be connected with the necessary thermodynamic state of the whole gas and, consequently, with its \(S\); the other—random—with the motion of individual molecules that is necessary from the standpoint of mechanics and random from the standpoint of thermodynamics.

This is precisely what the two basic premises of the calculation of \(W\) consist in. The logical distinction between the individual methods of combinatorial determination of \(S\) reduces to a different choice of the premises of the calculation of \(W\) as the basis for the subsequent construction of the formal-mathematical apparatus by means of which the numerical value of \(W\) is computed.

4

Let us now characterize all the essentially different methods hitherto proposed for the combinatorial determination of \(S\), from the point of view of the choice of the various premises of the calculation of \(W\).* From this point of view all methods can be reduced to the following cases:

A. Methods of classical statistics

1) The old, so-called classical statistics, usually taking the arrangement \(\{D\}\) as the combination that must necessarily be connected—

* The formal-mathematical and physical aspects of the methods listed below for both classical and quantum statistics (except for Fermi’s method) are discussed in detail in the review article by I. E. Tamm, “New Principles of Bose–Einstein Statistical Mechanics in Connection with the Question of the Physical Nature of Matter,” Uspekhi fizich. nauk, vol. VI, p. 112, 1926.

connected with \(S\), and the arrangement \([C]\) as the unit of enumeration \(W\). Hence the basic formula was derived, which we shall write as follows:

\[ \{D\}=W=\frac{N!}{\prod_i n_i!}[C]. \tag{2} \]

Rounding, for the case of equilibrium, the value \(W_{\max}\) by means of Stirling’s formula, we obtain:

\[ \{D\}=W=Z^N[C]. \tag{3} \]

However, the results obtained according to these formulas contradicted the facts in a whole series of cases. Therefore later corrections were introduced into classical statistics, of which we shall consider two as examples.

2) The first correction sought to eliminate the contradiction between \(S\), calculated by (1) and (2), and the property of entropy of being an additive quantity; this was achieved by declaring all molecules to be absolutely identical, and hence all \(N!\) permutations of molecules were declared to be completely indistinguishable from one another. On this basis (2) was divided by \(N!\):

\[ W=\frac{1}{\prod_i n_i!}. \tag{4} \]

But \(\prod_i n_i!\) is the number of arrangements of molecules in one arrangement. Thus formula (4) essentially means that the first correction reduces to taking the arrangement of molecules \(\{C\}\) as the combination connected with \(S\), and the arrangement of molecules \([P]\) as the unit of enumeration \(W\). Hence formula (4) must be written as follows:

\[ \{C\}=\frac{1}{W}=\prod_i n_i![P]. \tag{4a} \]

3) The second correction sought to eliminate the following contradiction: the entropy calculated by (1) and (2), from a certain moment, when the number of cells \(Z\) became greater than the number of molecules \(N\), ceased to change upon further rarefaction of the gas or raising of its temperature. In order to eliminate this physically inadmissible consequence, as the combination connected with \(S\), within individual microscopic parts of the system a distribution of molecules \(\{R\}\) was established while preserving the arrangement \([C]\) as the unit of enumeration \(W\). Hence for \(W\) formula (3) was directly obtained.

B. Methods of Quantum Statistics

However, the indicated corrections were not able to resolve all the contradictions that were revealed within classical statistics. Therefore in 1924 Bose proposed, and immediately after him Einstein developed, a new quantum statistics. Its characteristic ...

a feature common to all its variants is the adoption, as the unit of computation of \(W\), of the arrangement of molecules \([D]\); in this connection the arrangements of molecules must be regarded as equiprobable events. This equiprobability is achieved by postulating, just as in the case of the first correction, the absolute identity of the molecules of a homogeneous gas.

4) Initially (in 1924) Bose proposed a method in which, as the combination connected with \(S\), the grouping of molecules \(\{G\}\) figured (we shall call this method the initial one). The formula of this method is as follows:

\[ \{G\}=W=\frac{Z!}{\prod_n n!}[D]. \tag{5} \]

5) Subsequently (in 1925) the same Bose proposed a somewhat different method, in which the distribution of molecules \(\{R\}\) figured as the necessary combination (we shall call this method the completed one). Its formula is as follows:

\[ \{R\}=W=\frac{(N+Z-1)!}{N!(Z-1)!}[D]. \tag{6} \]

6) Finally (in 1926), Fermi put forward a new method, based on the additional premise of computing \(W\) by adopting a limiting number of molecules that can occur in one cell, namely—not more than one. The formula of Fermi’s method is as follows:

\[ \{R\}=W=\frac{Z!}{N!(Z-N)!}[D]. \tag{7} \]

It should be noted that, in its basic combinatorial premises, Fermi’s method does not differ from Bose’s completed method; the difference between the two methods lies on another plane—the plane of the third premise for computing \(W\); consideration of this new premise, together with Fermi’s method itself, we shall postpone until the end of the article, so as not to divert attention from what is most essential from the combinatorial point of view in the question of the two basic premises for computing \(W\), which constitute the essence of every combinatorial method.

However, the total number of all principally possible methods differing from one another in the choice of combinatorial premises for computing \(W\) is not limited only to these combinatorial methods of determining \(S\) that have been proposed so far. Since the number of orders of combination without \(E\) that differ essentially from one another is five, the number of all distinct pairs of combinatorial premises, and hence the number of methods of computing \(W\), is obtained by combining five elements two at a time and is equal to ten (see Table 1).

TABLE 1

Classification of methods for calculating $W$, constructed on the basis of various combinations of two combinatorial premises included in the scheme $RGDCEP$

Name of statistic	Name of method	Premises for calculating $W$	Premises for calculating $W$	Premises for calculating $W$	Premises for calculating $W$	Premises for calculating $W$
Classical statistic of arrangements $[P]$	—	$\{R\}\cdots$	$\cdots$	$\cdots$	$\cdots$	$[P]$
Classical statistic of arrangements $[P]$	—		$\{G\}\cdots$	$\cdots$	$\cdots$	$[P]$
Classical statistic of arrangements $[P]$	—			$\{D\}\cdots$	$\cdots$	$[P]$
Classical statistic of arrangements $[P]$	First correction				$\{C\}\cdots$	$[P]$
Classical statistic of combinations $[C]$	Second correction	$\{R\}\cdots$	$\cdots$	$\cdots$	$[C]$
Classical statistic of combinations $[C]$	—		$\{G\}\cdots$	$\cdots$	$[C]$
Classical statistic of combinations $[C]$	Ordinary method			$\{D\}\cdots$	$[C]$
Quantum statistic of placements $[D]$	Final Bose method	$\{R\}\cdots$	$\cdots$	$[D]$
Quantum statistic of placements $[D]$	Initial Bose method		$\{G\}\cdots$	$[D]$
Statistic of groupings $[G]$	—	$\{R\}\cdots$	$[G]$

From this table we see that the premises $DC$ constitute the basis of classical statistics in its usual form. The premises $CP$ and $RC$ determine the first and second corrections to this statistic; the premises $GD$ and $RD$ determine the initial and final Bose methods.

Comparing with one another all these methods, characterized by the corresponding combinatorial pairs, we state the following: when introducing the first correction we, as it were, descend along the ladder of the orders of combinations by exactly one step, for we construct the calculation of $W$ by passing from the premises $DC$ to the premises $CP$; at the same time the combination which in classical statistics was usually taken as acci-

random, in the present case is taken as necessary. On the contrary, by introducing Bose’s initial method, we as it were ascend our “ladder” by one step in comparison with classical statistics, for the arrangement which the latter took as necessary is taken in quantum statistics as random.

Finally, the second correction and Bose’s completed method, taking the distribution \([R]\) as the general necessary combination, establish various random combinations which, in their turn, are common to the second correction with the ordinary method of classical statistics and to Bose’s completed method with his initial method.

The remaining five combinatorial methods have so far found no application in physical statistics. The reason for this circumstance we shall easily understand from what follows.

Thus, in making the ascent along our “ladder,” we force combinations which at the lower steps appear as necessary to become random at the higher steps; in other words, we force the necessary moments contained in these combinations to reveal themselves as random. In the reverse movement from the higher steps to the lower, we force random moments to pass into necessary ones.

These mutual transitions of two opposite moments contained in each combination constitute the inner logical connection of all the already proposed and still possible combinatorial methods of determining \(S\). Therefore their classification may be expressed by means of the combinational series established by us \(RGDCEP\).

5

Let us proceed to clarify the quantitative relations following from the combinational series established by us. If we take as the necessary combination \(\{R\}\), we find that this combination is capable of being realized through strictly definite numbers of each of the five other combinations; in other words, in its numerical value \(\{R\}\) is equal to a definite number \([G]\), then to another, generally speaking, but also quite definite number \([D]\), and so on. Thus we obtain a certain series of quantitative relations, characterized by the relation:

\[ \{R\}\ldots [G, D, C, E, P], \]

where the numerical value \(\{R\}\) is considered successively with respect to each separate combination enclosed in square brackets. Let us call this series the distribution series \(R\).

In exactly the same way the remaining series of grouping \(G\), arrangement \(D\), and combination \(C\) are obtained, which are characterized by the corresponding expressions (see Table 2).

TABLE 2

Series of quantitative relations of various combinations included in the series \(RGDCEP\)

Series of combinations	Designation of combination	Number of random combinations	Number of random combinations	Number of random combinations	Number of random combinations	Number of random combinations
Series of distributions	\(\{R\}=\)	\(f(N,Z)[G]\)	\(\dfrac{(N+Z-1)!}{N!(Z-1)!}\,[D]\)	\(Z^N[C]\)	\(\dfrac{(N+Z-1)!}{(Z-1)!}\,[P]\)
Series of groupings	\(\{G\}=\)	—	\(\dfrac{Z!}{\prod_n \pi_n!}\,[D]\)	\(\dfrac{Z!\,N!}{\prod_n \pi_n!\prod_i m_i!}\,[C]\)	\(\dfrac{Z!\,N!}{\prod_n \pi_n!}\,[P]\)
Series of arrangements	\(\{D\}=\)	—	—	\(\dfrac{N!}{\prod_i m_i!}\,[C]\)	\(N![P]\)
Series of combinations	\(\{C\}=\)	—	—	—	\(\prod_i m_i!\,[P]\)

(We note once again that we have agreed to omit the relation between the arrangement \(E\) and the other combinations.)

Let us now verify the correctness of the basic formulas obtained for \(W\) by means of each of the 10 combinatorial methods. From the formal-mathematical standpoint, the basic formula for \(W\) must first of all satisfy the requirement that the probability of a compound event be equal to the product of the probabilities of the simple independent events which together form the given compound event. Thus, for example, after the insertion into a system of a separating partition in a gas that is in equilibrium, the product of the probabilities of the states of the separated parts \(W_1 \cdot W_2\) must be equal to the probability of the state of the entire system before its separation, \(W_{1+2}\):

\[ W_{1+2}=W_1\cdot W_2. \tag{8} \]

Applying formula (1), we obtain the mathematical expression for the additivity of entropy as a function of the state of the system:

\[ S_{1+2}=S_1+S_2. \tag{9} \]

The additivity of entropy is fully confirmed by all the data of thermodynamics; in the general case, the additivity of \(S\) consists in the fact that,

that if a system of an ideal gas is taken at constant temperature and specific volume, then the \(S\) of the system proves to be proportional exclusively to the number of molecules composing it:

\[ S=c\cdot N, \tag{10} \]

where \(c\) is a quantity constant under the given conditions and functionally dependent on the specific volume and temperature of the gas. Combining (1) and (10), eliminating \(S\) from them:

\[ W=\gamma^N. \tag{11} \]

where \(\gamma\) is equal to \(e^{\frac{c}{k}}\), and consequently, under the given conditions, is also a constant quantity depending only on the specific volume and temperature of the gas.

Formula (11) means that if a system of an ideal gas is taken at constant temperature and constant density, then the probability of the state of this system must be equal to some constant \(\gamma\) raised to the power equal to the number of particles composing the system. If, on the other hand, we divide the system by a partition into parts while keeping the external conditions unchanged, then the value of \(\gamma\) should not change under this division. Obviously we have arrived at this result directly from consideration of the basic formula of physical statistics [equation (1)] and the basic formula of probability theory [equation (8)]. Therefore we are entitled to accept equation (11) as a criterion for checking, from the quantitative side, combinatorial formulas derived by various methods for calculating \(W\). In the present article we restrict the verification of the correctness of the basic formulas for \(W\), obtained by one or another combinatorial method, exclusively to their correspondence to equation (11).

From Table 3 we see that for the case of a highly rarefied gas, when \(Z \gg N\) (this case is precisely the subject of our consideration), out of all 10 approximate formulas only two formulas of both Bose methods satisfy condition (11):

\[ W=\left(\frac{Z}{N}\right)^N. \tag{12} \]

Since under any division of the system by insulating partitions into macroscopic parts the ratio \(\frac{Z}{N}\) remains constant all the time, it follows that, from the formal-mathematical point of view, both Bose methods are correct within the limits of the requirement we have set—correspondence to condition (11). In all the remaining 18 formulas this requirement is not fulfilled, either because the exponent turns out not to be equal to \(N\), or because the base of the power does not remain constant when the gas is divided into its component parts and when it is formed from these parts. In particular, in the case of classical statistics

TABLE 3

Approximate formulas for the maximum value of \(W\) for two limiting cases of the relation between \(Z\) and \(N\)

Combinatorial premises	Formulas for \(W_{\max}\) when \(Z \gg N\)	Formulas for \(W_{\max}\) when \(Z \ll N\)
\(\{R\}\quad [P]\)	\(Z^{N}\)	\(\left(\dfrac{N}{Z}\right)^{Z} N^{N}\)
\(\{G\}\quad [P]\)	\(Z^{N}\)	\(Z^{Z}N^{N}\)
\(\{D\}\quad [P]\)	\(N^{N}\)	\(N^{N}\)
\(\{C\}\quad [P]\)	\(1\)	\(\left(\dfrac{N}{Z}\right)^{N}\)
\(\{R\}\quad [C]\)	\(Z^{N}\)	\(Z^{N}\)
\(\{G\}\quad [C]\)	\(Z^{N}\)	\(Z^{N}\)
\(\{D\}\quad [C]\)	\(N^{N}\)	\(Z^{N}\)
\(\{R\}\quad [D]\)	\(\left(\dfrac{Z}{N}\right)^{N}\)	\(\left(\dfrac{N}{Z}\right)^{Z}\)
\(\{G\}\quad [D]\)	\(\left(\dfrac{Z}{N}\right)^{N}\)	\(Z^{Z}\)
\(\{R\}\quad [G]\)	\(f(N)\)	\(f(N,Z)\)

Notes: 1. If, after deriving by an approximate method, with the aid of Stirling’s formula, the final formula contained the quantity \(e\) to one power or another, we omitted it in order to obtain the simplest possible form for \(W_{\max}\).

1. Since in the case of the first correction, based on the premises \(\{C\}\ [P]\),

\[ W=\frac{1}{\prod_i n_i!}, \]

in the present case we find not the maximum, but the minimum number of arrangements by means of which one combination can be realized.

\((W=Z^N)\), the exponential number is a function not of the specific, but of the total volume of the gas; hence follows the contradiction with the additivity of \(S\).*

6

There remains before us the still open question of why only two formulas of quantum statistics proved to be correct, whereas all the other formulas proved erroneous. An answer to this requires a preliminary examination of the connection between the different orders of combinations from the point of view of their essential features. First of all, it seems most essential to establish whether a given combination is in general capable of changing and, consequently, whether its features can vary quantitatively under the condition that the entire system as a whole is in a stationary state, i.e. in thermodynamic equilibrium; in other words, whether any of the possible permutations of molecules within the volume “occupied” by the gas can cause a change in the given combination.

If a combination is capable of changing under conditions constant for this gas, then such a combination will be varying, dependent on a definite permutation of molecules; but if within the given \(Z\) and \(N\) the combination does not change, then such a combination will be non-varying, independent of any permutations of any of the \(N\) molecules in the volume of \(Z\) cells. We shall call both of these cases two different divisions of combinations.

Considering further the varying combinations, we find that, since each of them changes by virtue of one or another permutation of individual molecules, it is natural that the classification of all these combinations must be constructed on the basis of considering the different features of permutations that underlie the emergence and change of the given combinations.

We can characterize permutations from three sides: vo-

* Let us note that, in order to construct a method based on the premises of \(RG\), it is first necessary to establish the general form of the function \(f(N,Z)\), which from the mathematical side reduces to the problem of partitions: in how many different ways can the number \(N\) be divided into \(Z\) summands, including zero among the summands. Without subjecting to a detailed investigation the form of the function \(f(N,Z)\), let us indicate only that for the case when the number of summands \(Z\) becomes greater than or equal to \(N\), the number of partitions becomes a function only of \(N\), and the given \(N\) is a constant quantity. Therefore, if we wished to construct a method of combinatorial determination of \(S\) on the premises of \(RG\), we would arrive at exactly the same contradiction of which quantum statistics is guilty. In fact: if \(S\) is determined as a function \(f(Z,N)\), then, obviously, \(N\) will correspond to the number of molecules, \(Z\) to the number of cells, and the number of methods of partition \(f(N,Z)\) to the number of all possible groupings of \(N\) molecules among \(Z\) cells. But then, from the known moment when \(Z \geq N\), the value of \(S\) will cease to change upon expansion of the volume of gas at a given temperature, i.e. upon increasing the number \(Z\), and this is a physical absurdity.

first, according to what is being permuted; second, according to how it is permuted; and, third, according to where it is permuted.

The displacement of molecules may be considered as occurring either with individual molecules or with whole groups of them, when at the final moment of its displacement the given group of molecules appears in exactly the same composition as at the initial moment, even though it may have broken up during the permutation itself. In the first case we have a single permutation; in the second case, a group permutation. The two cases represent two different orders of permutation.

Further, if the displacement of a molecule occurred without the reverse permutation into its place of another molecule precisely from the cell into which the given molecule fell, then we have an uncompensated permutation; if, however, there occurred simultaneously a reverse displacement of another molecule, so that in the end the molecules simply exchanged places, then the permutation will be compensated. It is obvious that the compensated permutation of two molecules is a combination of two directly opposite, and therefore mutually canceling, uncompensated permutations. If permutations occur between whole groups, then they will be compensated when both permuted groups contain the same number of molecules; with different numbers of molecules the permutations will be uncompensated. The two cases represent two different types of permutations.

Finally, if molecules are mixed within one and the same cell, so that none of them goes beyond the boundaries of the cell in which it is found, then we have an intracell permutation; if, however, molecules are permuted between cells, passing from one cell to another, then the permutation will be intercellular. We shall call the two cases two different classes of permutations.

Thus all possible permutations may be classified according to their orders (the feature “what”), according to their types (the feature “how”), and according to their classes (the feature “where”).

On the basis of the considered features of permutations, we construct a classification of combinations with the corresponding division into divisions, classes, types, and orders (see Table 4).

Only one \(R\) falls into Division I, for only its essential features, i.e. the numbers \(Z\) and \(N\), do not vary, so that only one definite \(R\) corresponds to one stationary state of the collective.

The remaining five orders fall into Division II; of them, only one \(P\) belongs to the 2nd class; the remaining four orders belong to the 1st.

With regard to these four combinations, let us note the following: as a direct consequence of dividing combinations according to the features of the permutations underlying them, we pass to [[unclear: text continues off page]]

B. KEDROV

TABLE 4

Classification of combinations constructed on the basis of the features of permutations

	R	G	D	C	E	P
Divisions	I	II	II	II	II	II
Dependence of combinations on permutations:	Do not depend	Depend	Depend	Depend	Depend	Depend
Classes		1	1	1	1	2
Character of permutations		Intercellular	Intercellular	Intercellular	Intercellular	Intracellular
Types		A	A	B	B	V
Character of permutations — essential moment in the characterization of molecules		Uncompensated — the total moment; molecules are depersonalized	Uncompensated — the total moment; molecules are depersonalized	Compensated — the unit moment; molecules are individualized	Compensated — the unit moment; molecules are individualized	Compensated — molecules are individualized
Orders	R	G	D	C	E	P
Character of permutations	—	Group	Single	Single	Group	Single

to the notions of individualization and of relative depersonalization of the molecules of a homogeneous gas.

Indeed: that change in the combination which occurred as a result of a compensated permutation means that, since the difference between the objects of the permutation does not concern the number of molecules (in number of molecules both objects are the same), then, from the point of view of the changes that have occurred in the system, the essential point is the fact of individual differences between the two objects, regardless of whether this difference concerns the arrangement of electrons in orbits inside the atoms or the so-called Vorgeschichte of each of the molecules. Hence—the individualization of molecules.

On the contrary, for combinations arising from uncompensated permutations, these really existing individual differences between the objects of permutation appear already as accidental and inessential, since the difference between both objects goes much deeper and consists above all in a different number of molecules, as the essential feature in the given connection of phenomena. With respect, however, to the number of molecules, the individual differences between separate molecules lose their decisive significance to such an extent that we not only can, but must, completely abstract from them in our investigation and, in this abstraction, regard the molecules as entirely depersonalized; this, of course, by no means signifies that in reality molecules lose absolutely every difference among themselves.

Thus the question of the individualization or depersonalization of molecules may be considered depending on the character of the permutations of molecules, when the individuality of the molecules being permuted appears now as an essential, now as an inessential moment of the occurring change in the state of the gas.

We have dwelt on this question because it acquires very great importance in the analysis of the so-called principle of the absolute identity of the molecules of an ideal gas, which will be discussed below.

Considering in Table 4 the sequence in which all orders of combinations are arranged, we establish that the given classification coincides with the series RGDCEP established by us earlier; consequently, by two paths that at first sight are independent, we have arrived at one and the same result. This coincidence occurred because the various features of permutations stand in the same relation of the accidental and the necessary to one another as do the corresponding combinations.

7

Starting from the analysis of the features of permutations, let us now clarify which of all the combinations is primary, initial in the sense of its direct connection with the mechanical displacement of separate molecules, and how from this simplest combination, in the process of gas formation (for example, when it is released into a vacuum during slow evaporation, or in the course of some very slowly proceeding reaction), secondary, more complex combinations arise; in short, let us consider the connection of the various combinations from the point of view of their sequential development.

First of all, we shall have to consider the sequential formation of various kinds of permutations. For brevity let us denote by the symbols p, c, l, d, and q the features of permutations that underlie the corresponding combinations, remembering that, in turn, at the basis of all permutations in general and, consequently, of all combinations depending on them, lies the mechanical displacement of separate molecules.

If we abstract from the most complex interaction of molecules, consisting in their continuous collisions, and take the point of view of individual molecules, then the displacement of each of them, considered independently of the displacement of other molecules, will appear as connected, in a necessary way, exclusively with the attribute \(d\), for in essence every displacement of an individual molecule in its pure form is a solitary, uncompensated permutation. Only when two or a greater number of necessary displacements of individual molecules intersect, accidentally with respect to each molecule, do other, more complex permutations arise.

Thus, at the basis of the emergence of all permutations there ultimately lies the solitary uncompensated permutation \(d\), in which, as in an elementary “cell,” there is embedded the possibility of the formation of all permutations in general and, consequently, of all “combinations” of molecules; but in its turn \(d\) characterizes the arrangement \(D\) in an essential way, constituting its immediate basis; therefore it is precisely the arrangement that must be regarded as the initial, simplest combination, from which all the remaining varying combinations develop.

This proposition appears with particular sharpness if we pass gradually from the motion of individual molecules, as quite independent, practically mutually independent mechanical systems, to their collective body, as a qualitatively new, more complex and developed form of motion of matter; we make this transition by increasing the number of molecules, when a definite interaction is gradually established among them; thanks to this interaction, the molecules within their collective body no longer exist as separate, entirely self-contained and independent mechanical systems; their mechanical displacement, considered in relation to the whole gas as a whole, has lost its former autonomy, being transcended by the thermodynamic state and motion of the entire gas and included in this motion only as its subordinate moment. In other words, in the process of development of the physical phenomenon the mechanical motion of the molecules has received its negation and has been preserved in the gas, as is said, only in a “sublated” form.

From the point of view of the attributes of permutations, the indicated development will be characterized as follows: at first, when the given volume contains so few molecules that in essence we do not yet have a genuine gaseous collective body in the full sense of the word, the probability of formation of groups of molecules and of compensated permutations will be vanishingly small, so that practically all existing permutations will be determined by the attribute \(d\).

Subsequently, as the number of molecules increases and the larger the size we establish for the elementary cells, the probability of several molecules falling into one and

the same cell and, consequently, the probability of the formation of groups of molecules becomes relatively greater and greater. For each individual molecule there still remains characteristic the change of the attribute \(d\), for its displacement has only one direct result, namely, a change in the number \(n_i\); but, in addition, a possible indirect result is a change in the number of groups \(\tau_n\); on the other hand, it is also possible that, by virtue of accidental coincidences of circumstances, in place of a destroyed group in one cell there arises a group with exactly the same number of molecules in another cell; then, within one grouping, there will occur a replacement of arrangements as accidental forms of its manifestation.

Let us note further that, at the initial stages of the formation of a gas that have been considered, the probability of the formation of compensated permutations, relative to the probability of the formation of uncompensated ones, remains at all times vanishingly small. This is understandable, since the former, in contrast to the latter, presuppose a definite, even if purely external, connection between the displacements of molecules, namely, that two or more displacements must necessarily have directions directly opposite to one another.

With a further increase in the number of molecules it naturally turns out that, when the elementary cell is sufficiently large, first of all single compensated permutations \(c\) begin to be realized; this is explained by the fact that \(c\) is connected with the exchange of molecules between their cells, whereas \(e\) requires this exchange by whole groups, and \(p\) requires that two molecules exactly exchange their places. The attributes \(e\) and \(p\) appear at the next, higher level of development of the phenomenon.

Thus there occurs a bifurcation of the attribute \(d\): on the one hand, its development gives rise to the attribute \(g\), which in a developed gas is an attribute necessary with respect to the arrangement \(D\) of the grouping \(G\); consequently, in this aspect \(d\) acts as the germ of an accidental combination. On the other hand, the coincidence of the displacements of two molecules, each of which displacement taken separately causes a change of the attribute \(d\) in a direction directly opposite to the other, leads to the formation of the attribute \(c\), accidental with respect to \(d\), and further—to \(e\) and \(p\); consequently, in this aspect \(d\), acting as the germ of an arrangement, reveals itself as a necessity; in other words, the necessary displacement of molecules and the attribute \(d\) directly connected with it, in the process of development of the phenomenon, has generated its accidental form in the form of the attributes \(c\), \(e\), and \(p\), which are germs accidental with respect to the arrangement of combinations.

Let us note that the germ of the distribution \(R\) in the form of the nonvarying attribute \(\tau\) appears already from the very beginning, for any number of molecules \(N\) and cells \(Z\); we call it a germ, however,

because there is as yet absent in the system that interaction between molecules which qualitatively determines their collective body and whose external manifestation is the self-distribution \(R\).

Schematically, the differentiation of the features of permutations, taking place at the very beginning of the process of gas formation, and its connection with the feature \(r\), independent of the permutations, may be represented as follows:

\[ \{r\}\ \cdot\ \cdot\ [g \leftarrow d \rightarrow c, e, p]. \]

Here, within the limits of features random with respect to \(r\), the successive bifurcation of the feature \(d\) is shown in the direction of the generation of \(g\), necessary in relation to it, and of the random features \(c, e\), and \(p\). The expression obtained basically coincides with the combinational series established by us above.

Thus the series \(RGDCEP\), purely logical at first glance, turns out to be a known reflection of a real “historical” process of development capable of taking place in nature itself; the separate stages of this series, i.e. the separate orders of combinations, turn out to be a reflection precisely of those real stages through which the gas passes successively in the process of its formation. In short, the logical sequence of combinations coincided with their “historical” sequence, with abstraction, of course, from all the inessential details of gas formation that always occur in setting up an experiment.

Finally, when the number of molecules in the given volume reaches a certain limit, there is formed a genuine gas collective with the interaction of molecules characteristic of it; then all combinations which in the “undeveloped” gas were represented only by the features of the corresponding permutations and existed as a kind of “embryo” attain their completed, developed form.

However, in a developed gas the sequence in which the features \(g, c, e\), and \(p\) arise from the single feature \(d\) becomes completely imperceptible, for all the corresponding combinations appear simultaneously as different random forms in which the necessary \(R\) is realized. At the same time, those changes of the feature \(d\) (i.e. of the system of numbers \(n_i\)) which carry within themselves the displacement of each molecule separately, in the overwhelming majority of cases mutually extinguish (compensate) one another, so that in these cases the feature \(d\) remains as if unchanged; by virtue of this, each compensated permutation abolishes that basic elementary process of displacement of individual molecules which, taken in its pure form, is connected only with a change in the feature \(d\). And since, as a result of the compensation of molecular displacements, an enormous multitude of collateral combinations arises, then, upon a superficial consideration of the gas, the impression is obtained that not \(D\), but rather \(C\), or even \(P\), are the перво

initial, and therefore a simpler, more ordinary and more massive random combination of molecules; this occurs because, on the surface of the phenomena taking place in a gas, these secondary random combinations catch the eye sooner than the elementary process hidden behind them and conditioning them, and the feature \(d\) directly connected with them.

But, considering the mutual connection of various combinations and features “historically,” we ascertain, first, that the simple mechanical displacement of individual molecules is that elementary “cell” which contains in embryo the possibility of the formation of ever more complex forms of motion of molecules within the gas; and, second, that the initial, and therefore simplest, combination is the distribution \(D\), connected at its basis directly with the mechanical displacement of individual molecules.

8

We are now already in a position to answer the question posed above: why the old statistics proved untenable, and how the new statistics succeeded in overcoming the difficulties with which its predecessor could not cope.

Let us subject to analysis the basic presuppositions of each of the methods of combinatorial determination of \(S\) from the point of view of how correctly it reflects the relation between the necessary thermodynamic behavior of a gas and the random mechanical behavior of individual molecules; in other words, how correctly it solves the basic problem of calculating \(W\).

Let us begin with the first presupposition, i.e., with the establishment of a combination essentially and necessarily connected with \(S\).

If the system as a whole is in equilibrium, then the combination which is necessarily, unambiguously connected with the values of the thermodynamic properties of the gas, and consequently with its \(S\), must not change until the equilibrium of the system is disturbed; in other words, under the given conditions it must possess invariant features. Of all the combinations, only one distribution \(R\), possessing the features \(Z\) and \(N\), fully satisfies this requirement; indeed, if the conditions that necessarily determine the thermodynamic state of the gas are given: the number of moles forming it \(m\), its volume \(V\), and energy \(E\), then these same conditions, expressed through the numbers \(N\) and \(Z\), necessarily and unambiguously determine at the same time the combination \(R\). Any other combination requires, for its necessary realization, additional conditions concerning the more detailed arrangement of molecules among the cells. Thus the totality of concrete conditions that gives the real possibility of the given thermodynamic state of the gas and the corresponding \(R\) proves at the very same time to be closed to the realization of any other, random in ...

with respect to \(R\) combinations; under the given conditions of equilibrium, all these combinations turn out to be random, continuously changing, capable of being both of one kind and another; only such a [[unclear: line cropped]] consequently, one \(R\) is necessary.

Mathematically this is manifested in the fact that if the necessary combination is taken to be \(R\), then the value \(W\), calculated by means of any unit of calculation, is a function only of \(Z\) and \(N\); at the same time, in no case, when any of the other combinations is taken as the necessary one, does the value represent such a function (see Table 2).

We pass to the second premise, i.e. to the establishment of the units of calculation \(W\).

As we have already seen, \(R\) is realized in the random form of five different orders; which of them should we take into account when calculating \(W\) so that in the result we obtain the desired formula (11)? If we wish logically to justify the choice of a unit of calculation \(W\), then formal mathematical considerations alone prove insufficient; likewise insufficient is a bare fact of the correspondence of the derived formula with formula (11).

A logical justification of the fact that \(W\) must be calculated by means of certain units, and not others, can be provided only by considering the basic problem of calculating \(W\), which, we repeat once more, consists in quantitatively expressing the law-governed connection between the necessary state of a gas and the random motion of its molecules. But since we consider the necessary behavior of the gas in relation to the random behavior of individual molecules, it naturally follows from this that the combination chosen as the unit of calculation of \(W\) must necessarily be connected precisely with the motion of individual molecules; only then will it appear, in relation to the whole gas, as an essential form of manifestation of its necessity. Such a combination is the arrangement \(D\). All the other combinations characterize not the simple initial mechanical motion of individual molecules, but a complicated case of this motion, since they include in themselves the moment of a more or less random connection between the displacements of individual molecules. Therefore all the other combinations cannot serve for the purposes of solving the basic problem of calculating \(W\).

Analyzing now the first premise of classical statistics, we ascertain that this statistics incorrectly attempted to determine the necessary behavior of the whole gas on the basis of the combination \(D\), taken as necessary, but objectively random with respect to the general state of the gas. We find the very same error in the first correction, where the random \(C\) was in fact taken as necessary, and in Bose’s original method, where the grouping \(G\), objectively random with respect to the state of the gas, was taken as such.

Let us note here that, if classical statistics nevertheless led to correct results within the known limits, this is explained exclusively by the following: at high gas density, when the numbers \(n_i\) are large, the number of combinations \(C\) entering into one—namely, into the most probable, uniform distribution of molecules—turns out to be a quantity of the same order as the total number \(C\) entering into \(R\), and also into the combination \(G\) intermediate between \(R\) and \(D\) [cf. formulas (2) and (3)]. Thus, in the case of a uniform arrangement of molecules at high gas density, both pairs of premises \(GC\) and \(DC\), where the necessary combination has been incorrectly chosen, numerically reduce to the premises \(RC\), and only for this reason, in this special case alone, do they lead to correct results. Under other conditions—at low gas density—when the premises no longer reduce numerically to \(RC\), the result consequently turns out to be incorrect.

In exactly the same way, the numerical coincidence of \(GD\) with \(RD\) at great rarefaction of the gas explains why, in this case, Bose’s original method gives the correct result despite an incorrect initial premise.

In contrast to the methods just analyzed, the second correction and Bose’s completed method establish, quite correctly, the distribution \(R\) as the necessary combination.

Analyzing the second premise in the various methods, we find that classical statistics, with its second correction, incorrectly chose the unit of calculation \(W\), taking \(C\) as the combination necessary with respect to the motion of individual molecules, whereas \(C\) is, in this connection, accidental, and appears only as the result of the intersection of necessary processes occurring with individual molecules. The same error is present to an even greater degree in the first correction, with its actual unit of calculation \(P\).

On the contrary, both of Bose’s methods quite correctly establish \(D\) as the unit of calculation \(W\).

It is precisely in this correct choice that the fundamental point consists which sharply separates the new quantum statistics from the old classical statistics and which, consequently, determines the boundary between two fundamental stages in the development of physical combinatorics.

Considering further the logical basis of the calculation of \(W\) as a whole, as the unity of both its principal premises, we find that the lawful connection between the necessary thermodynamic state of a gas and the random mechanical motion of molecules can be correctly expressed only by the correlation of the combinations \(\{R\}\) and \([D]\).

But since it is precisely the premises \(RD\) that constitute the logical basis of Bose’s completed method, it follows that this method, in the general case, correctly solves the combinatorial problem of determining \(S\) for an ideal gas.

In doing so, one must not forget that we have in mind only the first two basic premises for the calculation of \(W\), and so far we quite consciously do not raise the question of the Fermi method as a special case of the Bose method [see formula (7)], when, as a third premise, it is accepted that in one cell there may simultaneously be more than one particle.

Finally, considering the essence of the contradiction between formula (3), obtained by the method of classical statistics, and formula (11), confirmed by experiment, we state that the general cause of this contradiction consists in the fact that the logical basis of classical statistics incorrectly reflects the real relation between the accidental and the necessary moments of objective reality: first, as the necessary combination there has been chosen something accidental with respect to the state of the gas; second, as the unit of computation of \(W\) there has been chosen \(C\), which is not directly connected with the displacement of individual molecules, but represents the result of secondary permutations, accidental with respect to the [[unclear: cut-off word]] compensated permutations. Thus, in both premises classical statistics proved to be constructed on a random basis. This is its logical error. The formally mathematical contradiction between formulas (3) and (11) is only a consequence, only an external manifestation of this basic error.

9

It remains for us, if only briefly, to consider to what consequences for combinatorics the third premise of the calculation of \(W\) leads, the premise that underlies the Fermi method and represents a further development of Pauli’s principle. As has already been said, this premise consists in posing the question: is the number of particles that can simultaneously be in one cell limited or not, and, if it is limited, then by what limit? The Bose method answers this question negatively: the number of particles in a cell can in principle be indefinitely large (the case of a quantum and molecular gas). The Fermi method, on the contrary, answers positively: the number of particles in a cell cannot be greater than one (the case of an electron gas). From this originate two different channels of development of quantum statistics, united above all by the fact that their basic combinatorial premises are common.

Of special interest is the question of the physical justification of both answers to the third premise of the calculation of \(W\): why, in the case of one physical object, can a cell contain in principle an unlimited number of particles, whereas in the case of another object it can contain no more than one particle? Usually this question may be considered from the standpoint of what total charge—even or odd—the particles carry. In the case of an even charge, the Bose method is applied; in the case [[unclear: continuation cut off at right edge]].

odd—the Fermi method. Along with the character (charge) of the particles, it seems to us quite natural to raise the question of the nature and character of that space (phase or real) over whose cells the particles are distributed.

However, consideration of this question does not directly enter into the tasks of physical combinatorics and constitutes the subject of an independent, serious investigation, which we cannot undertake here.

Thus, let us accept as a result following directly from experiment that in one cell there cannot be more than one particle at the same moment. Then all our combinatorial relations will take on a much simpler form, since all combinations connected with intra-cell and group permutations will at once drop out of them.

The basic combinational series will appear in the following form:

\[ \{R\}\ .\ .\ .\ [D]\ .\ .\ .\ [C]. \]

The quantitative relations (series of combinations) following from this series will be expressed as follows:

\[ \{R\}=\frac{Z!}{N!(Z-N)!}\,[D]=\frac{Z!}{(Z-N)!}\,[C]; \]

\[ \{D\}=N!\,[C]. \]

Checking, from the formal-mathematical side, the various methods that have the common third premise of calculating \(W\), we find that the method constructed on the premises \(RD\), i.e. the Fermi method proper, leads to equation (12) of quantum statistics and consequently corresponds to control formula (11); the method constructed on the premises \(RC\), however, leads to equation (3) of classical statistics and consequently proves to be incorrect.

The general classification of all possible combinations is also greatly simplified, since instead of six of them only three remain. The corresponding classification is easily obtained by excluding intra-cell and group permutations from Table 4.

The successive development of the signs of permutations in the process of gas formation is likewise expressed much more simply:

\[ \{r\}\ .\ .\ .\ [d \to c]. \]

Finally, one more important point must be noted: by accepting the third premise of the Fermi method, we thereby establish the equiprobability of all arrangements. Indeed, the expression \(\{D\}=N!\,[\ ]\) means that, if a definite number \(N\) of particles is given, then any arrangement of them is always realized through \(N!\) combinations. Therefore, if we preserve the individuality of the particles and adhere to the proposition on equiprob-

ness of the combinations \(C\), then the probabilities of all arrangements \(D\) will turn out to be equal to one another and strictly proportional to the probability \(C\).

It is important for us to note this point in connection with the subsequent criticism of the so-called principle of the absolute identity of molecules; the latter appeared as the inevitable result of an attempt to substantiate, logically and physically, the equiprobability of \(D\), with the aim of choosing \(D\) as the unit for calculating \(W\). If, however, we proceed from the third premise of Fermi’s method, then in order to substantiate the equiprobability of \(D\) we shall not need to resort to establishing additional postulates in the spirit of the principle just mentioned.

10

In trying to provide a logical justification for the necessity of replacing the old statistics by the new one, a certain part of contemporary physicists falls into a definite methodological error when it establishes, as one of the starting points for quantum statistics, the principle of the absolute identity of the molecules of a homogeneous gas, seeing the logical error of the old classical statistics in the fact that the latter individualized molecules.

In its general form, the principle of the absolute identity of the particles forming the physical collective under investigation is set forth in such substantial literature as, for example, the book by P. A. M. Dirac, The Principles of Quantum Mechanics (GTTI, 1932).

Considering the question from the point of view of physical combinatorics, we shall allow ourselves to give the following explanation of why, in our view, the principle of the absolute identity of molecules has occupied such an important place in the new statistics. In our opinion, the introduction of this principle is essentially caused by the following concurrence of circumstances: we have already said that the decisive difference between the new statistics and the old one is the choice of \(D\) as the unit for calculating \(W\). But besides \(D\) there also exist other random combinations. Why, then, do the formulas for \(W\) give an incorrect result when one of the other combinations is taken as the unit for calculating \(W\)? Having not obtained the possibility of giving a logical answer to this question, some physicists have in fact taken another path, which, instead of explaining the difficulties, proposes simply to declare them nonexistent. In fact, instead of seeking an explanation of why it is necessary to choose \(D\) as the unit for calculating \(W\), it is sufficient to declare that all the other combinations—\(C\), \(E\), and \(P\)—do not exist in nature at all; then the choice of \(D\) is explained by itself, for only \(D\) turns out to exist.

But in order to destroy the obstructing combinations \(C\), \(E\), and \(P\), it is necessary to destroy the [[unclear: word beginning “compen-”]] common to all of them and lying at their basis.

of the logically substantiated combinatorial methods

...the compensated permutation of molecules; the latter, however, is connected with the individualization of molecules (see Table 4). Therefore this permutation can be eliminated only in advance, by declaring nonexistent the individual differences between the molecules that carry it; in other words, by turning the relative depersonalization of molecules associated with the arrangement \(D\) into their absolute identification.

But as soon as the principle of the absolute identity of molecules has been postulated, we can go from it in the reverse direction—through the statement of the absolute indistinguishability of compensated permutations and of the combinations associated with them—to the conclusion that, since all combinations are absolutely identical with one another, it has no physical meaning to speak of counting their number; on the contrary, all the combinations by means of which the given arrangement is realized must then be regarded as one single combination identical with itself, which also coincides with the arrangement. In other words, here the following operation is tacitly performed: once

\[ \frac{N!}{\prod_i n_i!} \]

combinations by means of which the given arrangement is realized are postulated to be indistinguishable, then on this basis the very existence of these combinations is also denied. But, as we have seen, this tacit operation is precisely the genuinely initial point which, in the final analysis, makes it necessary to postulate the principle of the absolute identity of molecules in order to create the possibility of a logical substantiation of quantum statistics.

After this it is easy to infer, first, that only \(D\) possesses real physical meaning and, second, that since any arrangement can be realized only through one single combination, all arrangements must therefore be mutually equiprobable.

In fact, however, here all combinations except \(D\) have simply been crossed out, so that nothing at all remains but to declare it the unit of calculation \(W\).

Thus one group of physicists, trying to find for quantum statistics the customary logical justification, found itself compelled to postulate the absolute identity of molecules as an initial principle, in order essentially to bypass the difficulties of the logical justification of the choice of the unit of calculation \(W\).

However, another group of physicists from the very beginning sensed the whole formalism and all the untenability of the principle of the absolute identity of molecules; thus, for example, I. E. Tamm, in the article to which we referred above, emphasizes that the indicated principle “entails extremely serious physically inadmissible consequences,” to the point that recognition of the principled indistinguishability of two compensated permutations of molecules is equivalent to a refusal “to apply to ma-

teria of the concept of “substance.” The way out of this contradiction, in the opinion of Ig. E. Tamm, consists in the fact that the principle of the absolute identity of molecules must be regarded as a characteristic not of the physical, but exclusively of the formal aspect of quantum statistics; therefore this principle is not at all obligatory for it; on the contrary, taking into account the physical aspect, on the analysis of which we cannot dwell here, it is apparently possible to obtain the results of Bose’s theory without renouncing the individualization of molecules.

The point of view set forth, held by that part of physicists who did not accept the principle of the absolute identity of molecules as an initial premise, is undoubtedly more correct. Developing this point of view, let us now examine the untenability of the indicated principle from the general methodological standpoint.

First of all, this principle contradicts reality; in actual fact, nature is not a bare, abstract unity in the form of the absolute identity and immutability of its primary elements, but a living unity, a living connection of the concrete manifold of individual bodies and phenomena, objects and processes, eternally developing and eternally changing.

“The electron is as inexhaustible as the atom, nature is infinite,” says Lenin. If, for a moment, we adopt the standpoint of the conception being criticized, then the question will inevitably arise before us: why do molecules turn out to be absolutely identical with one another? The various possible answers to this can be reduced to three main ones: first, the identity of molecules can be explained by the fact that they represent the ultimate, indivisible and indecomposable particles of matter; secondly, by the fact that even if they do represent complex systems, the conditions of development of each molecule of the given gas were absolutely identical, so that any possibility whatsoever of the appearance of any accidental individual deviations was excluded; and finally, thirdly, by the fact that if the conditions of development were not identical, then their difference could not have had any influence on the character of the molecules, for the development of the latter in all its details took place exclusively by virtue of laws immanently inherent in them and was absolutely independent of external influences.

All three answers stand in glaring contradiction to the facts of the development of matter and of the formation of the various forms of its motion; it is not the absolute identity of molecules, but Lenin’s proposition on the inexhaustibility of the atom that modern physics proves in each of its experimental investigations into the properties of atoms and molecules, in each of its strictly scientific attempts to penetrate into the essence of matter.

Thus those physicists who defend the principle of the absolute identity of molecules are taking an enormous step backward: away from the modern development of physics, which has proved the absence of immutable, absolutely identical particles from which supposedly ...

consists matter, to the mechanistic conceptions of the eighteenth and early nineteenth centuries, when atoms were seen as precisely such absolutely identical particles of matter.

Secondly, the conception under consideration essentially means that the general, necessary, and essential feature in the given connection (the number of molecules \(n_i\)) is declared to be the only one deserving attention, and therefore the only existing feature of the molecules; on the contrary, their individual differences, as singular, accidental, and inessential in the given connection, are declared to be absolutely devoid of any physical meaning and not really existing at all. As a result, it turns out that all the molecules of a given homogeneous gas exist only as such, deprived of their individual form, and consequently only as molecules in general.

Such a formulation of the question, which casts out of science accidental differences as unworthy of its attention and leaves to science only necessary processes in their pure form, likewise represents an enormous step backward: from contemporary physics, which by its objective development has fully confirmed the correctness of considering necessity in unity with chance, the general and the particular with the singular, the essential with the inessential—to the ossified Wolffian conception, which regards these categories as unconditionally excluding one another.

Thirdly, the principle of the absolute identity of molecules cannot be justified by the fact that we, owing to the imperfection of our instruments, are in principle deprived of the possibility of ever detecting differences between individual molecules and that, therefore, it has no physical meaning to speak of their difference. In other words, one must not, proceeding from our practical subjective impossibility at a given stage of scientific development to detect a certain phenomenon, draw on this basis the conclusion that this phenomenon does not exist in nature at all and that any conception of it is devoid of any objective meaning whatever. Such an approach to the study of nature—very old in essence and only renewed in form—is profoundly erroneous, for, as Engels says, human thought is “sovereign and unlimited in its tasks, in its purpose, in its possibilities, in its ultimate goal.”

The entire objective course of development of modern physics proves that in nature there are no barriers in principle insurmountable for human knowledge, and that what seems absolutely impossible at one stage in the development of science proves to be practically realized in the process of its further development.

Still less can one justify the tacit assumption that, since combinations are absolutely indistinguishable, then conse-

consequently, they do not exist at all in nature. Even if, contrary to all the facts confirming the infinite complexity of the structure of matter, we were for a moment to allow that molecules are absolutely identical, by this we would in no way prove that no compensated permutations exist in nature and that two identical molecules cannot, by virtue of precisely this identity, exchange their places. Since in this case, too, we still admit the existence of simpler uncompensated permutations, we must inevitably, in principle, admit the existence of more complex cases of their mutual cancellation with the formation of the corresponding combinations.

Thus the principle under criticism ceases to justify even itself, for the mere fact of postulating it proves insufficient for the hidden ultimate goal—to destroy \(C\), \(E\), and \(P\)—to be attained without new contradictions.

Finally, fourthly, by accepting the absolute identity of molecules, we thereby risk splitting the development of physical combinatorics as a unified science into two stages, absolutely opposed to one another, separated by a fundamental, theoretical-cognitive abyss. Indeed: since earlier it was precisely those permutations that, from the standpoint of the principle under criticism, are declared nonexistent in nature that were counted, the whole old statistics thus turns out, on the whole, to have been built on an absolutely invented, fictitious basis; consequently it turns out to be a fictitious, physically meaningless statistics, having nothing in common with the real physical process.

From this, naturally, follows the conclusion that the old statistics must be completely discarded as an entirely false theory.

Such are the general considerations that compel us to regard the principle of the absolute identity of molecules as erroneous.

All this taken together shows that the new quantum statistics has stumbled in the attempt to find for itself a logical foundation. This happened chiefly because, not being armed with the method of materialist dialectics, physicists were unable to give a correct analysis of the logical foundation of the new statistics, were unable to uncover the errors of the old statistics, and followed the usual path in such cases of postulating purely formal principles and axioms.

Owing to this, alongside the enormous step forward represented by quantum statistics as a whole, we at the same time have an attempt to drag science back toward conceptions and concepts long since overcome by the entire course of scientific development, a return to which at the present time is a serious brake on the further development of physical statistics, especially with respect to its logical foundation.

In contrast to this, we tried to show that compensated permutations, as an object of classical statistics, are quite real insofar as the displacements of individual molecules that form them are real, i.e., uncompensated permutations; the latter, however, are an object of quantum statistics. Therefore, in our opinion, there is no logical sense in abandoning the individualization of molecules altogether.

However, in the relation in which we determine the law of a gas on the basis of the mechanical motion of molecules, the compensated permutation is an inessential moment of reality. The error of the old statistics consists not at all in the fact that it constructed its justification on a fictitious basis, but in the fact that it took a moment of reality that is inessential in the given relation as its essential moment; of these two moments, the distinction between individual molecules is inessential, while their total number \(n_i\) is essential. Therefore we stated that only that statistics can lead to the correct result which abstracts from the individual differences of molecules and, consequently, carries out their depersonalization in a relative, but of course not in an absolute, sense. In its correct interpretation, quantum statistics is precisely to be regarded as constructed on this relative depersonalization of molecules.

Accordingly, as it seems to us, it is necessary: first, to replace the concept of the absolute identity of molecules by the concept of their relative depersonalization, understanding by this an abstraction from their individual differences; and, second, not to impose this concept on physical statistics in the form of a postulate fixed in advance, but to derive it as a logical consequence from the analysis of the fundamental problem of calculating \(W\) and of its basic premises.

Thereby the opposition between the new and the old statistics ceases to be absolute and assumes the character of a relative opposition between the essential and the inessential aspects of the phenomenon under study.

11

In revealing the logical errors of classical statistics, we must emphasize with complete sharpness that the view of things and processes from the standpoint of the interrelation of the accidental and the necessary is not an arbitrary point of view, peculiar only to our consciousness, i.e., a subjective point of view. In distinguishing and opposing to one another, in certain relations, the accidental and the necessary, the essential and the inessential moments of the phenomena we study, we only reflect in our consciousness the contradictory character of reality itself, breaking it down as a whole in our consciousness into

opposite aspects and bringing them subsequently into mutual connection.

In exactly the same way, one should not imagine that our transition from one pair of combinations, chosen as the prerequisite for constructing the combinatorial method, to another pair, just as the very gradation of combinations according to the degrees of their more or less accidental character relative to the necessary behavior of the gas as a whole, depends exclusively on our personal discretion, solely on which combination it is simply more convenient for us to take as necessary or as accidental rather than another. However, what in this case some physicists incorrectly call “convenience” follows not from our personal point of view, but from the correspondence of our theories to the nature reflected by them. If one or another theory proves well applicable to the explanation of a given phenomenon and leads to good results, then it may subjectively seem to the scholar who creates or uses this theory that he has simply managed to select a more “convenient,” more “simple,” and “economical,” and therefore also more scientific, way of approaching the facts under study. However, the objective side here is such that the method found for explaining or investigating the facts is more scientific not because it subjectively seems more convenient than other methods, but because it more correctly, more precisely and fully, as Lenin indicated, reflects objective reality. It is enough to imagine that some method, being “convenient” from a subjective point of view, gives incorrect results, for its very “convenience” to lose all meaning and for the method to be declared unscientific.

With regard to replacing the old statistics with the new one, we must emphasize that quantum statistics was introduced by no means because it is more convenient and simpler than classical statistics, as, in essence, Ya. I. Frenkel assumes in an analogous case (see Sorena, issue II, 1932), speaking of the relation between quantum and classical mechanics, but because it reflects reality more correctly and more deeply.

That is why we have tried to show in detail that the establishment of our classification of combinations corresponds to the real process of gas formation, when necessity actually passes into chance, and chance manifests itself in the process of development as necessity—in a word, when both appear as definite, mutually connected and mutually transforming opposite aspects of reality itself.

Thus the replacement of one pair of combinations, connected by the relation of the accidental and the necessary and lying at the basis of the calculation of $W$, by another pair, as well as the replacement of the concept of absolute identity by the concept of relative depersonalization, are not measures undertaken for reasons of convenience.

operations, but rather the correction of our physical theories in accordance with the objective relation of the necessary and the accidental, of the essential and the nonessential moments of reality itself.

12

Let us now formulate the general conclusions at which we have arrived as a result of our reasoning.

1) The basic task of the computation of \(W\) and of the combinatorial determination of \(S\) consists in reflecting, from the quantitative side, the lawful connection between the necessary behavior of the collective and the accidental behavior of the individuals forming it; in particular, in determining \(S\) for a homogeneous gas, the connection between the thermodynamic state of the gas and the mechanical motion of the molecules.

2) The general logical basis of all methods of computing \(W\) is the unity of the accidental and the necessary, which is concretized in the form of the unity of two mutually opposed combinations, one necessary and defining itself through the other—the accidental—as the form of its manifestation. The establishment of both combinations constitutes the two basic logical premises of the computation of \(W\).

3) The interconnection of the methods of computing \(W\) follows from their common logical basis and consists in the mutual transitions of the accidental and necessary moments contained in each of the combinations; the analysis of these transitions leads to the establishment of a rational classification of all combinations that can serve as premises for the computation of \(W\).

4) The classification of all combinations is established on the basis of the following three moments: a) the relation of the combinations as the relation of an accidental form and the necessity manifested in this form; b) the features characterizing those permutations of molecules which underlie the corresponding combinations; and c) those stages through which the gas passes in the process of its formation in nature.

5) The solution of the problem of the combinatorial determination of \(S\) consists, above all, in the correct choice, in accordance with reality, of the necessary and accidental combination, the unity of which must constitute the logical basis for the subsequent construction of the mathematical apparatus for the purpose of directly calculating the value of \(W\) and \(S\).

6) The combination connected in an essentially necessary way with the thermodynamic state of the gas is the distribution of molecules \(\{R\}\), which is characterized by the features \(Z\) and \(N\); \(R\) is the external manifestation of the interaction of molecules occurring within the gas, consisting of their continuous collisions.

7) The combination connected in an essentially necessary way with the...

with the mechanical displacement of individual molecules is the placement of the molecules \([D]\), which is characterized by the feature \(n_i\); therefore, of all the combinations, only \(D\) must be chosen as the unit for computing \(W\).

8) Of all possible combinatorial methods differing according to two basic premises, Bose’s completed method is the only correct method of combinatorially determining \(S\) for a homogeneous gas, for only it, being constructed on the ratio of the combinations \(\{R\}\) and \([D]\), is capable of correctly reflecting the lawful connection between the state of the gas and the motion of the molecules.

9) The third premise in computing \(W\) is the determination of the maximum number of particles that may be present simultaneously in one cell; in the case of Fermi’s method, proceeding from Pauli’s principle, this number cannot exceed unity. The establishment of the third premise is wholly determined by the character of the physical collective.

10) The third premise of Fermi’s method leads to the immediate consequence of the equiprobability of all placements, which makes it possible physically to justify the choice of \(D\) as the unit for computing \(W\).

11) Until now, the shortcomings and contradictions of classical statistics have scarcely been considered from the standpoint of a critique of the logical basis on which it was built in its combinatorial part. For our part, we have tried to uncover, behind the external formal side of the contradictions of the old methods, their essence, and to give a logical justification for the necessity of passing to a new method of computing \(W\).

12) The basic contradiction of modern quantum statistics consists in the fact that, while on the whole representing a revolutionary achievement of science, in the area of its methodology it makes a reactionary attempt at a backward movement toward the metaphysical conceptions of natural science of the eighteenth century, long since refuted by the entire course of the development of science.

13) In its correct interpretation, the new statistics does not absolutely discard the old, but overcomes its shortcomings by shifting the center of attention from the inessential side of the phenomenon to its essential side, namely: from the individual differences of molecules to their general, impersonal number.

14) This shift of attention gave some physicists occasion to take the path of a complete separation of the essential side of the phenomenon from the inessential and, in attempts to find a logical justification for the new statistics, to postulate the erroneous principle of the absolute identity of molecules. This erroneous conception can be overcome if, considering the essential side of reality in unity with its inessential side, the concept of absolute identity is replaced by the concept of the relative depersonalization of molecules.

In conclusion, let us note that the main purpose of the present article was to show that the correct method of scientific cognition of natural phenomena, in particular phenomena belonging to the domain of physical statistics, must, on the one hand, regard these phenomena as material processes, and physical theory as a more or less accurate reflection of these processes; and at the same time, on the other hand, must regard each phenomenon in its development, in its inner contradictoriness, in the mutual connection and transitions of its opposite moments into one another.

Proceeding from this, on the one hand, we have shown that one or another method of calculating \(W\) is not merely a mathematical device that makes it possible, by simply counting certain combinations, to calculate the value of \(S\), but is a definite reflection of the real connections existing in nature. Therefore, if one does not proceed from taking these connections into account, then on the basis of the consideration of subjective “convenience” alone it is impossible correctly to solve the problem of calculating \(W\).

On the other hand, we have also shown that the general philosophical categories of chance and necessity as applied to the specifically physical concept of thermodynamic probability \(W\) do not have a fixed, absolutely polar character in relation to one another, but are mobile and changeable, reflecting the mobility and changeability of reality itself.

In doing so, we have avoided an approach to the problem under consideration that reduces everything to a simple illustration of the correctness of the general propositions of dialectical materialism; on the contrary, we have tried to proceed from Lenin’s direct indication that the identity of opposites must be understood not as a sum of examples, but as a “law of cognition” (and as a law of the objective world).