New Principles of Bose–Einstein Statistical Mechanics in Connection with the Question of the Physical Nature of Matter

By E. Tamm.

§ 1. Introduction. — § 2. Some fundamental propositions of classical statistical mechanics. — § 3. A critical clarification of the principles of statistical mechanics. — § 4. The laws of ideal gases. — § 5. The laws of a quantum, or light, gas. — § 6. The physical meaning of the theory of Bose–Einstein. — § 7. The interference of molecules and the de Broglie theory. — § 8. Some consequences of the interference of molecules.

§ 1. Introduction.

In the summer of 1924 there appeared two papers by the Indian physicist Bose, followed by papers by Einstein, which substantially deepened Bose’s ideas and considerably broadened the range of their application. These works introduce an essentially new current into the development of thermodynamics and statistical mechanics. Although the new principles of statistical mechanics advanced by Bose and Einstein are at present in the process of intensive development and are far from having assumed a harmonious, completed form, they have already led to a number of important results which, in all probability, will become a lasting possession of theoretical physics. The chief of these results are as follows:

1) a number of contradictions have been revealed to which statistical mechanics leads in its former classical form, and a method for eliminating them has been indicated;

2) on the basis of new principles for calculating thermodynamic probability, corrections have been introduced into the classical formulas of the kinetic theory of gases, and a number of new phenomena have been predicted that can be subjected to experimental verification;

3) a profound physical connection has been elucidated between the new principles for calculating probabilities and one of the most interesting new quantum theories—the theory of de Broglie (L. de Broglie), which promises to shed unexpected light on the physical nature of matter;

4) for the first time a derivation is given—although a purely formal one, nevertheless free of internal contradictions—of Planck’s famous formula for the spectrum of a black body, which is the cornerstone of all quantum theory.

In the present article, without following the historical sequence in the development of the ideas, we shall try briefly to outline the physical meaning and the principal conclusions of the Bose–Einstein theory, and at the same time we shall try to show that the necessity of eliminating the contradictions that had accumulated in classical physics almost inevitably leads to the fundamental propositions of the new theory.

§ 2. Some fundamental propositions of classical statistical mechanics.

In almost all of what follows we shall confine ourselves to the consideration of an ideal monatomic gas, i.e. a gas consisting of identical monatomic molecules between which there are no forces of cohesion or repulsion, so that their interaction is limited to elastic collisions. The state of the gas is characterized by assigning, for each of its molecules, three coordinates of position \(q_1=x,\ q_2=y,\ q_3=z\), and three components of velocity \(v_1=v_x,\ v_2=v_y,\ v_3=v_z\). Instead of the components of velocity it is convenient to introduce the components \(p_\alpha=mv_\alpha\ (\alpha=1,2,3)\) of the momentum (or quantity of motion) of the molecule, whereby a considerable simplification of the subsequent calculations is achieved1. The exposition of the theory is greatly simplified by the introduction of geometrical terminology. We shall call phase space the space of six dimensions whose axes correspond to the three coordinates of the position of the molecule \(q_\alpha\) and to the three “coordinates of momentum” \(p_\alpha\), so that the position and velocity of each molecule is represented by a definite point of phase space. For brevity we shall say that a molecule is located at a given point of phase space if its coordinates \(q_\alpha\) and \(p_\alpha\) coincide with the coordinates of this point.

Exact knowledge of the state of the gas is inaccessible to us, i.e. exact knowledge of the coordinates of each molecule in phase space. By observation we can determine only the macroscopic state of the gas, i.e. the distribution of its molecules over separate regions or cells of phase space. Let us divide the entire phase space into separate cells, i.e. into very small but nevertheless finite six-dimensional parallelepipeds, and we shall say that the macroscopic state of the gas is completely determined if the number of molecules \(n_k\) located in each of the cells of phase space is known (\(n_k\) is the number of molecules in the \(k\)-th cell).

The state of a gas would be determined much more precisely if we knew not only the total number \(n_k\) of molecules located in each cell, but also which particular gas molecules are located in the given cell. If these data are known, i.e. if it is known in exactly which cell each of the gas molecules is located, then we shall say that the microscopic state of the gas is known. Consequently, the displacement of a molecule within the cell occupied by it does not change the microscopic state of the gas; in what follows we shall regard such displacements as inessential1.

Each macroscopic state of a gas, characterized by the set of numbers \(n_k\), can be realized in \(W\) essentially different ways, where

\[ W=\frac{N!}{\prod_k n_k!}, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (1) \]

and \(N=\sum_k n_k\) is the total number of all gas molecules2. Indeed, the macroscopic state of the gas will not change under an arbitrary permutation of molecules one for another. The number of such permutations is \(N!\). From the number of these permutations one must exclude all those permutations in which molecules located in one and the same cell exchange places, for these permutations, according to our assumption, are inessential. As a result we arrive at formula (1).

This formula plays a fundamental role in classical statistical mechanics, for it determines the thermodynamic probability of the given macroscopic state of the gas: the thermodynamic probability \(W\) of a state is precisely the number of essentially different ways of realizing it. It is necessary, of course, that each of the ways of realization be equally probable. This requirement will be satisfied if we agree once and for all to divide phase space into cells of equal volume3.

The fundamental thermodynamic quantity—entropy \(S\)—is connected with the thermodynamic probability by the well-known Boltzmann relation

\[ S=k\lg W, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2) \]

where \(k\) is Boltzmann’s universal constant:

\[ k = 1.372 \cdot 10^{-16}\,\frac{\mathrm{erg}}{\mathrm{degree}} . \]

The physical meaning of this equation consists in reducing the second principle of thermodynamics, according to which physical processes are associated with an increase of entropy, to the almost self-evident assertion that, in the overwhelming majority of cases, physical processes are associated with the transition of bodies from less probable to more probable states.

§ 3. Critical refinement of the principles of statistical mechanics.

Equations (1) and (2) serve as the foundation of the entire classical statistical theory of the ideal gas; however, as has become clear in recent times, they require substantial additions and modifications.

First of all, in introducing the concept of a phase cell, we left open the question of its dimensions1, requiring only that all cells be of equal size. Meanwhile, a change in the magnitude of the cells entails a change in the numbers \(n_k\), and hence also in the probability \(W\) and the entropy \(S\) of the gas. In order to assign to the entropy a completely definite value, it is therefore necessary to fix precisely the magnitude of the cells. Classical theory could not solve this problem and found a way out of the difficulty in the circumstance that, if the volume of the cells does not exceed a certain maximum limit, then further subdivision of the cells leads only to a change of an additive constant in the expression for the entropy2. Since classical thermodynamics considers only changes in the magnitude of entropy in physical processes, and regards the absolute magnitude of entropy as determined only up to some additive constant, classical statistical mechanics could confine itself to the requirement that, in calculating thermodynamic probability, the size of the cells not exceed the aforementioned limit. However, the development of thermodynamics at the beginning of the present century and, in particular, the establishment of the famous postulate of Nernst, which states that at absolute zero of temperature the entropy of every homogeneous body3 is equal to zero, led to the necessity of fixing the absolute value—

...not entropy, but $\alpha$ became so as well, and the sizes of the phase cells. This was accomplished by transferring into the theory of the ideal gas certain propositions at which quantum theory had arrived in considering ensembles of periodic systems. At the present time the following postulate of quantum theory may be regarded as generally accepted: the phase space of an ideal gas (as of any other body) has a discrete, not a continuous, structure, and breaks up into a series of elementary cells. The concept of an elementary cell has a real physical meaning and is by no means a conventional term introduced merely for computational convenience, as it appeared from the point of view of the classical theory. In the case of an ideal gas the volume of these cells is equal to $h^3$, where $h$ is Planck’s universal constant.

Having thus fixed the sizes of the cells, we have not, however, yet removed all the shortcomings of that definition of thermodynamic probability which is contained in formula (1). The point is that from equations (1) and (2) there follows the physically inadmissible consequence that at a sufficiently high temperature the entropy of a gas in its equilibrium state reaches a constant value and thereafter no longer changes with a further rise in temperature. Indeed, when the temperature of a gas is raised, i.e. when the energy, and consequently also the momentum, of its molecules is increased, the volume of that region of phase space over which its molecules are distributed increases. At a sufficiently high temperature the number of cells in this region becomes so large that, in the equilibrium, i.e. the most probable, state of the gas, no more than one molecule corresponds to each of these cells. Thus the number of molecules $n_k$ in each cell becomes equal either to zero or to unity, and therefore, by formula (1), $W$ assumes the constant value1 $W = N!$.

In order to remove this difficulty, it is necessary to modify the very definition of thermodynamic probability contained in formula (1).2 In deriving this formula we proceeded from the assumption that the macroscopic state of the gas is determined by the distribution of its molecules over the elementary cells, i.e. by specifying the numbers $n_k$. It is obvious that we fixed the state of the gas too precisely. Indeed, cells of size $h^3$ ($h = 6.545 \cdot 10^{-27}$ erg sec.) are so small that it is not accessible to our observation to investigate the distribution of molecules over individual cells; experimentally, the distribution of molecules over phase...

space can be defined only within much cruder and approximate limits. Let us therefore divide the phase space into regions, in each of which there is a large number of elementary cells of volume \(h^3\). Let \(z_i\) be the number of cells in the \(i\)-th region, whose boundaries are determined by the limits of the coordinates: from \(q_{ai}\) to \(q_{ai}+\Delta q_{ai}\) and from \(p_{ai}\) to \(p_{ai}+\Delta p_{ai}\) \((a=1,2,3)\); the volumes of the different regions may be unequal. Henceforth we shall regard the state of a gas as macroscopically determined if the distribution of its molecules over the regions (and not the cells) of phase space is known, i.e. if for each region the number \(N_i\) of molecules contained in it is known (the capital \(N_i\) refers to the distribution over regions, the lower-case \(n_k\) to the distribution over cells). Then the thermodynamic probability of the macroscopically determined state of the gas will be expressed by the formula [²]

\[ W=\frac{N!\prod_i\left(z_i^{N_i}\right)}{\prod_i N_i!},\ldots\ldots\ldots\ldots\ldots\ldots (3) \]

which is to replace formula (1). Indeed, formula (3) determines the number of possible ways of realizing the macroscopic state of the gas, i.e. the number of microscopic states corresponding to the given distribution of molecules over phase regions, since the \(N_i\) molecules can be distributed among the individual regions in

\[ \frac{N!}{\prod_i (N_i!)} \]

different ways; moreover, each of the \(N_i\) molecules that have fallen into a given region may be in any one of the \(z_i\) cells contained in it, i.e. may occupy in it \(z_i\) essentially different positions.

Since formula (3) contains a series of factors of the type \(z_i^{N_i}\), and since the sizes of the regions, unlike the sizes of the elementary cells, are arbitrary, the objection raised against equation (1) is inapplicable to this formula.

At first sight it might seem that, by replacing equation (1) with equation (3), we shall have to introduce changes also into all those propositions and conclusions at which classical statistical mechanics arrived starting from formula (3), and which, as is known, agree fully with the facts. This, however, is not so; the previous results remain in force, and only the erroneous assumption used in deriving them from equation (1) is eliminated. The point is that, in deriving the laws of ideal gases from formula (1), one always made the by no means always justified assumption¹) that the numbers \(n_k\) are so large that facto-

¹) Thus, for example, at high temperatures, as we have seen, the \(n_k\) are equal either to 0 or to 1.

their factorial can be expressed with sufficient accuracy by Stirling’s approximate formula:

\[ n_k! = \left(\frac{n_k}{e}\right)^{n_k};\quad \lg n_k! = n_k \lg n_k - n_k \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) \]

It can be shown that, if this assumption is correct, then formula (1) reduces to formula (3)¹). Thus, in deriving the laws of ideal gases, formula (1), through the introduction of an unfounded assumption, was always unconsciously replaced by formula (3); an erroneous course of reasoning corrected the shortcomings of an erroneous initial formula.

However, both formulas for \(W\)—(1) and (3)—have one more extremely essential common defect: both contradict the postulate of the additivity of entropy. Let us imagine that to a given volume of gas we add a completely identical second volume of the same gas, of equal temperature, equal number of molecules, with the same equilibrium distribution of them in phase space, etc., and consequently also of equal entropy. Combining these two volumes of gas into one (removing, for example, the partition separating them), we obtain a doubled volume of gas possessing, according to the fundamental laws of thermodynamics, a doubled value of the entropy. Meanwhile, from equation (3) and equation (2) it follows that

\[ S = k\left\{\lg N! + \sum_i \lg\left(\frac{z_i^{N_i}}{N_i!}\right)\right\}, \]

where the sum extends over all cells of the phase region. In doubling the volume of the gas, we do indeed double the value of this sum, since to each term of the sum there is then added an equal supplementary term referring to the correspondingly situated phase cell of the added volume. However, the first term of the right-hand side of the expression just given is by no means doubled when the volume is doubled, since when the number of molecules is doubled \(\lg N!\) increases by more than a factor of two. Consequently, the definition of entropy contained in equations (2) and (3) contradicts the postulate of the additivity of entropy. The replacement of equation (3) by formula (1) does not remove this contradiction.

¹) Let the \(i\)-th region contain \(z_i\) cells with numbers from \(k = r\) to \(k = r + z_i - 1\), or, what is practically the same thing, to \(k = r + z_i\), with each of these cells containing \(N_i/z_i\) molecules. The corresponding terms of formula (1) can, with the aid of equation (4), be represented in the following form:

\[ \frac{1}{\prod_{k=r}^{k=r+z_i} (n_k!)} = \frac{1}{\left(\left(\frac{N_i}{z_i}\right)!\right)^{z_i}} = \frac{1}{\left(\frac{N_i}{e z_i}\right)^{N_i}} = \frac{z_i^{N_i}}{\left(\frac{N_i}{e}\right)^{N_i}} = \frac{z_i^{N_i}}{N_i!}, \]

which coincides with the corresponding term of formula (3).

In recent years there has arisen an entire literature devoted to the question of ways of eliminating this contradiction (Sackur, Planck, Tetrode, Nordheim, Ehrenfest and Trkal, Schrödinger, and others; for a brief survey of the literature see the works of the last two authors [5] and [4]). In essence the matter reduces to removing from the expression obtained for the entropy the first term, equal to \(k \lg N!\), i.e., to removing from the expression for the probability \(W\) the factor \(N!\), and to finding a justification for this removal\(^1\).

Some of the authors mentioned motivate the removal of the factor \(N!\) simply by the need to eliminate the indicated contradictions, and we shall not discuss their works here; other investigators, and in particular Planck [6], try to give this step a statistical justification. The essence of Planck’s reasoning may be set forth as follows. All molecules of an ideal gas are completely identical and indistinguishable from one another. Meanwhile, in deriving formulas (1) and (3), we assumed that when two molecules are interchanged, the microscopic state of the gas, generally speaking, changes. Planck proposes to abandon the individualization of molecules and to regard the permutation of identical molecules as having no significance; it turns out that in this way the indicated contradictions with the postulates of thermodynamics can be eliminated. In doing so, however, Planck, for certain fundamental reasons, abandons the method of statistical mechanics which we have used throughout the preceding discussion, and resorts to considering an imaginary ensemble of identical volumes of gas (Gibbs’s method).

We, however, shall not follow Planck along this path\(^2\), but, remaining on the ground of our method of calculating thermodynamic probability, shall show that the consistent application of Planck’s idea of abandoning the individualization of molecules inevitably leads to the Bose–Einstein theory [4].

How will formulas (1) and (3) be modified if we abandon the individualization of molecules? It may seem that, since in this case the permutation of molecules with one another should not be considered as changing the state of the gas, the value of the probability \(W\) previously calculated by us will simply have to be divided by the number of possible permutations of \(N\) molecules, i.e. by \(N!\), which would remove the contradiction with the postulates of thermodynamics. The matter, however, is not quite so simple, as is clear at least from the following considerations, which

\(^1\) The removal of the factor \(N!\) is also necessary because only under this condition does the entropy of a saturated gas at low temperatures \(T\) (at which the heat capacity of the condensate may be neglected) become equal to \(\dfrac{r}{T}\), as thermodynamics requires (\(r\) is the specific heat of vaporization) [4].

\(^2\) A critique of Planck’s theory is given by Schrödinger [4].

we shall, for simplicity, apply to formula (1); the essence of these considerations, in a somewhat modified form, is also applicable to formula (3). Let us replace, as some of the above-mentioned authors propose, formula (1)

\[ W=\frac{N!}{\prod_k n_k!}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1) \]

by the formula

\[ W=\frac{1}{\prod_k n_k!},\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1a) \]

which is obtained from (1) by division by \(N!\). By definition, the thermodynamic probability \(W\) is the number of possible ways of realizing a given state of the gas, i.e. it is an integer and not less than unity, whereas formula (1a) gives a fractional expression for \(W\)! This contradiction is explained by the fact that, in giving up the individualization of molecules, we have divided the number of possible ways of realizing the given state \(W\) by the total number of possible permutations of the molecules, \(N!\), forgetting that in deriving the expression for \(W\) there had already been excluded those permutations of molecules with one another which, in our former terminology, do not create an essentially new distribution. The number of such permutations, reducible to the permutation of molecules situated in one and the same cell, is equal to \(\prod_k n_k!\). Consequently, when one gives up the individualization of molecules, expression (1) must be divided not by \(N!\), but by

\[ \frac{N!}{\prod_k n_k!}, \]

which leads to the equality \(W=1\). Indeed, if one does not individualize the molecules, then it is obvious that the state determined by assigning the number of molecules \(n_k\) in each of the phase cells can be realized in only one single way.

Of course, also in the case where one starts from the expression for \(W\) given by equation (3), and not (1), one likewise cannot, when abandoning the individualization of molecules, divide this expression simply by \(N!\). In this case, however, it is simpler not to introduce the necessary changes into equation (3), but to carry out anew the counting of the thermodynamic probability \(W\), as Einstein does [2].

Thus, let us abandon the individualization of molecules and, accordingly, in contrast to before, let us assume that the microscopic state of the gas is completely determined by assigning for each cell the total number of molecules \(n_k\) contained in it, independently of which particular molecules are in it. The macroscopic state of the gas, as also in the derivation of formula (3), will be determined by assigning for each phase region the number contained in it

molecules \(N_i\). The thermodynamic probability \(W\) of a macroscopic state is equal to the number of corresponding microscopic states, i.e. to the number of such ways of distributing \(N\) identical molecules among the cells of phase space in which to each phase region \(i\), i.e. to each group \(Z_i\) of cells, there corresponds a specified number of molecules \(N_i\). The number of possible distributions of \(N_i\) identical molecules among \(Z_i\) cells is, as is known, equal to\(^1\)

\[ \frac{(N_i+Z_i-1)!}{N_i!(Z_i-1)!}. \]

Carrying out all possible distributions of molecules in each of the phase regions, we obtain the total number of possible distributions. Thus,

\[ W=\prod_i \frac{(N_i+Z_i-1)!}{N_i!(Z_i-1)!}. \tag{5} \]

This is the desired Bose–Einstein formula, lying at the basis of the new statistical mechanics and, apparently, free from all internal contradictions.\(^2\)

Thus, from the purely formal side, the new Bose–Einstein statistics is characterized by the renunciation of the individualization of molecules, to which the necessity of eliminating a number of contradictions that had accumulated in statistical mechanics inevitably leads. However, this is only the purely formal side of the matter. As we shall see in § 6, the new statistics has a profound physical meaning, and the adoption of formula (5) entails the necessity of a fundamental revision of the basic views on the nature of matter.

In conclusion let us note that not only the classical methods for calculating statistical probability, but also certain propositions of thermodynamics itself require revision and correction. Thus, for example, the entropy of a gram-molecule of an ideal gas, according to classical thermodynamics, is expressed as follows:

\[ S=c_v \lg T+R\lg v+\mathrm{const}. \]

\(^1\) Proof. Arrange in arbitrary order in a single row \(Z_i\) boxes (corresponding to cells) and \(N_i\) balls (corresponding to molecules). We shall take the number of molecules in phase cell number \(n\) to be equal to the number of balls that immediately follow, to the right, the \(n\)-th box from the beginning of the row. Carrying out all possible permutations of the boxes and balls, we obtain all possible distributions of molecules among the phase cells. In doing this, however, it is necessary to make sure that on the left the row always begins with a box, for if there is a ball on the left, this means that it has not fallen into any of the boxes. Thus we must carry out all possible rearrangements of \(N_i+Z_i-1\) elements (the extreme left box remaining in place), divided into two groups of mutually identical elements consisting of \(N_i\) and \(Z_i-1\) specimens. The number of such permutations is equal to the expression given in the text.

\(^2\) It was first applied by Bose [1] to the case of a quantum (light) gas (see below); subsequently Einstein [2] transferred it also to the theory of material gases.

In principle it is always possible to make the specific volume \(v\) of the gas so small that \(S\) takes a negative value. Meanwhile, according to Boltzmann’s equation (2), entropy is an essentially positive quantity, since by definition \(W\) cannot be less than unity. It is evident that the laws of the classical thermodynamics of ideal gases, like the thermodynamics of radiation, are valid only for large \(T\) and \(v\); at low temperatures and high densities they require substantial changes.

§ 4. Laws of Ideal Gases.

Starting from the Bose–Einstein formula (5) and from the definition of entropy (2), one can derive the new laws of ideal gases in exactly the same way as is done in the classical theory. For this purpose it is first necessary to determine what the distribution of its molecules over the cells of phase space will be in the equilibrium state of the gas.

It is clear that in the equilibrium state the gas molecules will be uniformly distributed over the volume they occupy, and that all directions of the molecular velocities will be equally probable. Therefore our problem reduces to determining the distribution of molecules by energy. Up to now the form and size of the cells of phase space introduced by us into consideration have remained indefinite, and our problem can be facilitated by a proper choice of the cells. The volume \(\Omega\) of each six-dimensional phase cell

\[ \Omega=\iiint\iiint dx\,dy\,dz\,dp_x\,dp_y\,dp_z \]

depends on the limits of integration, i.e. on the position of the boundaries of the cell. Let us agree to choose the boundaries of the cells so that the points of each cell correspond to all those states of molecules whose kinetic energy \(\varepsilon\) lies within definite limits from \(\varepsilon\) to \(\varepsilon+\Delta\varepsilon\). Correspondingly, in each cell we include the entire volume \(V\) occupied by the gas, so that the expression for \(\Omega\) will take the form

\[ \Omega=V\iiint dp_x\,dp_y\,dp_z. \]

Passing to the determination of the extent of the phase cells along the momentum axes \(p_x\), \(p_y\), and \(p_z\), let us note that molecules possessing the same energy \(\varepsilon_i\) also possess momenta \(p_i\) equal in numerical magnitude; consequently, the points of phase space representing them lie on the surface of a sphere drawn from the origin with radius \(p_i\) \((p=\sqrt{p_x^2+p_y^2+p_z^2})\). Molecules whose energies lie within the pre-

intervals from \(\varepsilon_i\) to \(\varepsilon_i+\Delta\varepsilon_i\), are represented by phase points lying between two concentric spherical surfaces of radii \(p_i\) and \(p_i+\Delta p_i\). The volume of this spherical layer is equal to \(4\pi p_i^2\Delta p_i\).

Thus, the volume of the six-dimensional phase element containing all points corresponding to states of a molecule with energy from \(\varepsilon_i\) to \(\varepsilon_i+\Delta\varepsilon_i\) (or momentum from \(p_i\) to \(p_i+\Delta p_i\)) is equal to

\[ Q_i=V\iiint dp_x\,dp_y\,dp_z=V\cdot 4\pi p_i^2\Delta p_i. \]

Since the volume of each elementary phase cell is equal to \(h^3\), it follows that this phase element contains \(Z_i\) cells, where

\[ Z_i=\frac{Q_i}{h^3}=\frac{4\pi V_i^2 p\Delta p}{h^3}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (6) \]

Since \(p=mv\), and \(\varepsilon=\dfrac{mv^2}{2}\), where \(m\) is the mass and \(v\) the velocity of the molecule, we have

\[ p^2=2m\varepsilon \quad \text{and} \quad p\Delta p=m\Delta\varepsilon\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (7) \]

Substituting this into equation (6), we finally obtain

\[ Z_i=\frac{4\pi V\sqrt{2m^3\varepsilon_i}}{h^3}\Delta\varepsilon_i\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (8) \]

In this way we have divided phase space into elements so that in each \(i\)-th element, containing \(Z_i\) cells, lie all phase points corresponding to energy values from \(\varepsilon_i\) to \(\varepsilon_i+\Delta\varepsilon_i\). Let us now set ourselves the task of determining the number of molecules \(N_i\) located in this element (i.e., possessing energy \(\varepsilon_i\)) when thermodynamic equilibrium of the gas has been established. The equilibrium state is the most probable state and, therefore, corresponds to the maximum value of the probability \(W\), or, what is the same, of the entropy \(S\).

From equations (2) and (5) we obtain

\[ S=k\sum_i\{\lg(N_i+Z_i-1)!-\lg N_i!-\lg(Z_i-1)!\}. \]

In view of the fact that \(N_i\gg1\) and \(Z_i\gg1\), we may replace \(Z_i-1\) by \(Z_i\) and apply Stirling’s formula (4), as a result of which we obtain:

\[ S=k\sum_i\{(N_i+Z_i)\lg(N_i+Z_i)-N_i\lg N_i-Z_i\lg Z_i\}\ .\ .\ .\ (9) \]

In determining the values of the numbers \(N_i\) corresponding to the maximum of the entropy \(S\), it is necessary to remember that the total number of molecules

\[ N=\sum_i N_i\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (10) \]

and the total energy of the gas

\[ E=\sum_i \varepsilon_i N_i . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (11) \]

are given quantities. As is known, under the imposition of conditions (10) and (11), the maximum of \(S\) is determined by the equation:

\[ \frac{1}{k}\delta S-a\cdot\delta N-\beta\delta E=0 \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (12), \]

where \(\delta\) is the variation sign, and \(a\) and \(\beta\) are constant Lagrange multipliers, whose values will be determined in the subsequent course of the calculations1.

Varying equations (9), (10), and (11), we obtain:

\[ \delta S = k\sum_i \{\lg(N_i+Z_i)-\lg N_i\}\delta N_i = k\sum_i \lg\left(\frac{N_i+Z_i}{N_i}\right)\delta N_i, \]

\[ \delta N=\sum_i \delta N_i \quad \text{and} \quad \delta E=\sum_i \varepsilon_i\delta N_i . \]

Substituting these values into equation (12), we obtain:

\[ \sum_i \left\{ \lg\left(\frac{N_i+Z_i}{N_i}\right)-a-\beta\varepsilon_i \right\}\delta N_i=0, \]

or, equating to zero the coefficients of \(\delta N_i\),

\[ \lg\left(\frac{N_i+Z_i}{N_i}\right)=a+\beta\varepsilon_i, \]

whence

\[ N_i=\frac{Z_i}{e^{a+\beta\varepsilon_i}-1} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (13) \]

To determine the value of the constant \(\beta\), we substitute (13) into equation (9):

\[ S = k\sum_i \left\{ N_i\lg\left(\frac{N_i+Z_i}{N_i}\right) + Z_i\lg\left(\frac{N_i+Z_i}{Z_i}\right) \right\} = \]

\[ = k\sum_i \left\{ N_i(a+\beta\varepsilon_i) - Z_i\lg\left(1-e^{-a-\beta\varepsilon_i}\right) \right\} = \]

\[ = k \left\{ aN+\beta E-\sum_i Z_i\lg\left(1-e^{-a-\beta\varepsilon_i}\right) \right\} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (14) \]

and make use of the well-known thermodynamic relation defining the concept of absolute temperature \(T\):

\[ \frac{1}{T} = \left(\frac{dS}{dE}\right)_{V=\mathrm{const}.}, \]

which in the present case takes the form

\[ \frac{1}{T}=\frac{\partial S}{\partial E} +\frac{\partial S}{\partial \alpha}\frac{d\alpha}{dE} +\frac{\partial S}{\partial \beta}\frac{d\beta}{dE}. \]

It is easy to show on the basis of equations (10), (11), and (14) that

\[ \frac{\partial S}{\partial \alpha}=\frac{\partial S}{\partial \beta}=0 \]

and that, consequently,

\[ \frac{1}{T}=k\beta \]

or

\[ \beta=\frac{1}{kT}. \tag{15} \]

As for the value of the constant \(\alpha\), it is a function of \(T\), \(N\), and \(V\) (through \(Z_i\), see equation (8)) and is determined by equation (10) after substituting into it equations (13) and (15):

\[ N=\sum_i \frac{Z_i}{e^{\alpha+\frac{\varepsilon_i}{kT}}-1}. \tag{16} \]

Finally, the total energy of the gas, as a function of temperature, volume, and number of molecules, is determined by equation (11) after substituting into it equations (13) and (15):

\[ E=\sum_i \frac{\varepsilon_i Z_i}{e^{\alpha+\frac{\varepsilon_i}{kT}}-1}. \tag{17} \]

The system of equations (14), (17), and (18) derived by us, to which we shall also add the formula determining the pressure of the gas1;

\[ p=T\left(\frac{\partial S}{\partial V}\right)_{E=\mathrm{const}}. \]

Differentiating equation (14) with respect to \(V\), we see that the coefficients of

\[ \frac{\partial \alpha}{\partial V} \quad \text{and} \quad \frac{\partial \beta}{\partial V} \]

identically vanish, and that therefore

\[ \left(\frac{\partial S}{\partial V}\right)_E =-k\sum_i \frac{\partial Z_i}{\partial V}\lg\left(1-e^{-\alpha-\beta\varepsilon_i}\right) =-\frac{k}{V}\sum_i Z_i\lg\left(1-e^{-\alpha-\beta\varepsilon_i}\right). \]

\[ p=\frac{2}{3}\frac{E}{V}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (18) \]

is quite sufficient for studying all the macroscopic properties of a gas in thermodynamic equilibrium. As is easy to see, these equations, in their form, differ substantially from the corresponding equations of the classical theory1. However, in quantitative terms, in most cases they lead to results differing only extremely slightly from the customary laws of ideal gases. In this respect one may draw a certain analogy between the new statistics and the theory of relativity: both theories, in their methodological and fundamental aspects, constitute substantial progress, eliminating a number of internal contradictions and inconsistencies that had accumulated in classical physics, while in quantitative terms the changes introduced by them into the classical laws amount, in most cases, to insignificant corrections.

To verify the validity of this assertion, let us write equation (16) in the following form

\[ N=\sum_i \frac{Z_i e^{-\alpha-\frac{\varepsilon_i}{kT}}}{1-e^{-\alpha-\frac{\varepsilon_i}{kT}}}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (19) \]

\(e^{-\alpha}\) must be less than unity, for otherwise, as follows from equation (13), the number of molecules in the first phase cell with energy \(\varepsilon_i=0\) would be a negative number2. Let us introduce the notation \(\lambda=e^{-\alpha}\) and, together with Einstein, call \(\lambda\) the “measure of degeneration of the gas”; the meaning of this term will become clear later. Since \(\lambda<1\), equation (19) can be expanded in a series in powers of \(\lambda\):

\[ N=\sum_i Z_i e^{-\frac{\varepsilon_i}{kT}}\left(1+\lambda e^{-\frac{\varepsilon_i}{kT}}+\ldots\right)\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (20) \]

Replacing summation over \(i\) by integration over \(\varepsilon\) and carrying out the integration by parts, we find that

\[ \sum_i Z_i \lg\left(1-e^{-\alpha-\beta\varepsilon_i}\right)=-\frac{2}{3}\frac{E}{kT}. \]

Formula (18) follows directly from the equations given above.

Taking into account equation (8), we see that for small \(\lambda\), for which one may restrict oneself to the first term of the expansion, the distribution of molecules with respect to energy (or with respect to velocities) corresponds to the usual exponential Maxwell law.

Restricting ourselves in equation (20) to the first term of the expansion and replacing summation over \(i\) by integration over \(\varepsilon\), we obtain:

\[ \lambda=\frac{N}{V}\left(\frac{h^2}{2\pi mkT}\right)^{3/2}\ldots\ldots\ldots\ldots (21) \]

Expressing the ratio \(N/V\) through the molecular concentration \(\eta\) (\(\eta\) is the number of gram-molecules in \(1\ \mathrm{cm}^3\)) and the mass of the molecule \(m\) through the molecular weight of the gas \(M\) (\(M=N_0m\), where \(N_0\) is Avogadro’s constant), and substituting into equation (21) the numerical values of the universal constants, we obtain:

\[ \lambda=3.16\cdot 10^3\frac{\eta}{(MT)^{3/2}}\ldots\ldots\ldots\ldots (21\text{-}a) \]

At atmospheric pressure and \(T=273^\circ\)

\[ \eta=\frac{1}{22.4\cdot 10^3}; \]

under these conditions, for the lightest gas—hydrogen (\(M=2\))—we obtain \(\lambda=1.1\cdot 10^{-5}\); for heavier gases \(\lambda\) will be still smaller, so that indeed under normal conditions \(\lambda\ll 1\).

In an analogous way, restricting ourselves to the first terms of the expansion in a series, one may obtain a formula representing a modified Clapeyron law,

\[ p=\eta kT(1-0.1768\lambda)\ldots\ldots\ldots\ldots (22), \]

and an expression for the mean energy of a molecule

\[ \bar{\varepsilon}=\frac{3}{2}kT(1-0.1768\lambda)\ldots\ldots\ldots\ldots (23) \]

As is evident from the formulas given\(^{1}\), the corrections introduced by the new theory into the laws of ideal gases are, under normal conditions, in fact vanishingly small; the influence of “degeneracy,” characterized by the parameter \(\lambda\), attains any significant magnitude only for light gases at low temperatures and high pressures. Thus, for example, for helium at the critical point the value of \(\lambda\) reaches the order of one tenth; however, in this critical region helium can no longer be regarded as an ideal gas, and the influence of degeneracy is masked by the more significant influence of the molecular forces of cohesion.

\(^{1}\) In Einstein’s [2] formulas (22) and (23) are given with different numerical coefficients at \(\lambda\), apparently owing to a misprint.

§ 5. Laws of the Quantum or Light Gas.

When considering the formula for the energy of an ideal gas (17), to which the new theory leads, one is struck by its similarity to the well-known formula of Planck, which determines the distribution of energy in the spectrum of “black radiation,” i.e. in the spectrum of radiation in thermodynamic equilibrium. This similarity is not accidental, and the theory of ideal gases set forth in the preceding paragraph, as it turns out, can be transferred directly to the theory of radiation.

Indeed, from the standpoint of the theory of light quanta, a space filled with radiant energy must be regarded as a space filled with “atoms of light,” or light quanta, moving chaotically in all directions with the velocity of light \(c\). Each light “atom,” or quantum, is a portion of energy \(\varepsilon\) localized in an extremely small volume; to each quantum there corresponds a light wave with oscillation frequency \(\nu\), determined by the well-known relation

\[ \varepsilon = h\nu, \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (24) \]

where \(h\) is Planck’s constant. Thus radiation may be regarded as a “quantum” or “light” gas, to which one may try to apply the laws of ideal gases. Let us consider what modifications must be introduced for this purpose into the formulae of the preceding paragraph.

Formula (6) is applicable, evidently, also to a light gas with only one modification: whereas the state of a molecule of an ideal material gas is completely determined by specifying the three coordinates of position and the three components of momentum, light quanta may also differ in their polarization, i.e. in the polarization of the light wave corresponding to them. In accordance with the fact that polarization may be either right circular or left circular1, we shall have to double the number of elementary cells in the phase region representing the state of the light gas, and put

\[ Z_i=\frac{8\pi V p_i^{\,2}\Delta p_i}{h^3}. \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (6a) \]

Formula (7) requires a more substantial modification. First of all, what is to be understood by the mass of a light quantum? According to the theory of relativity, to the energy \(\varepsilon = h\nu\) there corresponds the mass \(m=\dfrac{\varepsilon}{c^2}=\dfrac{h\nu}{c^2}\),

where \(c\) is the speed of light. Since quanta move with the same velocity \(c\) \({}^{1}\), the quantity of their motion, or their momentum \(p\), is equal to \(p=mc=\dfrac{h\nu}{c}\). Thus, in the case under consideration equation (7) will take the form

\[ p=\frac{h\nu}{c};\qquad \Delta p=\frac{h\Delta\nu}{c}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (7a) \]

Substituting into equation (6a), we obtain

\[ Z_i=\frac{8\pi V\nu_i^{\,2}\Delta\nu_i}{c^3}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (8a) \]

Passing to the subsequent formulae, we shall see that equations (9) and (11) remain valid also for a quantum gas, while equation (10) drops out altogether, since the number of quanta in the radiation field in the presence of matter is not constant: during absorption and subsequent emission of quanta by atoms, the number and dimensions of the quanta may change arbitrarily. Thus, in seeking the maximum of the entropy \(S\), condition (10) drops out, and correspondingly in equation (12) and in all subsequent formulae one must put

\[ a=0\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (25) \]

In particular, equation (17), on the basis of (24), (25), and (8a), takes the form

\[ E=\sum_i \frac{8\pi Vh}{c^3}\, \frac{\nu_i^{\,3}\Delta\nu_i}{e^{\frac{h\nu_i}{kT}}-1} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (17a) \]

which coincides completely with Planck’s formula.

The derivation of Planck’s formula set forth above, due to Bose \([1]\), constitutes a major success of quantum theory: before Bose, all derivations of this quantum formula suffered from an internal contradiction, for all of them, to one degree or another, were based on the results of the classical, non-quantum theory of radiation.

\({}^{1}\) This does not contradict the theory of relativity, according to which motion with the limiting velocity \(c\) is impossible only for material bodies possessing a definite mass also in the state of rest. The whole difference between light quanta and electrons reduces to the fact that the latter are portions of energy connected with substantial carriers and, as such, continue to exist also in the state of rest, whereas quanta represent a pure form of motion, not connected with substantial carriers and existing only insofar as motion exists (the mass of a quantum in the state of rest is equal to zero).

Taken together, all that has been set forth reveals a profound analogy, not previously suspected, between the properties of gases and the properties of radiation1.

§ 6. Physical Meaning of the Bose–Einstein Theory

Immediately after the appearance of the first papers of Bose–Einstein, Ehrenfest and other physicists raised a number of objections to the new theory. The first of these amounts to the following: as had earlier been proved by Einstein himself on the basis of fluctuation theory [7], the validity of Planck’s formula for black radiation is inseparably connected with the existence of an interaction of light quanta, analogous to the interference of light waves. Meanwhile, the derivation of Planck’s formula just presented would seem to be based on the assumption that there is no interaction of light quanta. The second, no less essential objection, I shall permit myself to formulate—in application to the exposition given above, which differs somewhat from Bose–Einstein’s exposition—as follows.

The refusal to individualize molecules cannot be justified by saying that, with our crude methods of observation, we cannot distinguish molecules from one another, for it entails extraordinarily serious, physically inadmissible consequences. Indeed, to acknowledge that the arrangement in which molecule \(a\) is in the 1st cell and molecule \(b\) in the 2nd cell is, in principle, indistinguishable from the inverse arrangement (\(b\) in the 1st cell and \(a\) in the 2nd) means, in essence, to abandon the principle of the identifiability of the elements of matter, i.e., to abandon the application to matter of the concept of substance.

What, then, is the way out of these contradictions? It is that, although this was not brought out in the preceding exposition, the Bose–Einstein theory in fact presupposes the existence of a special kind of interaction between the molecules of an ideal gas (and hence also between light quanta); taking this interaction into account, it is apparently possible, by the ordinary classical route and without abandoning the individualization of molecules, to obtain all the results of the Bose–Einstein theory. In other words, the refusal to individualize molecules characterizes this theory only from a purely formal point of view. The following considerations will help convince us of the correctness of these assertions.

Up to now we have considered only the macroscopic distribution of the molecules of an ideal gas over regions of phase space, determined by the numbers \(N_i\). Let us now pass to the consideration

microscopic distribution of molecules among elementary phase cells, determined by the numbers \(n_k\); moreover, without detriment to the generality of the reasoning, we shall restrict ourselves to considering the group of cells forming part of the \(i\)-th region of the phase domain.

The character of the microscopic distribution will become known to us if we determine how many cells of the region under consideration contain zero, one, two, etc., molecules; in other words, if we determine the values of the numbers \(p_i^r\), where \(p_i^r\) is the number of cells of the \(i\)-th phase region containing \(r\) molecules. It is obvious that

\[ \sum_{r=0}^{S} p_i^r = Z_i \quad \text{and} \quad \sum_{r=0}^{S} r p_i^r = N_i \qquad \ldots \ldots \ldots (26)\ \text{and}\ (27) \]

The probability of the given state, determined by the system of numbers \(p_i^r\), is, obviously1,

\[ W_i = \frac{Z_i!}{\prod p_i^r!} \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots (28) \]

Indeed, to specify the numbers \(p_i^r\) means to divide the \(N_i\) molecules into \(Z_i\) groups so that each \(p_i^r\) of these groups contains \(r\) molecules. All possible microscopic distributions corresponding to the given system of numbers \(p_i^r\) are obtained by carrying out all possible \(Z_i!\) permutations of these \(Z_i\) groups of molecules among the \(Z_i\) cells. Taking into account that each \(p_i^r\) of these groups are identical among themselves2, we obtain formula (28).

The most probable system of numbers \(p_i^r\) corresponds to the maximum of the probability \(W_i\), or, what is the same thing, to the maximum of \(\lg W_i\). Suppose that the majority of the numbers \(p_i^r\) are so large that their factorials can, with sufficient accuracy, be represented by Stirling’s formula (4). Then

\[ \lg W_i = Z_i \lg Z_i - \sum_{r} p_i^r \lg p_i^r, \]

and the condition for a maximum will be written in the following form:

\[ \delta \lg W_i = - \sum_{r} \left(\lg p_i^r + 1\right)\delta p_i^r = 0. \]

Since \(Z_i\) and \(N_i\) are given numbers, from equations (26) and (27) we obtain the additional conditions

\[ \delta Z_i=\sum_r \delta p_i^r=0 \]

and

\[ \delta N_i=\sum_r r\,\delta p_i^r=0. \]

Thus, the maximum condition in its final form will take the following form (compare equation (12)):

\[ -\delta \lg W_i+\alpha_i\delta Z_i+\gamma_i\delta N_i =\sum_r\left(\lg p_i^r+1+\alpha_i+\gamma_i r\right)\delta p_i^r=0. \]

Equating to zero the coefficients of \(\delta p_i^r\) and introducing the notation

\[ a_i=e^{-(\alpha_i+1)}, \]

we obtain:

\[ p_i^r=a_i e^{-\gamma_i r}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (29) \]

To determine the values of the constants \(a_i\) and \(\gamma_i\), we substitute this expression into equations (26) and (27) and apply the well-known formula for the sum of an infinite geometric progression:

\[ Z_i=a_i\sum_r e^{-\gamma_i r}=\frac{a_i}{1-e^{-\gamma_i}} \]

\[ N_i=a_i\sum_r r e^{-\gamma_i r}=\frac{a_i e^{-\gamma_i}}{(1-e^{-\gamma_i})^2}, \]

whence

\[ a_i=Z_i(1-e^{-\gamma_i})\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (30) \]

and

\[ N_i=\frac{Z_i e^{-\gamma_i}}{1-e^{-\gamma_i}}=\frac{Z_i}{e^{\gamma_i}-1}. \]

Comparing the last expression with equations (13) and (15), we see that

\[ \gamma_i=a+\frac{\varepsilon_i}{kT}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (31) \]

Equations (29)—(31) give an exhaustive answer to the question of the character of the most probable distribution of molecules among phase cells and deserve the closest attention.

According to classical theory, the most probable distribution of molecules among the individual cells of each phase region is uniform (i.e., all \(p_i^r=0\), except for \(p_i^r\) with index \(r=\dfrac{N_i}{Z_i}\)). Meanwhile,

according to the theory of Bose–Einstein, the largest group of cells is entirely devoid of molecules (the maximum \(p'_r\) occurs at \(r=0\)), whereas the comparatively numerous cells that contain a number of molecules significantly exceeding the mean value of \(r\), equal to \(\dfrac{N_i}{Z_i}\). Everything is as if there existed between the molecules of an ideal gas forces of interaction and attraction, promoting the rapprochement of molecules and their accumulation within individual cells. However, in order to be convinced that the renunciation of the individualization of molecules, by which the Bose–Einstein theory is characterized from the formal side, physically does indeed amount to the introduction of a special kind of intermolecular interaction, it is necessary to turn to the theory of fluctuations.

According to the kinetic theory of matter, gas molecules are never distributed throughout the whole volume of the gas with complete uniformity; in individual, sufficiently small regions of the gas, for random reasons, there sometimes arises an excess of molecules, and in other regions—a deficiency; these are gradually smoothed out, only to arise again in other places, and so on. These random but inevitable deviations from the mean uniform distribution are called fluctuations.

Like these fluctuations of distribution, analogous fluctuations of temperature, chemical composition (in a mixture of gases), and so on also occur.

If the expression for the entropy of a gas (or its thermodynamic probability) is known, one can calculate the mean magnitude of the fluctuations. Let us note that, according to the classical theory of ideal gases, the mean square magnitude of the relative deviation \(\dfrac{\delta N}{N}\) of the number of molecules in a given volume1 from the normal (mean) number of molecules in this volume is equal to [*]

\[ \overline{\left(\frac{\delta N}{N}\right)^2}=\frac{1}{N}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (32) \]

On the other hand, starting from Planck’s formula, it can be shown by means of purely thermodynamic reasoning that the magnitude of the energy fluctuations \(\delta E_i\) in the field of equilibrium (“black”) radiation is determined by the following equation:

\[ \overline{\left(\frac{\delta N_i}{N_i}\right)^2} = \overline{\left(\frac{\delta E_i}{E_i}\right)^2} = \frac{1}{N_i}+\frac{1}{Z_i}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (33). \]

Here \(E_i\) is the mean value of the energy contained in the volume \(V\) under consideration and corresponding to radiation of a definite frequency

\(\nu_i\), \(N_i\) is the mean number of quanta of this frequency \(\left(N_i=\dfrac{E_i}{h\nu_i}\right)\), while \(Z_i\) is numerically equal to the number of the corresponding elementary phase cells, determined by equation (8a). It is characteristic that the fluctuation of the energy of radiation is composed of two constituent parts: the first term of formula (33) corresponds to random changes in the number of quanta in the volume \(V\) and is wholly analogous to the fluctuation of the number of molecules of an ideal gas [equation (32)]. The second term of formula (33), however, is due to the “interference of light quanta,” i.e. to the interference of the light waves corresponding to them. Indeed, on the basis of the wave theory of light, by direct calculation one can show that the interference of waves of frequency \(\nu_i\), the result of which depends on the random phase difference between the interfering waves, entails a fluctuation of the energy of radiation whose mean square value is equal to \(\dfrac{1}{Z_i}\) [7].

It is extremely remarkable that, as follows [2] from the basic formulas of the Bose–Einstein theory, the fluctuation of the distribution of molecules (and of energy) of an ideal gas, from the point of view of this theory, is determined not by the classical formula (32), but by formula (33), originally derived for the fluctuation of radiation. In applying this formula to a gas, we need only modify accordingly the meanings of the quantities entering into it: \(N_i\) will denote the mean number of gas molecules contained in the volume \(V\) under consideration and possessing a definite energy of motion \(\varepsilon_i\); \(Z_i\) is the corresponding number of elementary phase cells, determined by equation (8); and, finally, \(E_i\) is the mean energy in the volume \(V\) corresponding to the molecules \(N_i\) \((E_i=\varepsilon_i N_i)\). Thus the fluctuation in the number of molecules according to the new theory is composed of two constituent parts, exactly as is the fluctuation of the energy of radiation. If the interaction of the molecules were absent, then \(\overline{\left(\dfrac{\delta N_i}{N_i}\right)^2}\) would be equal to \(\dfrac{1}{N_i}\), as is the case according to the classical theory [see equation (32)]; the additional term \(\dfrac{1}{Z_i}\) is wholly analogous to the interference fluctuations of radiation. The only possible interpretation of this additional term consists in the supposition that an interaction really exists between the molecules of an ideal gas, and that this interaction is of such a character as if it were due to the interference of a special kind of waves, inseparably connected with the molecules of the ideal gas.

What, then, is the nature of these mysterious waves? And, in general, is it possible and necessary to introduce into physics an entirely new conception of a special kind of waves inseparably connected with each material particle? Does not the very comparison of the interaction of molecules with the interference of waves have the character of a purely formal analogy?

§ 7. Interference of molecules and de Broglie’s theory.

The notion of the interference of waves inseparably connected with material particles, or, as we shall say for brevity, the notion of the interference of these particles, is entirely foreign to classical physics. In the very latest years, however, a promising theory of de Broglie1 has arisen (for the most complete exposition see [8]), which, proceeding from quite different foundations, leads to the same conceptions of the “wave field” of material particles as does the theory of Bose-Einstein.

De Broglie proceeds from the assumption that with every material particle (i.e. with every electron or proton) there is inseparably connected an oscillatory process, whose frequency \(\nu\) depends on the energy \(\varepsilon\) of the particle and is determined by Bohr’s condition

\[ h\nu=\varepsilon. \]

Introducing the mass of the particle \(m\) and recalling that, according to the theory of relativity,

\[ \varepsilon=mc^2, \]

we obtain

\[ h\nu=mc^2, \ldots \ldots \ldots \ldots \ldots \ldots (34), \]

where the mass of the particle, as is known, depends in the following way on its velocity \(v\):

\[ m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}. \]

De Broglie supposes that, from the point of view of an observer at rest relative to the particle, oscillations of frequency

\[ \nu_0=\frac{m_0c^2}{h} \]

occur inside and near the particle synchronously (analogously to a system of standing waves). If this assumption is accepted, then it can be shown that for an observer relative to whom the particle is moving with velocity \(v\), this oscillatory process will appear in the form of a wave of frequency

\[ \nu=\frac{\nu_0}{\sqrt{1-\frac{v^2}{c^2}}}, \]

propagating in the direction of motion of the particle with phase velocity

\[ V=\frac{c^2}{v} \]

(since \(v < c\), then \(V > c\)). This wave carries no energy with it and therefore cannot be perceived by any physical instrument. The existence of this wave is manifested, however, in the circumstance that waves of different particles having the same frequency \(\nu\) and velocity \(V\) can interfere with one another, with each particle moving along an orthogonal trajectory (i.e., using the optical term, along a ray) of the resulting wave field. Thus the interference of waves entails a deflection of particles from a rectilinear path, i.e. it manifests itself in a special kind of interaction of particles1.

The greatest merit of de Broglie lies in the fact that, on the basis of his theory, he was for the first time able to give a successful and extraordinarily ingenious attempt at a rational interpretation of the basic postulate of the quantum theory of periodic systems (the postulate of stationary states). At the same time Einstein [2] notes, without giving the corresponding calculations, that de Broglie’s theory leads to the same expression for the fluctuation of the number of molecules in a given volume of an ideal gas as he had previously obtained on the basis of considerations of an entirely different kind [equation (33)]. This coincidence of the results of two completely different theories is undoubtedly a weighty confirmation of the correctness of the fundamental propositions both of de Broglie’s theory and of Bose–Einstein statistical mechanics.

Thus we may regard as proved the first of the assertions stated by us at the beginning of the preceding paragraph: that the physical basis of Bose–Einstein statistics is the assumption of an interference interaction of the molecules of an ideal gas. It is highly probable that the second of the assertions stated by us will also prove to be entirely valid, and that, consequently, when the interference interaction of molecules is taken into account, the results of the Bose–Einstein theory will be obtained on the basis of classical statistics, without renouncing the individualization of molecules. This is supported at least by the fact that the above-mentioned derivation of the fluctuation formula (33) from de Broglie’s theory serves as evidence. Unfortunately, no general proof of this assertion has yet been given by anyone; however, a very interesting step in this direction has recently been made by Lande [9], to a brief account of whose work we shall now turn.

As is known, the total energy of a system of interfering waves is not equal to the sum of the energies of each of these waves taken separately. If \(\varepsilon_i\) is the energy that each of the waves would have in the absence of interference, then, in order to determine the total energy \(u_i\) of the system

waves, one must pass from the energy \(\varepsilon_i\) to the amplitudes \(\sqrt{\varepsilon_i}\), add the amplitudes, taking into account the phase of each individual wave, and only then compute the total energy \(u_i\). Lande proceeds from the assumption that an analogous dependence also exists between the total energy \(u_i\) of a group of molecules (or light quanta) contained in one and the same phase cell\(^1\), and the energy \(\varepsilon_i\) which each molecule would have separately in the absence of interference. In other words, Lande introduces the concept of the “amplitude” \(\sqrt{\varepsilon_i}\) of a molecule (or quantum) and of its “phase” \(\varphi\). All molecules located in one and the same cell possess the same amplitude, while their phases are distributed according to the law of chance. The resulting amplitude of the group of molecules is equal to

\[ \sqrt{\varepsilon_i}\left(e^{i\varphi_1}+e^{i\varphi_2}+\ldots\right), \]

while their total energy \(u_i\) is equal to the square of the modulus of the amplitude:

\[ u_i=\varepsilon_i\left|\sum_\alpha e^{i\varphi_\alpha}\right|^2 . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (35) \]

Let us consider a portion of the phase region corresponding to molecules with energy \(\varepsilon_i\).\(^2\) Let the total number of molecules in this portion be \(N_i\), the number of cells \(Z_i\), and let, according to the laws of classical statistics, these molecules in the equilibrium state be uniformly distributed over all \(Z_i\) cells, so that each cell contains \(\dfrac{N_i}{Z_i}\) molecules. The total energy of the molecules contained in each individual cell will be determined by formula (35), where the sum entering into this formula will consist of independent terms of the type

\[ e^{i\varphi}=\cos\varphi+i\sin\varphi . \]

The probability that the absolute value of the sum of \(\dfrac{N_i}{Z_i}\) independent quantities \(e^{i\varphi}\) will lie between \(R\) and \(R+dR\) is equal to \(d\Omega\):\(^3\)

\[ d\Omega=2Ce^{-\mu R^2}R\,dR=Ce^{-\mu R^2}d(R^2), . \ . \ . \ . \ . \ . \ . \ . \ . \ (36) \]

where

\[ R=\left|\sum_\alpha e^{i\varphi_\alpha}\right|, \]

\(^1\) According to Lande, only those molecules interfere with one another which are contained in one and the same cell. Let us also note that the wave field of molecules, in contrast to the wave field of light quanta, has a scalar character (the amplitude of a molecule is a scalar, not a vector), which also agrees with de Broglie’s theory.

\(^2\) \(\varepsilon_i\) is the energy which each molecule located in this portion would possess in the absence of interference.

\(^3\) See Markov, Calculus of Probabilities, § 33.

where \(C\) and \(\mu\) are constant parameters. We shall, however, together with Lande, abandon the idea of a continuous manifold of possible values of the quantities \(R\), and shall introduce an additional condition, characteristic of quantum theory, by assuming that the energy of each cell \(u_i\) can be equal only to an integral number of portions \(\varepsilon_i\). In other words, let us assume [see equations (35)] that \(R^2\) can take only integer values. Moreover, let us assume, as is always done in analogous cases, that the probability \(\Omega(r)\) of the integer value \(R^2\) equal to \(r\) is equal to the probability, calculated from the ordinary “non-quantum” formula (36), of all values of \(R\) lying within the limits from \(r\) to \(r+1\):

\[ \Omega(r)=C\int_{R^2=r}^{R^2=r+1} e^{-\mu R^2}\,d(R^2) =\frac{C}{\mu}\left(e^{-\mu r}-e^{-\mu(r+1)}\right) =\frac{C}{\mu}e^{-\mu r}(1-e^{-\mu}). \]

Taking into account that \(\sum_{r=0}^{r=\infty}\Omega(r)\) must be equal to unity, we find that \(\frac{C}{\mu}=1\) and that, consequently,

\[ \Omega(r)=e^{-\mu r}(1-e^{-\mu}). \]

Such, therefore, is the probability that \(R^2=r\), or, according to equation (35), that \(u_i=r\varepsilon_i\). Denoting by \(p_i^r\) the number of all cells in the phase region under consideration whose energy is equal to \(r\varepsilon_i\), we obtain

\[ p_i^r=Z_i\Omega(r)=Z_i e^{-\mu r}(1-e^{-\mu}). \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (37) \]

This equality, in its form, is completely analogous to equalities (29) and (30), and coincides with them completely if we put

\[ \mu=\gamma_i=a+\frac{\varepsilon_i}{kT}. \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (38) \]

Equations (29) and (30), according to Bose–Einstein theory, determine the number \(p_i^r\) of cells containing \(r\) molecules; identifying them with equation (37), we thereby assert that \(r\) is only the apparent number of molecules in these cells, and that in fact in each of the \(p_i^r\) cells, as in the other cells of the phase region under consideration, there are \(\frac{N_i}{Z_i}\) molecules, but that the total energy of these molecules in each of the \(p_i^r\) cells, owing to interference, is equal not to \(\frac{\varepsilon_i N_i}{Z_i}\), but to \(\varepsilon_i r\). Thus, according to Lande, \(r\) is not the number of molecules in a cell, but a measure of the total energy of these molecules.

We shall not dwell on Lande’s not entirely successful attempts to justify equation (38); let us note only that, although the cited

...arguments are far from sufficient for constructing a rational statistical mechanics of molecules and quanta that interfere with one another; nevertheless, they undoubtedly indicate the correct path toward the solution of this problem. On the basis of the similarity of equation (37) to equation (29), we may with increased confidence repeat the assertion that the entire difference between the positions and conclusions of the new statistical mechanics and classical theory lies wholly in taking into account the interference of molecules. To this new and, at first sight, paradoxical conception of the interference of molecules, quite independently of de Broglie’s theory, we have been led by the necessity of eliminating a whole series of internal contradictions that had accumulated in the classical theory.

§ 8. Some Consequences of the Interference of Molecules.

Thus, with the motion of material particles there are inseparably connected waves of a special kind, just as light waves are connected with the motion of a light quantum: between matter and light there exists a far deeper analogy than could have been supposed until recently. What, then, are the conditions under which one can directly observe the interference or diffraction of “molecular” waves?

Starting from equation (34), one can show that the length of the “molecular” waves \(\lambda\) decreases with increasing molecular velocity and, under normal temperature conditions, is extremely small—smaller than the diameter of a molecule \(\sigma\). Since the diffraction of waves becomes noticeable only at openings or screens whose dimensions are comparable with \(\lambda\), it is clear that with the aid of artificial screens the diffraction of “molecular” waves cannot be observed. However, at low temperatures (about \(56^\circ\mathrm{K}\) for \(\mathrm{H}_2\), and \(40^\circ\mathrm{K}\) for \(\mathrm{He}\)) \(\lambda\) becomes comparable with \(\sigma\), and therefore diffraction of “molecular” waves should be observed on neighboring gas particles, which play the role of small screens. In other words, if a group of molecules moving in straight lines encounters on its path another molecule, which for simplicity we shall regard as stationary, then the waves of the moving molecules will undergo a deflection to the side, similar to the deflection of light waves when they are diffracted by a small screen of diameter \(\sigma\). Since molecules in their motion follow the direction of the “molecular” waves, it follows that the moving molecules, in addition to deflections under the influence of collisions with the stationary molecule, will also experience deflections under the influence of the diffraction of the “molecular” waves. Thus the diffraction of these waves must cause a decrease in the mean free path of the molecules and, consequently, a decrease in the viscosity of the gas (which, as is known, is proportional to the mean free path). This conclusion

is in complete agreement with the experimental fact that the viscosity of hydrogen drops sharply at low temperatures, at which the length of the “molecular” waves becomes comparable with the diameter of the molecules [²].

From the standpoint of the theory of “molecular” waves, the physical meaning of the well-known “Gibbs paradox”1 is also easily explained. As is known, the mutual diffusion of two chemically different gases entails an increase in their entropy, and the magnitude of the increase in entropy depends only on the number of gram-molecules of each of the diffusing gases, and does not at all depend on how much these gases differ from one another in their chemical nature. If, however, one passes to the limit and assumes that both diffusing gases are completely identical with each other, then the increase of entropy upon diffusion, as is known, reduces to zero, for in this case diffusion produces no change whatever in the state of the gas. Thus we arrive at the paradoxical conclusion that, in comparing the chemical nature of two gases, or in general of any two substances, there are and can be no continuous relations, so that one may speak either of their complete identity or of non-identity.

This circumstance, which is wholly incompatible with customary physical views, becomes, however, entirely intelligible from the standpoint of the theory set forth by us. Indeed, interference of two systems of waves can take place only under the condition of complete (or almost complete) identity of the lengths of these waves and of the velocities of their propagation. “Molecular” waves satisfy this condition only if they belong to molecules of identical mass and equal velocity. Consequently, interference interaction occurs only between identical molecules, and disappears completely even when there is an exceedingly small difference in the nature of the interacting molecules. In this fact lies the physical cause of the Gibbs paradox.

In conclusion we shall also mention an interesting paper by Jordan [¹⁰], who showed that the new statistical theory entails the necessity of modifying our ideas concerning the laws of molecular collisions, the laws of radiation, absorption, and scattering of light by atoms, and so forth. We shall confine ourselves to a consideration of the first of the phenomena mentioned, and shall consider the totality of such collisions in which two molecules collide, possessing velocities \(v_1\) and \(v_2\), and after the collision their velocities acquire the values \(u_1\) and \(u_2\). Jordan shows that, contrary to the generally accepted view up to the present, the number of collisions of the kind under consideration must depend chiefly—

…of the views set forth depends not only on the number of molecules in the initial state (velocities \(v_1\) and \(v_2\)), but also on how many molecules are at the given moment in the final state (with velocities \(u_1\) and \(u_2\)). Pauli [11] already several years ago, on the basis of entirely different considerations, came to an analogous conclusion concerning the number of collisions of light quanta with free electrons1.

References

1. S. Bose. Zs. f. Phys. 26, 178, 1924, 27, 384, 1924.
2. A. Einstein. Berl. Ber. 1924, p. 261; 1925, pp. 3, 13.
3. P. u. T. Ehrenfest. Encykl. d. math. Wissensch. IV, 2, II, Heft [[unclear: issue number]].
4. Schrödinger. Berl. Ber. 1925, p. 434.
5. P. Ehrenfest u. V. Trkal. Ann. d. Phys. 65, 609, 1921.
6. M. Planck. Ann. d. Phys. 66, 365, 1921, Berl. Ber. 1925, p. 49.
7. H. Lorentz. Theories statistiques en thermodynamique. 1916.
8. L. De-Broglie. Annales de Phys. 3, 22, 1925.
9. A. Landé. Zs. f. Phys., 33, 571, 1925.
10. P. Jordan. Zs. f. Phys. 33, 563, 1925.
11. W. Pauli. Zs. f. Phys. 18, 272, 1923.
12. Westphal. Zs. f. Phys., 33, 557, 1925.
13. A. Smekal. Zs. f. Phys. 33, 613, 1925.

1. In a work being prepared for publication, the author of the present article proposes to set forth a general method by means of which the conclusions mentioned can be justified more rigorously than was done by Pauli and Jordan. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩

2. Groups with an equal number of molecules can be regarded as identical among themselves only insofar as we do not individualize the molecules. Therefore formula (28) is not applicable in the classical theory. ↩↩↩↩↩

3. This postulate in fact applies only to condensed, i.e. non-gaseous, bodies, since gaseous bodies cannot exist at absolute zero. From the validity of this postulate there follows, however, the requirement that the expression for the entropy of an ideal gas should also reduce to zero at a temperature equal to zero, ↩↩