Electrical Fluctuations and Thermal Radiation*)

S. M. Rytov

If I venture to speak at a meeting dedicated to the memory of L. I. Mandelstam with a report on my work on the theory of electrical fluctuations and thermal radiation, I do so because this report gives me an opportunity to recall some of L. I.’s scientific interests and certain features of his creative work.

Both fluctuation phenomena and electrodynamics always occupied a large place among L. I.’s scientific interests. In particular, I should like to draw attention to one question that is directly connected with the role of fluctuations in electrodynamics and to which L. I. returned more than once for various reasons. This is the transition from microphenomena to an averaged macroscopic picture and, correspondingly, from the Lorentz equations for the microfield to the equations of phenomenological electrodynamics.

This question has a didactic aspect, which undoubtedly attracted L. I.’s attention. But in a number of his works L. I. showed that, in the relation between micro- and macropictures, there are new circumstances that are essentially important in many cases. Fluctuations drop out of mean linear quantities, but they enter into mean quadratic quantities. The resulting difference between the mean energy of the microfield and the macroscopic energy—the energy of the mean field—is of interest not only from a pedagogical point of view, but also for the correct theory of certain electromagnetic phenomena.

Not having the opportunity to set forth those works of L. I. in which this question essentially occupies the central place, I shall confine myself merely to listing them.

*) Report delivered on December 15, 1954, at a joint meeting of the Division of Physico-Mathematical Sciences and the Division of Technical Sciences of the Academy of Sciences of the USSR, dedicated to the 10th anniversary of the death of Academician Leonid Isaakovich Mandelstam.

This is, first, L. I.’s work dating back to 1907–1908 on the scattering of light, and his polemic with Planck that arose from it.^1 For the intensity of scattered light, i.e., for the mean square value, it was necessary to take into account the deviations of the true field from the mean more carefully than Planck had done; and then, as L. I. showed, in a homogeneous medium no damping of a light wave due to its scattering is obtained.

This is, second, the question of the physically possible values of the dielectric constant, the correct solution of which is likewise connected with taking into account the relation between micro- and macroquantities. Unfortunately, precisely this aspect, which especially occupied L. I., was not sufficiently brought out in one of the works carried out at his suggestion.

Finally, one must also point to L. I.’s 1940 work on the refractive index of media with free and bound electrons, in which the question of the proper averaging of electromagnetic microfields is likewise central.^2

I have dwelt specially on the connection between micro- and macropictures in electrodynamics because this question plays a large role in the work set forth below.

Turning to certain features of L. I.’s scientific creativity, one should recall his characteristic striving to seek connections and common foundations in phenomena which, outwardly, would seem heterogeneous and alien to one another. This approach, this cast of thought, runs through all his theoretical works, touching the most diverse areas of physics. At the same time, it was not characteristic of L. I., and seemed to him purely conventional, to divide science into theoretical and applied. In order to solve a serious technical problem he was able to find means from the entire arsenal of theoretical physics. This applies, in particular, to the radio engineering of ultrahigh frequencies, in which L. I. became interested in the last years of his life. As is known, one of the earliest and exceedingly elegant solutions of problems concerning the excitation of waves in tubes belongs to him.^3

Everything mentioned above undoubtedly influenced both the formulation of the question of a theory embracing, from a unified point of view, a broad class of electrical fluctuation phenomena, and the treatment of this question in the work that forms the subject of my report. To what has been said it should be added that the foundations of the method developed in this work were outlined by M. A. Leontovich and me in connection with certain experimental investigations of electrical fluctuations begun at the P. N. Lebedev Physical Institute of the Academy of Sciences of the USSR on the initiative of L. I. Mandelstam.

But, in turning to the exposition of the substance of the matter, I shall allow myself not to follow the path which in fact led to the formu-

new problems and to methods for solving it, and I shall try to outline both as they appear now, as a result of work already carried out.

Physicists and engineers, in particular radio engineers, are well aware of the great importance that fluctuation phenomena have acquired at the present time. This has happened, of course, not because fluctuations have now become stronger than before, but because measuring technology has advanced far ahead and in many cases has approached the fluctuation limit of sensitivity or accuracy of measuring instruments. For example, fluctuations in quartz radio generators are now already of practical importance in the question of the ultimately attainable stability of the frequency of oscillations, i.e., in the question of using such generators as standards of frequency or time.

However, in what follows we shall not be speaking of all electrical fluctuations, but only of those whose cause is the thermal motion of electrons, ions, and, in general, microcharges in matter. In radio engineering and radiophysics, random deviations of voltages and currents from their mean values have been given the name “noise.” Thus we shall be concerned only with thermal noise, caused by the thermal motion of microcharges. It is precisely this noise, whose power depends on temperature, that determines the lowest level of the total fluctuation noise possible in a given circuit at a given temperature. It is therefore of interest not only when it itself is the object of observation (as is the case, for example, in radio astronomy), but also as that fundamental limit approached by radio engineering in its efforts to reduce the noise factor.

The idea that the thermal motion of microcharges in bodies should cause random variations of the electrical quantities characterizing macroscopic systems is not new. It arose some forty-five years ago, when the theory of Brownian motion was being intensively developed. But only in 1927 did the means of radio engineering make it possible experimentally to detect and measure this electrical thermal noise—random, spontaneous pulsations of voltage on a wire resistance.^4

The well-known theorem of classical statistics—the theorem on the equipartition of energy among degrees of freedom—made it possible at once to indicate what the mean energy of the thermal fluctuations of the voltage on a capacitor or of the current in an inductance coil is. This energy is equal to \(kT/2\), where \(k\) is Boltzmann’s constant and \(T\) is the absolute temperature of the device. But for radio-engineering apparatus, which always possesses spectral selectivity and transmits only some band of the harmonic spectrum, it is important not only, and even not so much, to know the total energy of the noise as its distribution over the spectrum. This question was solved at the same time, in 1927, simultaneously

with the experimental discovery of electrical thermal noise, observed initially in the range of very low (acoustic) frequencies.

Nyquist developed the spectral theory of thermal noise^5. He based it on the notion of a random electromotive force acting in an electrical circuit and causing fluctuations of currents and voltages in it. This notion had been introduced by de Haas–Lorentz as early as 1913^6, by analogy with the Langevin random force that causes the Brownian motion of a suspended particle^7. Relying on thermodynamics and statistics, Nyquist gave a formula for the spectral intensity of the random e.m.f. This intensity is proportional to the absolute temperature and to the value of the active resistance. Thus, the fluctuational e.m.f. acts only in active resistances; its sources should be imagined as being included in those branches of the circuit that possess losses.

The spectral distribution of the random e.m.f., if the active resistance does not depend on frequency, is uniform. Translated into the language of time, this means that the fluctuational e.m.f. possesses the so-called $\delta$-correlation in time: the random values assumed by it at arbitrarily close, but noncoincident, moments of time are not correlated with one another. The situation is analogous with respect to spatial correlation: between random e.m.f.’s acting in different resistances, generally in nonoverlapping segments of a circuit, there is no correlation.

The notion of a random e.m.f. and Nyquist’s formula completely solve the problem of thermal fluctuations of currents and voltages in any linear electrical circuit. It is enough to write Kirchhoff’s equations for this circuit for alternating current, taking into account in them all the random e.m.f.’s acting in each of the branches containing active resistance; then, by solving the equations, we obtain expressions for the currents and voltages in terms of the random e.m.f.’s. Since the correlation properties and intensities of these e.m.f.’s are known, it is then no longer difficult to compute the statistical characteristics of the noise—the mean squares of the fluctuations of currents and voltages, their correlation functions, and so on.

Thus we have here a fairly general spectral theory of thermal noise, a theory based, on the one hand, on thermodynamics and statistics, and, on the other, on the electrodynamics of quasi-stationary currents. In this latter point lies its principal limitation. The theory is tied to electrical circuits that can be characterized by lumped parameters—capacitances, inductances, resistances—that is, it applies to quasi-stationary circuits, to sys-

topics whose dimensions are small in comparison with the wavelength.

Meanwhile, radio engineering has systematically and successfully advanced over recent decades toward ever shorter waves. Centimeter waves today already belong to a fully mastered domain. It is well known that in this range of ultrahigh frequencies the entire appearance of radio equipment has been transformed. Coils, capacitors, and wires have been replaced by reservoirs and channels of an entirely different kind for electromagnetic energy—cavities and tubes with metallic walls, the so-called cavity resonators and waveguides. Of course, these changes, the causes of which there is no need to dwell on here, do not in the least remove the question of thermal noise. In the materials from which the new structures are made there are microcharges; they undergo thermal motion and, consequently, the primary source of electrical fluctuations has not disappeared. But the conditions for detecting thermal noise in this microwave radio engineering have changed to the same extent as they have changed for the regular signals for which it was created. Initially—before conversion to lower frequencies—we are dealing here not with a voltage on capacitor plates and not with a current in a wire, but with a chaotic electromagnetic field. It is created inside waveguides and cavity resonators by electrical thermal fluctuations in the material of their walls, or else enters from outside in the form of thermal electromagnetic radiation from external sources, for example, as in radio astronomy, from celestial bodies.

But regardless of the “internal” or “external” origin of thermal noise, what is involved is a chaotic electromagnetic field (in particular, a radiation field), and moreover in such a frequency range that the basic premise of the quasistationary theory is violated—the dimensions of the devices here are of the same order as the wavelength. Of course, it is not excluded a priori that the concepts and methods of the quasistationary theory may, in some sense and to some extent, be preserved, but in any case this requires clarification and justification.

What else did physics have at its disposal in order to try to answer the new demands of radio engineering?

Since thermal radiation is of interest, it is natural to turn to the classical theory of thermal radiation, whose foundations were laid in the middle of the last century by Kirchhoff and which was crowned in 1900 by the hypothesis of light quanta and by Planck’s formula for the distribution of energy in the spectrum of a black body. The object of this theory is precisely the chaotic field of electromagnetic waves emitted by bodies owing to the thermal motion of microcharges.

However, the classical theory of radiation was constructed as an optical theory. At the center of its attention was the range of infrared and shorter waves, i.e. waves which, as a rule, are very small in comparison both with the dimensions of the radiating bodies and with all sorts of bodies or inhomogeneities of the medium encountered during propagation. Precisely for this reason the classical theory of thermal radiation, applying thermodynamics and statistics to the electromagnetic field, could confine itself, and did confine itself, to that asymptotic, limiting form of the theory of the wave field which is called geometrical, or ray, optics. It operates with the concepts of light rays, of the flow of energy through bounded areas, and so on; all its laws apply to the case when the wavelengths are very small in comparison with the dimensions of bodies. This premise is good for optics. In apparatus for ultrahigh radio frequencies it is likewise not fulfilled.

Thus, until quite recently, for electrical fluctuations of thermal origin there existed a theory in two extreme cases. There was a theory for wavelengths large in comparison with the dimensions of bodies, and for waves small in comparison with the dimensions of bodies. These two limiting cases are so different in their place on the frequency scale that the corresponding theories turned out to be extremely disconnected, unlike one another in their concepts, in their approach, and in their method, to such an extent as if they referred to things completely unrelated. Indeed, what commonality can be discerned between Kirchhoff’s equation for alternating current with a random e.m.f. on the right-hand side and Kirchhoff’s law for the radiation of a heated body? Is it only the author’s surname?

Of course, the understanding that thermal radiation—light, infrared waves, and so forth—is the result of the chaotic motion of microcharges in bodies did not arise today. As early as 1908 Lorentz derived the Rayleigh–Jeans law—one of the fundamental laws of the classical theory of radiation—by considering the thermal motion of electrons in a metal.

Thus, in the period when the quasi-stationary theory of thermal noises was being created, physicists understood very well that the glow of an incandescent wire and the electrical noise at the output of an amplifier which it produces when connected to its input are related phenomena, phenomena with a common cause. Nevertheless, there existed no general theory of electrical fluctuations that would encompass both of these phenomena in a unified way, and in which the relation between wavelength and the dimensions of bodies would not be restricted. This was unsatisfactory both from the general physical point of view and from the standpoint of the practical tasks of ultrahigh-frequency radio engineering, where wavelengths comparable with the dimensions of bodies occur everywhere.

In recent years it has proved possible to fill this gap and to construct a sufficiently general theory of electrical fluctuations1. I shall now try briefly to outline its content, as well as some of its results and possibilities.

Of course, this is a statistical theory, but in its electromagnetic part it rests not on the theory of quasistationary circuits and not on geometrical optics, but on general electrodynamics. Thanks to this, the restrictions concerning the relation of the wavelength to the dimensions of bodies are removed. In the limiting case of short wavelengths it yields the laws of the classical theory of thermal radiation, and in the limiting case of long wavelengths—the formulas of the quasistationary theory of thermal noise.

In these limiting cases, as is well known, a macroscopic description of the electrical properties of the medium is sufficient. Likewise, in the general case it proves possible not to resort to overly detailed conceptions of the electrical microstructure of bodies, but to describe their electromagnetic properties with the aid of the parameters of phenomenological electrodynamics—complex permittivities \(\varepsilon\) and \(\mu\). Of course, such a description is legitimate only under the condition that the wavelength in the medium is much greater than the microinhomogeneities of the medium, say, the distances between molecules or atoms; but this condition is fulfilled even for visible light and ultraviolet, i.e. it is fulfilled even in the quantum region. Thus, the general theory of electrical fluctuations is based on the application of statistics to general electrodynamics, and in the description of the electromagnetic properties of bodies this is phenomenological electrodynamics.

However, if one were to go over completely to these positions and immediately begin with Maxwell’s equations, in which everything is already averaged and only mean, macroscopic fields appear, then there could be no question of any account of fluctuations. Hence it is clear that the starting point must be the electrodynamic equations for microfields—the Lorentz equations.

How is statistics to be introduced into these equations? The quasistationary theory suggests how this can be done. The electromotive force is an integral quantity; it is a linear integral of the intensity of the so-called extraneous field. The latter is introduced into electrodynamics as a certain equivalent field that makes it possible, in electrical language, to describe all possible forces acting on the charges of the system under consideration but caused by sources not belonging to this system. These include, too, various kinds of forces operating at the expense of alien sources of energy—mechanical, chemical, and so on. It is natural to extend this general understanding of e.m.f. also to a random thermal e.m.f. We then arrive at the notion of a random extraneous field, which is distributed over

to the entire volume of the bodies, includes all chaotic forces acting on microcharges and associated with their thermal motion, and which causes in the bodies electrical fluctuations—random pulsations of the densities of charge and current. It was precisely this path that was chosen by M. A. Leontovich and me when we tried to clarify the question of the influence of electrical noise on the skin effect2, i.e., a question which, in essence, still does not go beyond the limits of the quasistationary theory. The generalization to arbitrary frequencies is carried out as follows.

By its very meaning, the random extraneous field must simply be added to the electric field of the system under consideration when we are interested in the forces acting on microcharges. Thus the intensity of the extraneous field enters Ohm’s law for the microcurrent density in the sum with the intensity of the electric field. Through this differential Ohm’s law the intensity of the extraneous field is then introduced into the Lorentz equations, so that as a result we obtain inhomogeneous equations of the microfield. Equations containing in their right-hand sides the fluctuational extraneous field.

The solution of these equations, obtained in any concrete problem according to the general rules for solving boundary-value problems, gives the intensities of the electric and magnetic fields in the form of linear expressions in the intensity of the random extraneous field. If there are no regular sources, i.e. the mean value of the extraneous field is zero, then the electromagnetic field is also zero on the average, i.e. it is purely fluctuational.

Perhaps a small remark of a general nature will clarify the mathematical side. The transition from the quasistationary to the general theory means a transition from ordinary differential equations (such as Kirchhoff’s equations for alternating current) to partial differential equations (the Lorentz equations), and correspondingly from a concentrated random force to one distributed in space. With respect to statistics, this is the transition from the correlation theory of random functions to the correlation theory of stationary random fields. The correlation theory proves sufficient, since only energy quantities are of interest, i.e. moments of second order (mean values of expressions bilinear or quadratic with respect to field intensities). Moments of first order (the mean field intensities themselves) are zero in the absence of regular sources.

From what has been said it is clear that the central question for the entire theory is the question of the correlation function of the random extraneous field. Since the theory is constructed from the very beginning spectrally for the harmonic components in time, what is involved is a spatial correlation function, with the aid of

which can be used to calculate the spectral intensities of all energetic quantities.

I shall not go into the arguments that establish the form of the correlation function. These are general considerations connected with spatial symmetry and therefore dependent on whether the medium is isotropic or anisotropic. But the most important point is the assumption that already at distances of the order of interatomic distances the correlation of the random extraneous forces is lost. Thus, the radius of their correlation is a microquantity, and for most questions of the macroscopic theory it may simply be taken equal to zero, i.e. one may assume a spatial \(\delta\)-correlation of the fluctuating extraneous field. This assumption is sufficient, for example, for all questions concerning the thermal radiation of bodies into external space, which is usually of the greatest practical interest.

The general considerations indicated determine the correlation function of the random extraneous field only up to a coefficient. It can be found in various ways, but their essence is one and the same—an appeal to a limiting case. One must solve, by means of fluctuational electrodynamics, some concrete problem that also admits a solution by classical methods. A direct comparison of the two results then gives the value of the desired coefficient. Of course, one must use such a concrete problem in which the limitations of the classical theory would be bypassed. This can be done, and the coefficient is determined in a completely general way.

Summarizing the content of the theory, one may say the following. The determination of the mean energetic quantities characterizing the thermal fluctuations of the electromagnetic field falls into two stages. The first is the solution of a purely electrodynamic boundary-value problem with inhomogeneous field equations. The second is the compilation of the energetic quantities of interest to us (energy density, energy flux) and their statistical averaging with the aid of the correlation function of the random extraneous field.

To this it should be added that, knowing the intensities of the electric and magnetic fields, we, of course, also know other quantities: charge densities, currents, integral quantities such as the total current strength—that is, we can answer any question to which electrodynamics is in general capable of giving an answer.

Let us now dwell on some results of the application of the theory.

Of course, the applications of greatest interest are those that lie beyond the possibilities both of the classical theory of thermal radiation and of the quasi-stationary theory of noise, i.e. problems concerning bodies comparable in size with the wavelength. Two such problems have already been solved: the thermal radiation of an infinite

a circular cylinder and radiation of a sphere, whose radii may be in any ratio to the wavelength of interest to us (classical theory requires that these radii be much greater than the wavelength)8.

The solutions of these problems contain some interesting points (for example, the radiation from a unit surface of a sphere comparable with the wavelength turns out to be 1.5–2 times greater than from a unit surface of a large sphere), but in this brief exposition there is no possibility of going into details.

In the named problems the conditions are simplified by the symmetry of the bodies: the energy flux of fluctuation radiation is distributed uniformly in all radial directions. Still more interesting are those cases when the body is not so symmetric, but represents, for example, a plate or an aperture in a screen, a wire or slot antenna. In such cases the radiation will no longer be uniform, but will possess a directional pattern depending on the shape and dimensions of the radiator and on the spectral interval under consideration, i.e., on the wavelength. Recently, together with M. L. Levin, one problem of this kind was also solved, namely, the thermal radiation of a thin rectilinear antenna of finite length was considered10.

In this problem there is also a simplifying circumstance, affecting both its electrodynamic part and its statistical part. This is the assumption of the thinness of the wire. First, it makes it possible to pass from the general field equations to the equations of the theory of thin wire antennas, developed several years ago by M. A. Leontovich and M. L. Levin11. Second, the thinness of the wire makes it possible to introduce, instead of an extraneous field distributed over the volume of the wire, a linear Nyquist electromotive force distributed only along the length of the antenna. Without going into concrete results here either, it should only be noted that they exhaustively outline the whole picture of the thermal radiation of an antenna, the directionality of this radiation, the distribution of the noise current along the antenna, and so on.

Another group of questions in which the general theory of electrical fluctuations finds a natural and practically interesting application is thermal radiation not into free space, but in cavities and channels with metallic walls—cavity resonators, waveguides, and coaxial lines, i.e., thermal radiation in ultrahigh-frequency radio equipment. Here, too, a number of concrete problems have been solved, for example, on thermal radiation of a partition in a waveguide, on radiation of the walls of the waveguide itself, etc.8.

The general result here is a peculiar form of Kirchhoff’s law, which I have proposed calling the waveguide law and which relates the power of thermal radiation passing-

... passing through the cross-section of the channel, with the absorption coefficients of the radiating body for the various types of waves possible in the given channel.^8

It should be noted that, as long as the radiators are uniformly heated bodies, the waveguide form of Kirchhoff’s law can be obtained, apart from the general theory of electric fluctuations, on the basis solely of energy considerations and the theorem on the distribution of energy over degrees of freedom. It is precisely for this reason that one special case of the waveguide Kirchhoff law was derived earlier by Nyquist,^5 and radio engineers had already been using it. But as soon as the question is posed of nonuniform heating (which is, of course, more interesting from the practical point of view), the solution of the problem requires the use precisely of the general theory. Its method is applicable also under these conditions, since the form of the correlation function of the external field can remain unchanged. If the inhomogeneities of the material and of the temperature have macroscopic scales, i.e., if the dimensions of these inhomogeneities are large in comparison with interatomic distances, then one may still use a $\delta$-correlation, and only the coefficient, depending on the temperature and the dielectric constants, will now be a function of position. The result is a waveguide form of Kirchhoff’s law, generalized in accordance with the indicated formulation of the problem.^12

As has already been indicated, the first stage in the solution of any question on thermal radiation is the solution of the electrodynamic boundary-value problem. Thus the answer already includes all the interference and diffraction phenomena taking place under the given geometrical conditions. Suppose now that, for the body under consideration, a ready solution of the diffraction problem is already available; for example, the diffraction field is known which is obtained if the given body is irradiated by a wave emanating from a dipole situated somewhere in the external space. Then, as M. L. Levin has recently shown, one can obtain, with the aid of fluctuation electrodynamics and the reciprocity theorem, very simple formulas in which the intensity of the thermal radiation of a body at some external point is expressed through the auxiliary field produced by a dipole placed at that point. The determination of the energy of thermal radiation and of other mean quadratic quantities is thereby reduced to quadratures; moreover, for well-conducting bodies the calculation is simplified still further, and in the limiting cases of short and long waves becomes completely elementary.

In this way M. L. Levin obtained formulas for the short-wave thermal radiation of well-conducting flat plates and of bodies of revolution, as well as formulas for the radiation of spheroids in the opposite case of long waves. He further found the fluctuation field near a well-conducting plane;

In the approximation of geometrical optics they calculated the fluctuation field in the focus of a parabolic mirror and at the center of a spherical one. By the same method he clarified the question of thermal noises induced in an arbitrary wire antenna by external radiators, in particular by the surface of the earth, above which the antenna is located.

This list covers only what was easiest to calculate. The multitude of solutions of diffraction problems already available in the literature is, so to speak, waiting in line for the fluctuation electromagnetic field for the corresponding bodies to be extracted from them. One may name, for example, the radiation of a disk and of an elliptic cylinder, the fluctuation field in the near zone for a circular cylinder and for a sphere, and so on.

The distinction between the fluctuation field of the near zone and fluctuation radiation is, of course, not accidental. The classical theory of thermal radiation, relying on geometrical optics, deals only with the wave field. If, as classical theory assumes, the wavelength is vanishingly small, then at any finite distance from the source we are already in its wave zone. General electrodynamics inevitably broadens the notion of an electromagnetic field of thermal origin. Random elementary sources, which are distributed in the volume of a body and are described with the aid of an extraneous field, create not only electromagnetic waves, but also a quasistationary field, which decreases more rapidly with distance. Inside the body, i.e. in the absorbing medium, the wave and quasistationary fields are mixed and, strictly speaking, inseparable. But outside the body, in the surrounding nonabsorbing medium or in vacuum, they behave sharply differently. The elementary wave fields, when superposed, give the radiation of the given body, i.e. the total chaotic wave field. The elementary quasistationary fields, however, do not create an energy flux, but form a total quasistationary thermal field that rapidly decreases with distance from the surface of the body. This field, as it were lining the surfaces of bodies, is just as alien to the classical theory of thermal radiation as it is entirely natural and comprehensible from the point of view of fluctuation electrodynamics.

The energy density of the quasistationary thermal field increases rapidly as one approaches the surface of a body and, beginning at distances of the order of a wavelength, predominates over the energy density of the radiation. It follows from this that cavities whose dimensions are smaller than the wavelength are filled mainly precisely with the quasistationary thermal field. It further follows from this that any device possessing frequency-selective absorption, when brought to the surface of a heated body closer than one wavelength, must immediately register a strong increase in fluctuation

stresses. But this is not the only manifestation of the quasistationary thermal field. It has a substantial effect in an entirely different phenomenon, which was recently investigated on the basis of the theory set forth here by E. M. Lifshitz3.

He considered the ponderomotive forces of interaction of bodies caused by the fluctuational electromagnetic field in the space between the surfaces of these bodies. Let, for example, two half-spaces, filled for simplicity with one and the same substance at the same temperature, be separated by a plane gap. Fluctuational electrodynamics makes it possible to find the thermal electromagnetic field in this gap and then to calculate the mean value of the corresponding component of the Maxwell stress tensor, i.e., the force of attraction per unit surface area of the bodies. This was done by E. M. Lifshitz3.

He thus obtained an expression for the force of “molecular cohesion” between macroscopic bodies, applicable for arbitrary densities of the bodies and arbitrary temperatures. This expression automatically takes into account the predominance of the quasistationary thermal field in the case of narrow gaps and, conversely, the predominance of the wave field, i.e., retardation effects, for a large width of the gap. At low temperatures \((\hbar \omega \gg kT)\), the only scale for the width of the gap is provided by those wavelengths which are most strongly absorbed by the substance under consideration. The force of cohesion turns out to be inversely proportional to the cube of the distance between the surfaces if the latter is small in comparison with these characteristic lengths of the absorbed waves, and to the fourth power of the distance if it is large.

The theory of macroscopic cohesion forces had previously been constructed starting from an elementary law of van der Waals forces between atoms or molecules, which restricted the result in advance to the case of rarefied media. A purely phenomenological theory based on fluctuational electrodynamics removes this restriction. On the contrary, starting from the expression for the macroscopic force of cohesion, one can, in the case of rarefied media, draw the inverse conclusion—about the law of pair interaction of individual neutral atoms and molecules. Such a path, paradoxical as it may seem at first glance, proves to be simpler than a direct quantum-mechanical calculation for two neutral particles, in which the interaction law is obtained only in high orders in calculations by the perturbation method.

Let us return to thermal radiation. Interference phenomena in this radiation were mentioned earlier. Such phenomena are encountered in practice in apparatus for ultra-high radio frequencies, when the thermal radiation entering a waveguide is partially reflected at the other end because of an incompletely matched load. Standing waves are produced in the waveguide

waves, whose contrast depends on the passband of the receiver, or, speaking in optical language, on the degree of monochromaticity of the observed radiation. Similar phenomena occur whenever the dimensions of the space provided to the thermal field are not sufficiently large, just as optical interference arises in thin plates. In optics, for a given monochromaticity of light, interference is blurred and disappears as the thickness of the plate increases. In exactly the same way, in the microwave range, for a given passband of the apparatus, interference phenomena disappear when the space filled with thermal radiation (in the example given—the length of the waveguide) becomes too large. The energy density then becomes uniform, and the asymptotic laws of the classical theory of radiation come into force.

Thus, for a sufficiently selective (narrow-band) receiver the field of thermal radiation, even equilibrium radiation, is not homogeneous and isotropic. It contains interference maxima and minima, whose explanation lies beyond the bounds of the classical theory of radiation, but which are fully accounted for by the general theory.

Of course, it is not possible here to exhaust all the questions that arise in connection with fluctuation electrodynamics. We have not touched, for example, on that simpler form of the theory which is applicable in the case of a sufficiently strong skin effect. Nor have we touched on the question of thermal radiation in inhomogeneous and anisotropic media. Finally, we have not dealt with questions perhaps most directly connected with the relation between the micro- and macroscopic pictures in electrodynamics.

This is, first of all, the extension of the theory to fluctuations of nonthermal origin. For the theory presented, only one thing is important—the correlation function of the extraneous field. As applied to thermal fluctuations, it proves possible to determine its form within the framework of phenomenological concepts, without going into statistical electronics. For nonthermal random electrical processes, it is apparently impossible to avoid statistical microtheory. But if its result can be expressed in terms of the correlation function of some extraneous field, then all further questions reduce to the scheme already developed.

The second question concerns the thermal electromagnetic field not in external space, but inside strongly absorbing media, where the separation of the field into wave and quasistationary components loses its meaning. Connected with this is the ambiguity of dividing electromagnetic energy into the energy of radiation and into the part attributable to ...

...relating to the internal energy of the medium. Here, too, complete clarity can most likely be achieved only by taking the microstructure of the medium into account.

Some of the questions listed above have already been considered in sufficient detail. For example, a modified form of the theory has been given for the case of well-conducting bodies (strong skin effect), and equilibrium radiation in transparent anisotropic and magnetoactive media has been studied^8. Other questions have so far been investigated only in part. Among them one may mention one problem on which F. V. Bunkin is now working.

This is the problem of the thermal radiation of a magnetoactive medium—for example, an ionized gas—placed in a magnetic field. Such conditions occur in the Earth’s ionosphere and in the atmosphere of the Sun, especially in sunspots. For the propagation of waves in such media under real conditions, geometrical optics is applicable. However, it does not answer the question of how a volume element of such a medium radiates, or how the energy is distributed between the two polarizations for different directions of observation. Fluctuation electrodynamics, if the correlation function of the extraneous field is generalized to the case of an anisotropic medium, makes it possible to clarify questions of this kind and thereby to obtain a theoretical basis for polarization radio-astronomical observations.

In conclusion I shall permit myself to say a few words about matters which have not yet been done at all. The entire semi-phenomenological approach, which has proved so productive as applied to electric fluctuations, naturally suggests extension to other fields. An analogous treatment is quite conceivable for thermal fluctuations of density, pressure, temperature, and other parameters characterizing the state of a material medium. For the corresponding random extraneous forces, here too caused by thermal motion, the correlation radius must be microscopic, i.e. zero on macroscopic scales. Thus there emerges a certain fluctuation hydrodynamics, operating with random extraneous pressures and heat sources which possess spatial \(\delta\)-correlation and whose intensities depend on the phenomenological characteristics of the medium. If a theory of this kind can be constructed, it will make it possible to hope for certain new results concerning, for example, the molecular scattering of light by a liquid in which the dispersion of elastic waves is strongly manifested. Of course, for the time being these are only preliminary considerations, and if they are mentioned here it is only because they too arose under the influence of the ideas of L. I. Mandelstam—ideas which for a long time to come, and for many people, will serve as guiding ones.

References

1. L. I. Mandelstam, Collected Works, vol. I, Publishing House of the USSR Academy of Sciences, 1948, articles 4, 5, 9, and 10.
2. L. I. Mandelstam, Collected Works, vol. II, Publishing House of the USSR Academy of Sciences, 1947, article 46.
3. L. I. Mandelstam, Collected Works, vol. II, Publishing House of the USSR Academy of Sciences, 1947, article 50.
4. J. B. Johnson, Nature, 119, 50 (1927); Phys. Rev. 29, 367 (1927); 32, 97 (1928).
5. H. Nyquist, Phys. Rev. 29, 614 (1927); 32, 110 (1928).
6. G. L. de Haas-Lorentz, Die Wissenschaft, 52, p. 86, Braunschweig (1913).
7. P. Langevin, Comptes Rendus 146, 530 (1908).
8. S. M. Rytov, Theory of Electrical Fluctuations and Thermal Radiation, Publishing House of the USSR Academy of Sciences, 1953.
9. M. A. Leontovich and S. M. Rytov, ZhETF 23, 246 (1952).
10. M. L. Levin and S. M. Rytov, ZhTF 25, 151 (1954).
11. M. A. Leontovich and M. L. Levin, ZhTF 14, 481 (1944); Izv. AN SSSR (ser. fiz.) 8, 157 (1944); M. L. Levin, DAN 54, 599 (1946); Izv. AN SSSR (ser. fiz.) 11, 117 (1947).
12. S. M. Rytov, ZhETF 27, 571 (1954).
13. E. M. Lifshitz, DAN 47, issue 4, 643 (1954).

1. 2 UFN, vol. LV, issue 3. ↩

2. 9. ↩

3. E. M. Lifshitz. ↩↩