BIBLIOGRAPHY

V. G. Levich. Introduction to Statistical Physics, Gostekhizdat, Moscow–Leningrad, 1950, 417 pp., price 19 rubles 70 kopecks.

The book under review was compiled on the basis of a course of lectures delivered by the author at the V. I. Lenin Moscow State Pedagogical Institute in 1940–1949.

The books published up to the present time were intended mainly for the university curriculum and did not take into account the needs of the contingent of students studying statistical physics in an abridged program. The book under review largely fills this gap.

V. G. Levich’s book differs substantially from all previously published textbooks on statistical physics. The author believes that “at the present time an exposition of statistical physics without taking account of the quantum properties of atomic systems is not possible” (p. 7), for the results of classical statistics, when applied to concrete physical systems, are often in serious contradiction with experiment. Since, however, the study of statistical physics precedes the course in quantum mechanics, the author’s “vicious circle” (p. 7) arises: “an exposition of statistical physics requires acquaintance with quantum mechanics; to study quantum mechanics it is necessary to know statistical physics.”

The author sees the way out of this situation in abandoning the traditional exposition of statistical physics in two parts (first classical statistics, and then statistics of systems obeying quantum laws) and, from the very beginning, giving the reader the information from quantum mechanics that is essentially necessary in order to construct a unified exposition of classical and quantum statistics. This constitutes the great value of the book, which fully corresponds in its scientific level to the requirements placed on a modern course in statistical physics. In general, the author succeeded in overcoming the difficulties of such a construction of the course and in setting forth, with pedagogical skill, one of the most difficult sections of theoretical physics. For the construction of the course a quasi-classical approximation is used, in which the description of systems in a classical manner is supplemented by a representation of the discrete energy states of the system. The author consciously departs from complete rigor and logical impeccability in the sequence of exposition, which are impossible in the quasi-classical approximation. He sees the justification for this in achieving the “practical aim—to set forth, insofar as possible in an accessible form, the fundamental questions of statistical physics that are of both principal and practical interest” (p. 8).

The book consists of 16 chapters and contains the general foundations of statistical physics and a large number of applications of statistical methods to a wide variety of questions in classical and modern physics: from

ideal gases to the theory of liquid helium II and the statistical theory of nuclear processes. Thus, by means of concrete examples, the great significance of statistical methods in physics is emphasized. However, for a conscious assimilation of all the applications considered, the reader is required to have a prior acquaintance with these questions within the framework of a sufficiently serious course in general physics.

The exposition of the kinetic theory of gases (Chapter III) is preceded by an introductory chapter and Chapter II, devoted to the foundations of probability theory. In § 1 of the introduction, the discussion concerns the tasks and subject matter of statistical physics. One cannot but note the serious error of the author: speaking of the “fierce ideological struggle” with representatives of the reactionary school of energetics, he notes the “special role” of L. Boltzmann in this struggle. But it is known that the decisive blow to energetics was dealt by V. I. Lenin, and not by Boltzmann, who fought from the positions of inconsistent materialism. The author does not even mention the role of Lenin in the struggle against energetics. In general, in a book on statistical physics, and in particular in the introduction, more attention should have been paid to the exposition of general methodological questions that are of great importance for educating students in a materialist worldview (statistical and dynamical laws, their interrelation, and other questions). The education of a worldview is a task no less important than the communication of certain factual knowledge. Only in § 10, having passed to the kinetic theory of gases, does the author mention that, in the aggregate of an enormous number of molecules, “laws of a special type are manifested..., which have received the name of ‘statistical laws’” (p. 50).

The author correctly points out the fundamental significance of Lomonosov’s work in the creation of kinetic theory. In our opinion, it would only have been necessary to analyze more fully the arguments with the help of which Lomonosov justified the existence of the motion and interaction of particles. In particular, it should have been indicated that Lomonosov arrived at these conclusions, so important for kinetic theory, by comparing such externally contradictory properties of gases as the elasticity of air and its great compressibility.

The introductory chapter is devoted to the necessary information from thermodynamics, mechanics, and quantum mechanics. This information is presented with excessive brevity. Although the author assumes that the basic propositions of thermodynamics are known (p. 9), in § 2 more attention should have been paid to the physical meaning of thermodynamic functions and the conditions of thermodynamic equilibrium, since later, especially in Chapters V and X, the author makes extensive use of them. Apparently, the author believes that a special course in thermodynamics should precede a course in statistical physics. It seems that at present there are not sufficient grounds for separating thermodynamics from statistical physics into separate courses, since everything not relating to the first law of thermodynamics cannot be satisfactorily set forth in a course of “pure” thermodynamics. In § 4 the author presents, in a very compressed form, some information from quantum mechanics. Excessive conciseness of exposition here is in no way justified, since subsequently the author makes extensive use of this information, which is, to a considerable extent, being presented for the first time. The energy of a rotator in quantum mechanics is not mentioned at all in the introduction, although in Chapter VI, in discussing the thermal properties of diatomic molecules, the expression for this energy is used in several places (§§ 41 and 44).

In Chapter II it would have been necessary to criticize indeterminism, which some foreign physicists regard as a conclusion allegedly following from the probabilistic description of phenomena in statistical physics.

In Chapter III, devoted to the kinetic theory of gases, the author introduces the distribution function of molecules by velocities and derives Maxwell’s distribution. A detailed analysis of the role of molecular collisions and the chaotic nature of molecular motion in establishing a stationary distribution

makes the not entirely rigorous conclusion sufficiently convincing. A detailed discussion of the features of the Maxwellian distribution should have been supplemented at least by a brief acquaintance with its experimental verification. The simple derivation of the mean relative speed, which makes it possible to refine the expression for the mean free path (p. 68) by introducing \(\sqrt{2}\), is successful. A few words should have been said about the experimental determination of the mean free path, all the more since on p. 70 molecular beams are discussed.

The numerical material on p. 49, characterizing a gas under normal conditions, contains a misprint in the “density” (instead of \(2.68\cdot 10^{19}\ \mathrm{1/cm^3}\), there is the figure \(6\cdot 10^{19}\ \mathrm{1/cm^3}\)). It is hardly appropriate to place in this chapter § 16 on the Boltzmann distribution in a gravitational field, since formula (16.3) for the distribution in a force field does not follow from the barometric formula (16.2). It would be more expedient to transfer this paragraph to Chapter VI, where the Maxwell–Boltzmann distribution is obtained from the Gibbs distribution (p. 149). In this chapter it would first have been necessary to analyze Pirogov’s important works devoted to the substantiation of the Maxwellian distribution.

In Chapter IV, difficult questions of statistical distribution are expounded with sufficient completeness, and at the same time accessibly.

In §§ 17 and 18 macroscopic systems consisting of a large number of quasi-independent, weakly interacting subsystems are considered, and the question is posed of finding the probability of a definite state of some subsystem moving according to the laws of quantum mechanics. At the same time, the role of weak interaction is clearly emphasized as the cause of changes in the quantum state and the impossibility, owing to the complex character of this interaction, of determining the state of a single subsystem. In these paragraphs the question of the relation between the time average and the ensemble average should have been posed more clearly. Liouville’s theorem, which the author does not use later, ought to have been presented in the chapter devoted to general questions of statistical physics.

The insufficient rigor of the derivation of the Gibbs distribution in § 20 is fully justified, since all the physical prerequisites of such a derivation are set forth. All the remaining paragraphs of this chapter, devoted to the study of the Gibbs distribution, the transition to classical statistics, and questions of relaxation, are successful.

In Chapter V, devoted to the connection between statistics and thermodynamics and, in particular, to the statistical substantiation of the second law, the important role of the function of states (the sum or integral of states) for finding thermodynamic functions is well shown. The author leads the reader to the conclusion that the generality of the laws of statistical mechanics makes it possible to apply it to the study not only of thermal but also of other properties of matter—electrical, magnetic, chemical, and so on.

It is necessary to note the thoroughness of the exposition in § 30 of the statistical character of the second law, with a detailed and sufficiently well-argued critique of idealistic and pseudoscientific “conclusions” about the “heat death” of the universe. It is also necessary to dwell separately on § 32, devoted to the third law of thermodynamics. In the traditional exposition of statistical physics, only a few words are usually devoted to this question for lack of space. In V. G. Levich’s book this question is presented with sufficient completeness, and the role of quantum energy states in the behavior of a system near absolute zero is clearly shown. The example with the CO molecule (p. 145) vividly illustrates the limits of applicability of the third law.

Let us note that in Chapters IV and V the author had the opportunity to point to important works by Soviet scholars: Shiller, Pirogov, and others, but did not do so.

The next ten chapters are devoted to applications of statistical methods to various systems. In Chapter VI—ideal gases—the Gibbs method is applied to ideal monatomic, diatomic, and polyatomic gases.

The limited character of the law of uniform distribution of energy over degrees of freedom is well illustrated by concrete numerical data for heat capacity (see, for example, Table 4, p. 158). The dependence of the heat capacity of diatomic gases on temperature, shown in Fig. 23 (p. 156), needs additional explanation, since it is not clear what $\theta_c$ means in units of $T_0$ on the abscissa axis (a characteristic temperature has not yet been introduced here). The author shows how taking account of the identity of particles makes it possible to obtain correct values of the entropy constant for gases and to explain the Gibbs paradox. Section 46 of Chapter VI is of unquestionable value, since it shows by numerical examples how thermodynamic quantities should be practically calculated in particular problems.

In Chapter VII, where nonideal gases are presented in the usual way, it would have been advisable to dwell in greater detail on the works of Mendeleev, Bogoliubov, Vlasov, and others mentioned in § 51. (Only a few words are said about Mendeleev’s works in § 72.) A mere enumeration of names does not give the reader a correct idea of the significance of these works. The important works of Nadezhdin should also be indicated. In Chapter VIII, devoted to the thermal properties of solids, the one-dimensional crystal model should have been supplemented by a consideration of the equation of state, energy, and heat capacity of a one-dimensional crystal, and only then should the transition have been made to § 54, where long waves in a three-dimensional crystal are considered. Thermal expansion of solids is too important a question for it to be given only a footnote on p. 238. It is necessary to discuss in greater detail the role of the anharmonicity of oscillations in thermal expansion. In Chapter X—“On Systems with a Variable Number of Particles”—§ 75 deserves mention; it expounds the theory of second-order phase transitions belonging to L. D. Landau. In Chapter XI, a special § 82 is devoted to the application of this theory to the explanation of the properties of barium titanate (work of V. L. Ginzburg).

An undoubted merit of the author is his successful exposition of the theory of liquid helium II. The electrical and magnetic properties of matter (Chapter XI) are treated with unusual thoroughness for a course in statistical physics.

The book ends with a chapter devoted to applications of statistical physics to atomic and nuclear systems. At the end of the book there is a fairly complete list of literature on statistical physics and related questions.

In conclusion, let us note that the author needlessly did not include in the text of the book the problems contained in his methodological manual on statistical physics for student problem-sessions. These problems should have been enlarged somewhat and included as exercises for the individual chapters of the course. This would only have increased the value of the book.

As is evident from all that has been said, V. G. Levich’s book is not without shortcomings. The necessary proportions in the presentation of the material have not always been found in it, and not all chapters are equally successful.

These shortcomings, however, do not lessen the overall value of the book.

V. G. Levich has achieved the goal he set himself—to write a course in statistical physics that gives the student correct notions of the present state of this scientific discipline and presents, in an accessible form, the fundamental questions of statistical physics and its numerous applications.

B. Yavorsky