§1. A satisfactory solution to the problem of establishing the connection between the principles of statistical physics and those of microscopic mechanics can be arrived at only when it is based on a single point of view in answering the main questions of the problem. A significant number of works on the subject treated only some part of the general problem: in most cases, they either obtained the equality of time average and phase average values (the so-called ergodicity problem), or tried to prove the ?-theorem (the irreversibility problem). The methods used to solve the different parts of the general problem and the assumptions made differed widely and bore no relation to each other.

We shall now outline the problems in brief; we shall describe the basic statements forming the foundation of statistical mechanics. The relation of these statements to the principles of microscopic mechanics is the subject matter of the above-mentioned general problem. These statements carry primarily the requirement that time average and ergodic average values should be equal; that is, any physical quantity characterizing a system considered in statistical mechanics should have a time average value equal to the average value of this quantity on the surface of a given energy (averaging on the surface being done with the usual, so-called ergodic measure, dΩ/grad ε, where dΩ is an element of the surface of a given energy). The fulfillment of this requirement amounts to what is known as the ergodicity problem.

In addition to the above, these statements include the requirement that the average values of physical quantities within a time interval should reach a given degree of approximation to the limit, as the time interval increases, regardless (or, in the overwhelming majority of cases, at least practically regardless) of the initial state of the system. This proximity to the limit must be attained within certain time intervals common to all physical quantities of a given type; these intervals being equal to those registered in practice. The exact meaning of this requirement — that convergence to the limit should be uniform with respect to the various physical quantities of a given group — will be clarified further (cf. Chapter V). For the time being it should only be pointed out that without the said requirement, the application of the calculated average ergodic value of a physical quantity to an experiment would have no basis. Indeed, no matter how long the time interval during which a quantity is observed to change might be, we could not feel confident that the average value for this interval comes anywhere near the calculated limit. Experience shows that, provided the time intervals are sufficiently great for a quantity of a given group (considerably greater than the so-called relaxation times), we can have this confidence. Furthermore, in practice, we can always let this confidence guide us without any extra analysis of the initial state of the system or any other quantities. Consequently, the above-mentioned requirement must be fulfilled with at least an overwhelming probability. This requirement lies beyond the customary definition of the ergodicity problem, but unless it is accepted, the applicability of a mathematical scheme to an experiment cannot be guaranteed.

Finally, among the above-mentioned basic statements, there is one regarding the existence of finite relaxation time, and another regarding the monotonic development of the relaxation process. This statement means that for every initial state of a given system, after a sufficiently long time — the relaxation time — the system under consideration will enter this or that state with a probability wi, which is independent of the initial state and proportional to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here si — the entropy of the state being discussed — is defined (by means of a certain generalization of the thermodynamic concept of entropy) as k ln ΔΓ, where ΔΓ is the measure of the region (using the ergodic definition of the measure) corresponding in the phase space to the state we are considering. The relaxation time depends on the type of the states under discussion; that is, it may vary for experiments measuring different quantities (the relaxation time with respect to temperatures, pressures, and so on). However, for any kind of state there will always exist a corresponding relaxation time; that is, a time following which the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be valid. With an overwhelming probability, this will be accompanied by the appearance of an equilibrium state (a maximum entropy state), which, together with the statement of the monotonic development of the entropy increase process, constitutes the subject matter of the H-theorem.

We shall point out here a characteristic feature of the foregoing description of relaxation processes: it has the form of a probabilistic statement. Both a description of the results of measurements after the relaxation time and, in general, a description of consecutive measurements of different physical quantities (particularly entropy) can only be given by a certain probabilistic scheme. It should be stressed that the probabilistic character of the series of measurements obtained is, in the case of the above questions — the fluctuation theory, the Brownian motion, the H-curve form, and so on — an absolutely reliable experimental fact, no less reliable than the probabilistic character of the series of tests obtained in any other, however well founded, application of probability theory. The series of results obtained by such measurements have, in consequence, a property common to all probabilistic objects — the non-existence of any algorithm that could determine the results of successive measurements. A formula, however complex, cannot in principle describe the successive changes in quantity measurements that are governed by the probabilistic law of the property distribution (the "Regellosigkeit" property of the probabilistic series).

In characterizing the principal statements that form the basis of statistical mechanics, that is, those requirements that are imposed on the construction of statistical mechanics (a construction that is in some way connected with the principles of microscopic mechanics), we are guided by our experience. We shall not consider here any approaches based on concepts of classical mechanics or the question of how these approaches agree with our experience. We shall only indicate that experience does provide such guidance, and that our statements are valid for all physical quantities that are measurable in the systems described by statistical mechanics.

A physical quantity is defined here as any quantity measured in reality in the systems studied by statistical mechanics; this definition is not theoretical, but empirical. A theoretical definition based on micromechanics cannot possibly be given at the beginning of our research. The connection between the concept of a physical quantity and the principles of classical mechanics is fully established by the observation that corresponding to a given result of a measurement of this quantity in the phase space of the system is a region with a measure different from zero. Similarly, the connection between the initial state concept used here and the principles of classical mechanics is fully established by reference to the fact that the initial state is always associated in the phase space of the system with a region with a measure different from zero. Such a definition of concepts does not necessarily follow from the principles of classical mechanics (allowing a state to be defined as a point in the phase space). But, as indicated already, what we discuss in this section is not the principles of classical mechanics, but the experiment, and so we can confine ourselves to this description of the connection between the concepts we introduce and those of classical mechanics. These concepts can be made to agree with classical mechanics if, for instance, the following practically irrefutable statement is accepted: every experiment determines a non-empty interval of the quantity being measured.

§2. The object of the present chapter is to investigate the possibilities offered by classical mechanics for producing a satisfactory construction of statistical mechanics (which, among other things, satisfies the requirements outlined in §1). Beginning with this section, and in contrast to §1, we shall discuss classical mechanics and, in doing so, we shall assume the classical approach.

In the very first decades following the appearance of the molecular kinetic theory, which was proposed to explain thermodynamic and kinetic processes in terms of mechanics, it became apparent that purely mechanical concepts were absolutely insufficient for this purpose and had to be supplemented by the introduction of assumptions of a probabilistic nature. Although the ergodic hypothesis was from the very first given a purely mechanical formulation, a mechanical interpretation of the entropy increase principle proved at once to be impossible. On the one hand, it was found impossible to develop not only a purely mechanical model of the probabilistic behavior of entropy, but even a model of its irreversible change, in accordance with the dogmatic interpretation of the second law of thermodynamics (similar to Helmholtz's and others theory of monocyclic systems — cf. Poincaré's resumé in Chapter XVII of his "Thermodynamics"). On the other hand, probabilistic assumptions were shown to be present in Boltzmann's proof of the H-theorem (these indications are contained in the well-known works criticizing the assumption of the number of collisions on which the proof is based). This was quite clearly demonstrated by P. and T. Ehrenfest in their well-known review. We shall only point out at this juncture that probabilistic assumptions arise already in the simplest notions of statistics and kinetics.

Let us suppose that the same macroscopic state is repeatedly reproduced. Knowing that the process may vary in its further development in different experiments, the assumption must be made that in the different experiments within the bounds of the same macroscopic state different initial microscopic states are achieved. Already at this juncture the question arises of the probabilistic law of distribution of different initial states.

At the same time, when we describe a certain individual, as we shall call it, system, not in different experiments but after a single given macroscopic experiment, by means of a given distribution in the phase space of one molecule (for example, in the μ-space, as in Boltzmann's proof of the ?-theorem), the description itself, particularly in its use of continuous distribution functions, contains certain hidden probabilistic assumptions. Indeed, a precise knowledge of the microscopic states of all the molecules of the system does not suffice to determine any continuous density. Let the density at a given point M be represented by the ratios ni/m(Δτi), providing that M is inside all Δτi and ni is the number of molecules inside Δτi, and let the tendency of m(Δτi) to zero be stopped at a moment when there are still many molecules inside Δτi. Then the limit will evidently be determined by the character of an arbitrarily selected sequence of regions Δτi (that is, by the extent to which these regions encompass the fixed points of molecule positions). To obtain a definite result we assume these fixed molecule positions to be distributed in the overwhelming majority of cases so uniformly that, with the form of the regions Δτibeing sufficiently simple (rectangular, for instance), the ratio ni/m(Δτi), starting from some moment as Δτi is undergoing a change, will be close to a constant, that is, to the limit. Besides, we make the assumption that the states of the system in the Γ-space (the phase space of the system as a whole) that correspond to roughly the same distribution of molecules in the μ-space (that is, to the same or roughly the same numbers of molecules in given small intervals of the μ-space equalling the mathematical expectation of the number of molecules for a given distribution function) are equally probable.

It is readily seen that the assumptions we make regarding the frequency of different cases and the probability of states are, in fact, assumptions about the distribution functions describing the probability of discovering different microscopic states inside the Γ-space region that is determined by the given macroscopic state. It is also evident that these assumptions are really statements of the uniformity of distribution inside the small intervals of the μ-space and inside the corresponding regions of the Γ-space. It is precisely assumptions of this kind that enable us, in particular, to use the conventional formula for the number of collisions in the kinetic theory.

§3. The possibility of a mechanical interpretation of thermodynamics and kinetics has, as is known, been the subject of numerous discussions. This possibility was rejected by Loschmidt, Zermelo, Barbary, Leepman, Lanard and others. There are no sufficient grounds for including the name of Poincaré in this list: Poincaré was never an absolute supporter of a mechanical interpretation of thermodynamics, but, as far as we know, he never raised any objections against the molecular kinetic theory that were erroneous (or easily removable by means of slight adjustments in the theory).

The discussions, especially those that arose in connection with the H-theorem, have brought about alterations and improvements in the original viewpoints of the founders of the molecular kinetic theory. But even the publication of the latest works by Boltzmann and Gibbs failed to give complete satisfaction to everybody. This may have been caused, in part, by the fact that many of Boltzmann's opponents have retained their original views. In particular, some dissatisfaction has found expression in the hopes, which many people pinned on the appearance of quantum mechanics, that the situation is getting better, although no one has ever succeeded in demonstrating clearly what are the weak points of the old classical theory. Nevertheless, the overwhelming majority of physicists seem to have accepted the viewpoint put forward by the Ehrenfests in their review. According to that viewpoint the corrected molecular kinetic theory, as stated in Boltzmann's latest works, after some slight improvements becomes logically satisfactory and free of any contradictions. The belief that Gibbs' statistical mechanics is logically well-composed appears to be even more widely accepted.

Let us formulate at this point the results we expect to obtain by analyzing the possibility of developing a satisfactory scheme of physical statistics based on classical mechanics.

1. All the criticism (by the above-mentioned authors, among others) so far levelled against the classical point of view can be completely discarded following the publication of Boltzmann's latest works and after some minor, sometimes almost terminological, improvements made by some other authors (for example, the Ehrenfests' remarks concerning the H-curve, or Jeans' research on the number of collisions; cf. §§4, 8). This statement was made in the Ehrenfests' review (a new form of objection to irreversibility, differing from the original one, requires the acceptance of additional postulates that are alien to the classical theory (cf. §§8,17), but in spite of which the principles of the classical theory and their interpretation remain unchanged). As to the old objections, it is only Zermelo's remark (which is touched upon in §14) that may retain its significance with respect to the most complete version of Boltzmann's ideas, but also only after some new arguments that cannot be found in Zermelo's work are added; if taken in its original form, this remark is not sufficiently well-grounded either.