§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...