## N S Krylov's monograph - Introduction

 In 1946 Nikolai Sergeevitch Krylov began to write a monograph explaining his revolutionary ideas on statistical mechanics. We give the introduction to N S Krylov's monograph below. Note that in the introduction Krylov refers to the contents of the six chapters he intended to write. However at his death he had only completed Chapter 1 and most of Chapter 2. Chapters 3 to 6 and part of Chapter 2 were, sadly, never written:-

### I. THE FOUNDATIONS OF STATISTICAL PHYSICS

Introduction

The object of this work is to investigate problems related to the general principles of the relaxation of statistical systems. These problems have arisen simultaneously with the problem of what is known as "laying the foundations of statistics," that is, establishing the connection between physical statistics and mechanics. The difficulties the problem involved were, evidently, of two major types of a totally different nature. In the first place, there were difficulties presented by the need to introduce probabilistic concepts, constituting an essential feature of statistical physics (for example, its basic statement: the H-theorem), into mechanics. In the second place, difficulties were caused by the necessity to describe in terms of mechanics those systems to which the results of statistical mechanics can be applied.

Closely connected with the first type of difficulties are the problem of the mechanical interpretation of irreversibility and, among other things, all the well-known objections to Boltzmann's treatment of the H-theorem, and all the attempts still being made at achieving a quantum-mechanical solution of this problem. Difficulties of the second type were encountered in the course of research on ergodicity, which has made such little headway in accomplishing the objective set by statistical mechanics - to find an effective criterion for a physical description of systems that are consistent with the mathematical definitions we introduce.

Despite the results, some of which are of exceptional value, obtained in attempting to overcome difficulties of either type, the problem of establishing the connection between statistics and mechanics should be regarded as being absolutely unsolved. This will be demonstrated in the first two (critical) chapters of the work.

The purpose of this work is, in considering together the two above-mentioned groups of difficulties, to give a solution to the problem that would explicitly introduce the basic concept of statistical physics - the concept of relaxation - and would make possible the quantitative evaluation of relaxation time. It would, besides, provide a method of determining those mechanical systems to which statistical mechanics is applicable, that is, it would give a criterion, expressed through the properties of the Hamiltonian of the system, permitting one to draw conclusions, at least in principle again, as to the applicability of the results of physical statistics to the given system. Finally, it should form a logically well built structure devoid, at any rate, of those contradictions that are characteristic, as will be shown further, of all the solutions that have ever been proposed to the problem taken in all its generality (that is, comprising the problem of introducing irreversibility, ergodicity, and finite relaxation time into the theory).

The complexity of the problem being largely due to the difficulty of giving its correct formulation, it would be unwise to attempt to give here a detailed idea of the content of this work.

The work consists of six chapters. The first chapter analyses the possibilities offered by classical mechanics for solving the basic problem under consideration, and discusses works on the subject based on classical mechanics. The second chapter contains a similar analysis in quantum mechanics. The third chapter considers the question of describing inexhaustively complete experiments and, in particular, conditions for the applicability of the statistical operator (density matrix) concept. In the fourth chapter we deduce some restrictions imposed on the possibility of measuring quantities in macroscopic systems by the condition that their given macroscopic characteristics should be retained. Most of the questions touched on in the third and fourth chapters deal with obtaining relaxation characteristics, the H-theorem, and so on, which are macroscopic statements, that is, statements that are not, as it may seem, related to the problem of the possibility of measurements. Therefore, lest it appear odd that these questions should arise when solving the problem set in the work, we shall emphasize at this point that the very essence of the problem lies in establishing the connection between macroscopic statements and micromechanics; and, as is known, physical sense can be imparted to macromechanical equations only where measurements are possible. The fifth chapter discusses general concepts of physical system relaxation, the H-theorem, and the time average values of physical quantities. The sixth chapter investigates the connection between the existence of relaxation and certain properties of the Hamiltonian of the system.

Although the solution of the said basic problem must be brought about by the work as a whole, each of the above-mentioned parts may be regarded as constituting a more or less self-contained unit. Owing to this structure, the discussion in each part becomes independent of the argumentation in the other parts and may be taken separately.

JOC/EFR November 2006

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