Mathematical Foundations of Statistical Mechanics

A. Ya. Khinchin

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486138732
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Phase space, ergodic problems, central limit theorem, dispersion and distribution of sum functions. Chapters include Geometry and Kinematics of the Phase Space; Reduction to the Problem of the Theory of Probability; and more.

CHAPTER I

INTRODUCTION

1. A brief historical sketch. After the molecular theory of the structure of matter attained a predominant role in physics, the appearance of new statistical (or probabilistic) methods of investigation in physical theories became unavoidable. From this new point of view each portion of matter (solid, liquid, or gaseous) was considered as a collection of a large number of very small particles. Very little was known about the nature of these particles except that their number was extremely large, that in a homogeneous material these particles had the same properties, and that these particles were in a certain kind of interaction. The dimensions and structure of the particles, as well as the laws of the interaction could be determined only hypothetically.

Under such conditions the usual mathematical methods of investigation of physical theories naturally remained completely powerless. For instance, it was impossible to expect to master such problems by means of the apparatus of differential equations. Even if the structure of the particles and the laws of their interaction were known, their exceedingly large number would have presented an insurmountable obstacle to the study of their motions by such methods of differential equations as are used in mechanics. Other methods had to be introduced, for which the large number of interacting particles, instead of being an obstacle, would become a stimulus for a systematic study of physical bodies consisting of these particles. On the other hand, the new methods should be such that a lack of information concerning the nature of the particles, their structure, and the character of their interaction, would not restrict the efficiency of these methods.

All these requirements are satisfied best by the methods of the theory of probability. This science has for its main task the study of group phenomena, that is, such phenomena as occur in collections of a large number of objects of essentially the same kind. The main purpose of this investigation is the discovery of such general laws as are implied by the gross character of the phenomena and depend comparatively little on the nature of the individual objects. It is clear that the well-known trends of the theory of probability fit in the best possible way the aforementioned special demands of the molecular-physical theories. Thus, as a matter of principle, there was no doubt that statistical methods should become the most important mathematical tool in the construction of new physical theories; if there existed any disagreement at all, it concerned only the form and the domain of application of these methods.

In the first investigations (Maxwell, Boltzmann) these applications of statistical methods were not of a systematical character. Fairly vague and somewhat timid probabilistic arguments do not pretend here to be the fundamental basis, and play approximately the same role as purely mechanical considerations. Two features are characteristic of this primary period. First, far reaching hypotheses are made concerning the structure and the laws of interaction between the particles; usually the particles are represented as elastic spheres, the laws of collision of which are used in an essential way for the construction of the theory. Secondly, the notions of the theory of probability do not appear in a precise form and are not free from a certain amount of confusion which often discredits the mathematical arguments by making them either void of any content or even definitely incorrect. The limit theorems of the theory of probability do not find any application as yet. The mathematical level of all these investigations is quite low, and the most important mathematical problems which are encountered in this new domain of application do not yet appear in a precise form.¹

It should be observed, however, that the tendency to restrict the role of statistical methods by introducing purely mechanical considerations, (from various hypotheses concerning the laws of interaction of particles), is not restricted to the past. This tendency is clearly present in many modern investigations. According to a historically accepted terminology, such investigations are considered to belong to the kinetic theory of matter, as distinct from the statistical mechanics which tries to reduce such hypotheses to a minimum by using statistical methods as much as possible. Each of these two tendencies has its own advantages. For instance, the kinetic theory is indispensable when dealing with problems concerning the motion of separate particles (number of collisions, problems concerning the study of systems of special kinds, mono-atomic ideal gas); the methods of the kinetic theory are also often preferable, because they give a treatment of the phenomena which is simpler mathematically and more detailed. But in questions concerning the theoretical foundation of general laws valid for a great variety of systems, the kinetic theory naturally becomes sometimes powerless and has to be replaced by a theory which makes as few special hypotheses as possible concerning the nature of the particles. In particular, it was precisely the necessity of a statistical foundation for the general laws of thermodynamics that produced trends which found their expression in the construction of statistical mechanics. To avoid making any special hypotheses about the nature of the particles it became necessary in establishing a statistical foundation to develop laws which had to be valid no matter what was the nature of these particles (within quite wide limitations).

The first systematic exposition of the foundations of statistical mechanics, with fairly far developed applications to thermodynamics and some other physical theories, was given in Gibbs’ well-known book.² Besides the above mentioned tendency not to make any hypotheses about the nature of particles the following are characteristic of the exposition of Gibbs.

1. A precise introduction of the notion of probability, which is given here a purely mechanical definition, is lacking with the resulting questionable logical precision of all arguments of statistical character.
2. The limit theorem of the theory of probability does not find any application (at that time they were not quite developed in the theory of probability itself).
3. The author considers his task not as one of establishing physical theories directly, but as one of constructing statistic-mechanical models which have some analogies in thermodynamics and some other parts of physics; hence he does not hesitate to introduce some very special hypotheses of a statistical character (canonical distribution, ch. 25, § 25) without attempting to prove them or even to interpret their meaning and significance.
4. The mathematical level of the book is not high; although the arguments are clear from the logical standpoint, they do not pretend to any analytical rigor.

At the time of publication of Gibbs’ book, the fundamental problems raised in mathematical science in connection with the foundation of statistical mechanics became more or less clear. If we disregard some isolated small problems, we have here two fundamental groups of problems representing a broad, deep, interesting and difficult field of research in mathematics which is far from being exhausted even at present. The first of these groups is centered around the so-called ergodic problem (ch. III), that is, the problem of the logical foundation for the interpretation of physical quantities by averages of their corresponding functions, averages taken over the phase-space or a suitably selected part of it. This problem, originated by Boltzmann, apparently is far from its complete solution even at the present time. This group of problems was neglected by the investigators for a long time after some unsuccessful attempts, based either on some inappropriate hypotheses introduced ad hoc, or on erroneous logical and mathematical arguments (which, unfortunately, have been repeated without any criticism in later handbooks). In the book of Gibbs these problems naturally are not considered because of the tendency to construct models. Only recently (1931), the remarkable work of G. D. Birkhoff again attracted the attention of many investigators to these problems, and since then this group of problems has never ceased to interest mathematicians, who devote more and more effort to it every year. We will discuss this group of problems in more detail in the ch. III.

The second group of problems is connected with the methods of computation of the phase-averages. In the majority of cases, these averages cannot be calculated precisely. The formulas which are derived for them in the general theory (that is, without specification of the mechanical system under discussion) are complicated, not easy to survey, and as a rule, not suited for mathematical treatment. It is quite natural, therefore, to try to find simpler and more convenient approximations for these averages. This problem is always formulated as a problem of deriving asymptotic formulas which approach the precise formulas when the number of particles constituting the given system increases beyond any limit. These asymptotic formulas have been found long ago by a semi-heuristic method (by means of an unproved extrapolation, starting from some of the simplest examples) and were without rigorous mathematical justification until fairly recent years. A decided change in this direction was brought about by the papers of Darwin and Fowler about twenty years ago. Strictly speaking these authors were the first to give a systematic computation of the average values; up to that time, such a computation was in most cases replaced by a more or less convincing determination of “most probable” values which (without rigorous justification) were assumed to be approximately equal to the corresponding average values. Darwin and Fowler also created a simple, convenient, and mathematically rigorous apparatus for the computation of asymptotic formulas. The only defect of their theory lies in an extreme abstruseness of the justification of their mathematical method. To a considerable extent this abstruseness was due to the fact that the authors did not use the limit theorems of the theory of probability (sufficiently developed by that time), but created anew the necessary analytical apparatus. In any case, the course in statistical mechanics published by Fowler³ on the basis of this method, represents up to now the only book on the subject, which is on a satisfactory mathematical level.⁴

In closing this brief sketch we should mention that the development of atomic mechanics during the last decades has changed the face of physical statistics to such a degree that, naturally, statistical mechanics had to extend its mathematical apparatus in order to include also quantum phenomena. Moreover, from the modern point of view, we should consider quantized systems as a general type of which the classical systems are a limiting case. Fowler’s course is arranged according to precisely this point of view: the new method of constructing asymptotic formulas for phase-averages is established and developed for the quantized systems, and the formulas which correspond to the classical systems are obtained from these by a limiting process.

Quantum statistics also presents some new mathematical problems. Thus, the justification of the peculiar principles of statistical calculations which are the basis of the statistics of Bose-Einstein and Fermi-Dirac required mathematical arguments which were distinct as a matter of principle (not only by their mathematical apparatus) from all those dealt with in the classical statistical mechanics. Nevertheless, it could be stated that the transition from the classical systems to the quantum systems did not introduce any essentially new mathematical difficulties. Any method of justification of the statistical mechanics of the classical systems, would require for quantized systems an extension of the analytical apparatus only, in some cases introducing small difficulties of a technical character but not presenting new mathematical problems. In places where we might have to use finite sums or series, we operate with integrals, continuous distributions of probability might be replaced by the discrete ones, for which completely analogous limit theorems hold true.

Precisely for these reasons in the present book we have restricted ourselves to the discussion of the classical systems, leaving completely out of consideration everything concerning quantum physics, although all the methods which we develop after suitable modifications could be applied without any difficulties to the quantum systems. We have chosen the classical systems mainly because our book is designed, in the first place, for a mathematical reader, who cannot always be assumed to have a sufficient knowledge of the foundations of quantum mechanics. On the other hand, we did not consider as expedient the inclusion in the book of a brief exposition of these foundations. Such an inclusion would have considerably increased the size of the book, and would not attain the desired purpose since quantum mechanics with its novel ideas, often contradicting the classical representations, could not be substantially assimilated by studying such a brief exposition.

2. Methodological characterization. Statistical mechanics has for its purpose the construction of a special physical theory which should represent a theoretical basis for some parts of physics (in the first place, for thermodynamics) using as few special hypotheses as possible. More precisely, statistical mechanics considers every kind of matter as a certain mechanical system and tries to deduce the general physical (in particular, thermodynamical) laws governing the behavior of this matter from the most general properties of mechanical systems, and eo ipso to eliminate from the corresponding parts of physics any theoretically unjustified postulation of their fundamental laws. The basic assumptions of statistical mechanics should be then (1) any general laws which hold for all (or at least for very general classes of) mechanical systems, and (2) representations of any kind of matter as a mechanical system consisting of a very large number of components (particles). Thus the purpose of statistical mechanics consists in deriving special properties of such many-molecular systems from the general laws of mechanics and in showing that, with a suitable physical interpretation of the most important quantities appearing in the theory, these derived special properties will give precisely those fundamental physical (and in particular, thermodynamical) laws governing matter in general and certain special kinds of matter in particular. The mathematical method which allows us to realize these aims, for the reasons explained in §1, is the method of the theory of probabilities.

Let us make some further remarks concerning the above described purpose of statistical mechanics.

1. The fact that statistical mechanics considers every kind of matter as a mechanical system and tries to derive all its properties from the general laws of mechanics, often leads to a criticism of being a priori mechanistic. In fact, however, all reproaches of such kind are based on a misunderstanding. Those general laws of mechanics which are used in statistical mechanics are necessary for any motions of material particles, no matter what are the forces causing such motions. It is a complete abstraction from the nature of these forces, that gives to statistical mechanics its specific features and contributes to its deductions all the necessary flexibility. This is best illustrated by the obvious fact that if we modify our point of view on the nature of the particles of a certain kind of matter and on the character of their interaction, the properties of this kind of matter established by methods of statistical mechanics remain unchanged by these modifications because no special assumption was made in the process of deduction of these properties.

The circumstance of being governed by the general laws of mechanics does not lend any specific features to the systems studied in statistical mechanics; as it has been said already, these laws govern any motion of matter, whether it has any relation to statistical mechanics, or not. The specific character of the systems studied in statistical mechanics consists mainly in the enormous number of degrees of freedom which these systems possess. Methodologically, this means that the standpoint of statistical mechanics is determined not by the mechanical nature, but by the particle structure of matter. It almost seems as if the purpose of statistical mechanics is to observe how far reaching are the deductions made on the basis of the atomic structure of matter, irrespective of the nature of these atoms and the laws of their interaction.

2. Since the mechanical basis of statistical mechanics is restricted only by those general laws which hold for any systems (or at least for very general classes of systems), of considerable interest for us (even before the assumption of a large number of components) are the results of the so-called general dynamics, a branch of mechanics whose purpose is the deduction of such laws which hold for all mechanical systems and can be derived from the general laws of mechanics alone. This theory, evidently of a considerable philosophical interest, is of comparatively recent origin. In the past it was usually assumed that the deductions which could be made from the general laws of mechanics were not sufficiently concrete to have any scientific interest. It developed later that the situation was different, and at present the constructions of general dynamics are attracting interest of more and more investigators. In particular, all the above mentioned investigations of Birkhoff and of the increasing number of his disciples belong to this theory. It is particularly interesting to us that the methods (and partially, problems) of general dynamics, even before any assumptions are made concerning the number of degrees of freedom of a system under investigation, show a definitely expressed statistical tendency. This fact is well-known to anyone who has studied investigations in this field with any amount of attention. Thus the fundamental theorem of Birkhoff is formally equivalent to a certain theorem of the theory of probability; conversely, the theory of stationary stochastic processes, which represents one of the most interesting chapters of the modern theory of probability, formally coincides with one of the parts of the general dynamics.

The reason for this can be easily recognized. The most important problem of general dynamics is the investigation of the dependence of the character of the motion of an arbitrary mechanical system on the initial data, or more precisely the determination of such characteristics of the motion which in one sense or another “almost do not depend” on these initial data. Such a quantity for a great majority of trajectories assumes values very near to a certain constant number. But the expression “for a great majority of trajectories” has the meaning that the set of trajectories which do not satisfy this requirement is metrically negligible in some metric, that is, has for its measure either zero or a very small positive number.

In this sense many propositions of general dynamics are of a peculiarly typical form. They state that for most general classes of mechanical systems the motion is subjected to certain definite conditions, if not for all initial data then at least for a metrically great majority of them. It is known, however, that propositions which can be formulated in such form, in most cases turn out to be equivalent to some theorems of the theory of probability. This theory from a formal point of view could be considered as a group of special problems of the theory of measure, namely such problems as most often deal with the establishment of a metrically negligible smallness of certain sets. It suffices to remember that the majority of propositions of the theory of functions of a real variable concerned with the notions of convergence “in measure”, “almost everywhere” etc., finds an adequate expression in the terminology of the theory of probability. Thus it can be stated that even general dynamics which represents the mechanical basis of statistical mechanics, is a science which is filled to a great extent with the ideas of the theory of probability and which successfully uses its methods and analogies.

As to the statistical mechanics, it is a science whose probabilistic character is noticeable in two entirely distinct and completely independent features: in the general dynamics as its mechanical basis, and in the postulate of a great number of degrees of freedom allowing a most fruitful application of methods of the theory of probability.

3. It remains to discuss the form in which methods and results of the theory of probability could be utilized in determining asymptotic formulas which express approximately the phase averages of various functions in the case of a large number of degrees of freedom (or for systems consisting of a large number of particles).

As previously mentioned, in most expositions these formulas are introduced without any justification. After having derived these formulas for some especially simple particular case (for instance, for a homogeneous mono-atomic ideal gas) the authors usually extend them to the general case either without any justification, or using some arguments of heuristical character. Perhaps a single exception from this general rule is represented by the method of Fowler. Darwin and Fowler, as was already mentioned, develop a special and very abstruse analytical apparatus for a mathematical justification of the method of obtaining asymptotic formulas, which they have created. Nowhere do they use explicit results of the theory of probability; instead, they build a separate logical structure, but, as a matter of fact, they are merely moving along an analytical path parallel to that which is used by the theory of probability in deriving its limit theorems. From here only one step remains in attempting to introduce a method which we consider as the most expedient: instead of repeating in a complicated formulation the whole long analytical process which leads to limit theorems of the theor...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• CHAPTER I - INTRODUCTION
• CHAPTER II - GEOMETRY AND KINEMATICS OF THE PHASE SPACE
• CHAPTER III - ERGODIC PROBLEM
• CHAPTER IV - REDUCTION TO THE PROBLEM OF THE THEORY OF PROBABILITY
• CHAPTER V - APPLICATION OF THE CENTRAL LIMIT THEOREM
• CHAPTER VI - IDEAL MONATOMIC GAS
• CHAPTER VII - THE FOUNDATION OF THERMODYNAMICS
• CHAPTER VIII - DISPERSION AND THE DISTRIBUTIONS OF SUM FUNCTIONS
• APPENDIX
• NOTATIONS
• INDEX
• A CATALOG OF SELECTED DOVER BOOKS IN SCIENCE AND MATHEMATICS