Statistical mechanics presents two fundamental problems for mathematics:
(1) the so-called ergodic problem, that is the problem of a rigorous justification of replacement of time-averages by space (phase)-averages;
(2) the problem of the creation of an analytic apparatus for the construction of asymptotic formulas.
In order to become familiar with these two groups of problems, a mathematician usually has to overcome several difficulties. For understandable reasons, the books on physics do not pay much attention to the logical foundation of statistical mechanics, and a great majority of them are entirely unsatisfactory from a mathematical standpoint, not only because of a non-rigorous mathematical discussion (here a mathematician would usually be able to put things in order by himself), but mainly because of the almost complete absence of a precise formulation of the mathematical problems which occur in statistical mechanics.
In the books on physics the formulation of the fundamental notions of the theory of probability as a rule is several decades behind the present scientific level, and the analytic apparatus of the theory of probability, mainly its limit theorems, which could be used to establish rigorously the formulas of statistical mechanics without any complicated special machinery, is completely ignored.
The present book considers as its main task to make the reader familiar with the mathematical treatment of statistical mechanics on the basis of modem concepts of the theory of probability and a maximum utilization of its analytic apparatus. The book is written, above all, for the mathematician, and its purpose is to introduce him to the problems of statistical mechanics in an atmosphere of logical precision, outside of which he cannot assimilate and work, and which, unfortunately, is lacking in the existing physical expositions.
The only essentially new material in this book consists in the systematic use of limit theorems of the theory of probability for rigorous proofs of asymptotic formulas without any special analytic apparatus. The few existing expositions which intended to give a rigorous proof to these formulas, were forced to use for this purpose special, rather cumbersome, mathematical machinery. We hope, however, that our exposition of several other questions (the ergodic problem, properties of entropy, intramolecular correlation, etc.) can claim to be new to a certain extent, at least in some of its parts.
For an extract from the Introduction to the book see Introduction to A I Khinchin's Statistical Mechanics