Axiomatics of Classical Statistical Mechanics

by Rudolf Kurth

Overview

This monograph constructs classical statistical mechanics as a deductive system, based on the equations of motion and the basic postulates of probability. The treatment consists chiefly of theorems and proofs that are expressed in a manner that reveals the theory's logical structure. Requiring only familiarity with the elements of calculus and analytical geometry, Axiomatics of Classical Statistical Mechanics is geared toward advanced undergraduates and graduate students in mathematical physics.
An opening chapter on mathematical tools makes the text as self-contained as possible. Subsequent chapters explore the phase flows of mechanical systems, the initial distribution of probability in the phase space, and both time-dependent and time-independent probability distributions. A final chapter covers statistical thermodynamics.

Product Details

ISBN-13:	9780486832753
Publisher:	Dover Publications
Publication date:	03/20/2019
Series:	Dover Books on Physics
Pages:	192
Product dimensions:	8.30(w) x 5.40(h) x 0.50(d)

About the Author

Rudolf Kurth taught at several institutions, including Southern Illinois University, Edwardsville, the Georgia Institute of Technology, and Durham College at the University of Durham, U.K. His other books include Dimensional Analysis and Group Theory in Astrophysics, Introduction to Stellar Statistics, and Elements of Analytical Dynamics.

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PREFACE

This book is an attempt to construct classical statistical mechanics as a deductive system, founded only on the equations of motion and a few well-known postulates which formally describe the concept of probability. This is the sense in which the word "axiomatics" is to be understood. An investigation of the compatibility, independence and completeness of the axioms has not been made: their compatibility and independence appear obvious, and completeness , in its original simple sense, is an essential part of the mos geometricus itself and, in fact, of the scientific method in general: all the assumptions of the theory have to be stated explicitly and completely. (I know there are other interpretations of the word "completeness", but I cannot help feeling them to be artificial.)

The aim adopted excluded some subjects which are usually dealt with in books on statistical mechanics, in particular the theory of Boltzmann's equation; it has not been and cannot be derived from the above postulates, as is shown by the contradiction between Boltzmann's H-Theorem and Poincare's Recurrence Theorem. By this statement I do not wish to deny the usefulness of Boltzmann's equation; I wish only to emphasize that it cannot be a part of a rational system founded on the above assumptions. Instead , it constitutes a separate theory based on a set of essentially different hypotheses.

My aim demanded that the propositions of the theory be formulated more geometrico also, that is, in the form "if ..., then ...", which, in my opinion, is the only appropriate form for scientific propositions. For convenience in derivation or formulation the assumptions were sometimes made less general than they could have been.

It was intended to make the book as self-contained as possible. Therefore the survey of the mathematical tools in Chapter II was included so that only the elements of calculus and analytical geometry are supposed to be known by the reader. In order to confine this auxiliary chapter to suitable proportions, however, full proofs were given only in the simplest cases, while lengthier or more difficult proofs were only sketched or even omitted altogether. Thus, in this respect, I have achieved only a part of my aim.

The set of the references given is the intersection of the set of writings I know by my own study and the set of writings which appeared relevant for the present purpose. A part of the text is based on investigations of my own, not all of which have yet been published.

It is a pleasure for me to remember gratefully the encouragement I received from Dr. D. ter Haar (now at Oxford University) and Professor E. Finlay-Freundlich (at St. Andrews University) when I made the first steps towards the present essay and had to overcome many unforeseen obstacles. Several colleagues at Manchester University were so kind as to correct my English. Particularly, I am indebted to Dr. H. Debrunner (Princeton) for his many most valuable criticisms of my manuscript. Last but not least, I gratefully acknowledge that it was A. J. Khinchin's masterly book which gave me the courage to think for myself in statistical mechanics, after a long period of doubt about the conventional theory. To them all I offer my best thanks.

Cheadle (Cheshire) Rudolf Kurth

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Table of Contents

Table of contents:
I. Introduction
II. Mathematical Tools
III. The Phase Flows of Mechanical Systems
IV. The Initial Distribution of Probability in the Phase Space
V. Probability Distributions Which Depend on Time
VI. Time-Independent Probability Distributions
VII. Statistical Thermodynamics
Index