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Il
STATISTICAL MECHAN ICS
A deductive treatment
,
By
~~JPENROSE /I I
PROFESSOR OF MATHEMATICS THE OPEN UNIVERSITY, LONDON
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. I OI..C.III".
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PERGAMON PRESS OXFORD TORONTO
LONDON SYDNEY
EDINBURGH PARIS
NEW YORK
BRAUNSCHWEIG
Contents vii
Chapter I
Basic Assumpti011~
1. Introduction 2. Dynamics 2.1. Ejrercises 3. Observation 3.1. Exercises 4. Probability 5. The Markovian postulate 5.1. Exercises 6. Two alternative approaches
Chapter II Probability Theory 1. Events 1.1. Exercises 2. Random variables
2.1. Exercise 3. Statistical independence 3.1. Exercises 4. Markov chains 4.1. Exercises 5. Classification of observational states 5.1. Exercises 6. Statistical equilibrium 6.1. Exercises 7. The approach to equilibrium 7.1. Exercises 8. Periodic ergodic sets 8.1. Exercises 9. The weak law of large numbers 9.1. Exercises
Chapter III
The Gibbs Ensemble
1. Introduction 2. The phase-space density 2.1. Exercise 3. The classical Liouville theorem
3.1 . Exercises 4. The density matrix 4.1. Exercises 5. The quantum Liouville theorem 5.1. Exercises
Contents
Chapter N
Probabilities from Dynamics
1. Dynarnical images of events 1 .l. Exercise 2. Observational equivalence 2.1. Exercise 3. The classical accessibility postulate 3.1. Exercises 4. The quantum accessibility postulates 4.1. Exercises 5. The equilibrium ensemble 5.1 Exercises 6. Coarse-grained ensembles 6.1. Exercises 7. The consistency condition 7.1. Exercises 8. Transient states 8.1. Exercise
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Chapter V
Boltanam Entropy
1. Two fundamental properties of entropy Composite systems 2.1. Exercise 3. The additivity of entropy 3.1. Exercises 4. Large systems and the connection with thermodynamics 4.1. Exercises 5. Equilibrium fluctuations 5.1. Exercises 6. Equilibrium fluctuations in a classical gas 6.1, Exercises 7. The kinetic equation for a classical gas 8. Boltmann's W theorem 8.1. Exercise 2.
Chapter VI 1.
Ststistical Entropy
The definition of statistical entropy
1.1. Exercises 2. Additivity properties of statistical entropy 2.1. Exercises 3. Perpetual motion 3.1. Exercise 4. Entropy and information 5. Entropy changesin the observer 5.1. Exercises
OTHERTITLES IN THE SERIES
Preface THEthesis of this book is that statistical mechanics can be built up deductively from a small number of well-defined physical assumptions. To provide a firm basis for the deductive structure, these assumptions have been converted into a system of postulates describing an idealized model of real physical systems. These postulates, which are listed immediately after this preface, thus .play a role in the theory similar to the role of the first and second laws in thermodynamics. Of these five postulates, the crucial one is the fourth, expressing the assumption that the successive observational states of a physical system form a Markov chain. This is a strong assumption, whose influence is felt throughout the book. It is possible, indeed, that this postulate is too strong to be satisfied exactly by any real physical system; but even so, it has been adopted here because it provides the simplest precise formulation of a hypothesis that appears to underlie all applications of probability theory in physics. Our treatment may thus be regarded as a first approximation to the more elaborate theory that would be obtained if this postulate were replaced by a less idealized statement of the same basic hypothesis. Our main concern is not so much to find out whether a real system can exactly obey all the postulates-although we do discuss this difficult question in Chap. IV, 4 7- but to show that a