Time's Arrow: The Origins of Thermodynamic Behavior

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Book Description
The Second Law of Thermodynamics has been called the most important law of nature: It is the law that gives a direction to processes that is not inherent in the laws of motion, that says the state of the universe is driven to thermal equilibrium. Its mathematical formulation is simple: The entropy of a closed system cannot decrease. Since the recognition that macroscopic phenomena have an atomic basis, it has remained a fundamental problem to reconcile the increase of entropy with the known reversibility of all the laws of microscopic physics. Professor Michael Mackey of McGill University here explores the dynamical basis for the Second Law, that is, he seeks to illuminate the fundamental dynamical properties required for the construction of a successful statistical mechanics. Aimed at physicists and applied mathematicians with an interest in the foundations of statistical mechanics, the book includes such new material as: a demonstration that the black body radiation law can be deduced from maximal entropy principles; a discussion of sufficient conditions for the existence of at least one state of thermodynamic equilibrium; a description of the behavior of entropy in asymptotically periodic systems; a necessary and sufficient condition for the evolution of entropy to a global maximum; and a presen- tation of the three main types of ergodic theorems and their proofs. He also explores the potential role of incomplete knowledge of dynamical variables, measurement imprecision, and the effects of noise in giving rise to entropy increases.


Table of Contents

Preface ix
Chapter 1. Starters
A. Thermodynamic Systems 1
B. Dynamics 2
C. Thermodynamic States 4
D. Boltzmann-Gibbs Entropy 5
E. Summary 8
Chapter 2. Maximal Entropy Principles
A. Microcanonical Ensembles 9
B. Canonical Ensembles 11
C. The Thermodynamic Connection 12
D. Boltzmann and Other Statistics 16
E. Blackbody Radiation 18
F. General Entropy Maxima 19
G. Maximal Entropy Principles 20
H. Summary 20
Chapter 3. Invertible and Noninvertible Systems
A. Markov Operators 21
B. Conditional Entropy 27
C. Existence of Equilibria 34
D. Summary 37
Chapter 4. Ergodicity and Unique Equilibria
A. Frobenius-Perron Operator 39
B. The Koopman Operator 43
C. Recurrence 45
D. Densities and Flows 46
E. Ergodicity 48
F. Summary 57
Chapter 5. Mixing
A. Mixing 58
B. Kolmogorov Systems 66
C. The Decay of Correlations 67
D. Summary 68
Chapter 6. Asymptotic Periodicity and Entropy Evolution
A. Asymptotic Periodicity 70
B. Asymptotic Periodicity Illustrated 75
C. The Weak Form of the Second Law 80
D. Asymptotic Periodicity and Correlations 86
E. Summary 88
Chapter 7. Global Evolution of Entropy
A. Exactness 89
B. The Strong Form of the Second Law 98
C. Summary 101
Chapter 8. Coarse Graining
A. Coarse Grained Entropy 103
B. Coarse Graining of Mixing Systems 106
C. Sampling 107
D. Summary 109
Chapter 9. Traces and Factors
A. Traces 110
B. Factors 112
C. Coarse Graining and Traces 113
D. Summary 114
Chapter 10. Open Discrete Time Systems
A. An Operator Equation 116
B. Loosely Coupled Systems 117
C. Strongly Coupled Systems 120
D. Asymptotic Periodicity and Additive Perturbations 123
E. Exactness and Additive Perturbations 129
F. Parametric Perturbations 131
G. Markov Operators and Deterministic Processes 136
H. Summary 138
Chapter 11. Open Continuous Time Systems
A. Wiener Processes and Stochastic Differential Equations 140
B. The Fokker-Planck Equation 142
C. Solutions of the Fokker-Planck Equation 145
D. The Behavior of Entropy 148
E. Phase Transitions and Additive Perturbations 151
F. Parametric Perturbations and Phase Transitions 156
G. Summary 158
References 159
Index 167

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