Complexities of Scales in Space and Time
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PhenomenaComplexities of Scales in Space and Time
Michael H. Peters, Ph.D.Virginia Commonwealth University
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In memory of Debbie and Harold
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Preface xiNotes on Notation xiii
Chapter 1. Introduction to Statistical Mechanics and the ClassicalMechanics of Interacting Particles 1
1.1 Introduction to Classical Statistical Mechanics and a Unificationof Equilibrium and Nonequilibrium Analysis 1
1.2 Molecular Interactions in Gases, Liquids, and Solids and theNature of Intermolecular (Interatomic) Interaction Forces 41.2.1 Modeling Born repulsive and van der Waals forces 81.2.2 Models for electrostatic interactions 8
1.3 Introduction to Classical Mechanics 181.3.1 Newtonian mechanics 19
1.4 Lagrangian Mechanics 251.5 Hamiltons Equations of Motion 271.6 Summary 29
Problems 30References 32Further Reading 32
Chapter 2. Phase Space and the Liouville Equation 33
2.1 Introduction: The Merger of Classical Mechanics andProbability Theory 33
2.2 Derivation of the Liouville Equation 342.3 The Liouville Equation in Terms of Generalized Coordinates and
Conjugate Momenta 402.4 Summary 48
Problems 49References 51Further Reading 51
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Chapter 3. Reduced Density Functions and the ReducedLiouville Equation 53
3.1 Overview: The Practical Importance of the ReducedLiouville Equation 53
3.2 Reduced Density Functions 543.3 The Reduced Liouville Equation 573.4 The Boltzmann Transport Equation 613.5 Boltzmanns Entropy 663.6 Reduced Liouville Equation in Generalized Coordinates 703.7 Summary 73
Problems 73References 76Further Reading 76
Chapter 4. Equilibrium Solution to the Liouville Equation andthe Thermodynamic Properties of Matter 77
4.1 Introduction to the Equilibrium Behavior of Matter 774.2 Solution to the Liouville Equation under Equilibrium Conditions 784.3 The Thermodynamic Functions U, P, and S 864.4 Determination of the Configurational Distribution Functions 894.5 Thermodynamic Functions for a Dilute Gas 93
4.5.1 Hard-sphere interactions 954.5.2 Point centers of attraction or repulsion 96
4.6 Configurational Integral Equation for Dense Gases and Liquids 984.7 Equilibrium Properties of Nonspherical Molecules 1024.8 Complex Equilibrium Systems: Multicomponent and Multiphase 1034.9 Chapter Summary and Note on the Lack of Proof of the Existence
and Uniqueness of the Equilibrium Solution 104Problems 106References 109Further Reading 109
Chapter 5. The General Equations of Change forNonequilibrium Systems 111
5.1 Introduction: The Scope of Transport Phenomena 1115.2 Conservation Equation for Any Dynamical Variable 1125.3 Mass Conservation Equation (the Equation of Continuity) 1145.4 Momentum Conservation 1175.5 The Energy Balance Equation 1235.6 Entropy Conservation 1265.7 Local Equilibrium Flows 131
Problems 136References 137Further Reading 137
Chapter 6. Closure of the Transport Equations 139
6.1 Introduction: Complexities of Scales in Time and Space 1396.2 Scaling of the Reduced Liouville Equation 1406.3 Regular Perturbative Expansion of the Reduced Liouville Equation
for Dense Gases and Liquids 1476.4 Perturbation Expansion for Dilute Gases 1556.5 Chapman-Enskog Solution to the Boltzmann Transport Equation 1586.6 Property Flux Expressions for Gases 1616.7 Chapter Summary and the Closed Forms of the Transport Equations 165
Problems 168References 168Further Reading 169
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This book is intended for upper-level undergraduates and first-yeargraduate students in science and engineering. It is basically an intro-duction to the molecular foundations of thermodynamics and transportphenomena presented in a unified manner. The mathematical and phys-ical science level sophistication are at the upper undergraduate level,and students who have been exposed to vector calculus, differentialequations, and calculus-based physics should be adequately preparedto handle the material presented. There is also sufficient advanced ma-terial in each chapter or topic for first-year graduate students in scienceor engineering.
The basic foundations of both (equilibrium) thermodynamics andtransport phenomena lie in the descriptions and treatment of largecollections of interacting molecules and atoms. From this viewpoint,there are only minor differences in fundamentally describing the be-havior of systems at equilibrium versus nonequilibrium, and these dif-ferences are immersed in the mathematical identification of length andtime scales appropriate for any particular system. The importance ofunderstanding the complexities of length and time scales is ubiquitousthroughout science and engineering and is not unique to thermodynam-ics and transport phenomena. As such, this book will provide systematicand methodological examples of the analysis of length and time scalesand its implications for physical behavior.
The book begins by providing the necessary background for themechanics of interacting particles in Chapter 1. Atomic and molecularinteraction forces are also presented in this introductory chapter.Chapter 2 develops the formal and general statistical representationof a collection of interacting particles through the derivation of theLi