New Perspectives in Thermodynamics

ISBN-13: 9783540159315 / Angielski / Miękka / 1986 / 262 str.

James Serrin

Opis

The material included in this book was first presented in a series of lectures de livered at the University of Minnesota in June 1983 in connection with the con ference "Thermodynamics and Phase Transitions." This conference was one of the principal events in the first year of operation of the Institute for Mathematics and its Applications (lMA) at the University of Minnesota. The Institute was founded under the auspices of the National Science Foun dation of the United States and the University of Minnesota and is devoted to strengthening and fostering the relation of mathematics with its various applica tions to problems of the real world. The present volume constitutes an important element in the continuing pub lication program of the Ipstitute. Previous publications in this program have ap peared as lecture notes in the well-known Springer series, and future ones will be part of a new series "IMA Volumes in Applied Mathematics." Preface Until recently it was believed that thermodynamics could be given a rigorous foundation only in certain restricted circumstances, particularly those involving reversible and quasi-static processes. More general situations, commonly arising in continuum theories, have therefore been treated on the assumption that inter nal energy, entropy and absolute temperature are a priori given quantities, or have been dealt with on a more or less ad hoc basis, with emphasis for example on various types of variational formulations and maximization rules.

Spis treści

I Foundations of Thermodynamics.- 1. An Outline of Thermodynamical Structure.- 1.1 Introduction.- 1.2 Classical Thermal Structure.- 1.3 The Formal Structure of Thermodynamics.- 1.4 The First Law.- 1.5 The Second Law.- 1.6 State Structure and Potentials.- 1.7 Reversible Processes.- 1.8 Special Systems.- 1.9 Concluding Remarks.- References.- 2. Foundations of Continuum Thermodynamics.- 2.1 Introduction.- 2.2 Work, Heat, and Empirical Temperature.- 2.3 The First Law of Thermodynamics.- 2.4 The Second Law of Thermodynamics.- References.- 3. Foundations of the Clausius-Duhem Inequality.- 3.1 Introduction.- 3.2 Thermodynamical Theories.- 3.3 The Existence of Specific Entropy Functions and Thermodynamic Temperature Scales.- 3.4 Properties of the Set of Clausius-Duhem Temperatures Scales.- 3.5 Properties of the Set of Specific Entropy Functions.- 3.6 Concluding Remark.- References.- 4. Recent Research on the Foundations of Thermodynamics.- References.- 5. A Third Line of Argument in Thermodynamics.- List of Sources.- 6. The Laws of Thermodynamics for Non-Cyclic Processes.- 6.1 Basic Definitions.- 6.2 The First Law and Energy.- 6.3 The Second Law and Entropy.- 6.4 Deterministic State Structures.- References.- II The Thermodynamics of Gibbs and Carathéodory.- What Did Gibbs and Carathéodory Leave Us About Thermodynamics?.- Apology.- 7.1 The Words.- 7.2 Statics and Dynamics: the Catenary.- 7.3 The Thermostatics of Gibbs.- 7.4 Gibbs on Thermodynamics.- 7.5 The Thermodynamics of Planck.- 7.6 Gibbs’s Rational Foundations of Thermodynamics: Gibbsian Statistical Mechanics.- 7.7 Bryan’s Rational Thermodynamics.- 7.8 Carathéodory’s Axioms.- 7.9 Carathéodory’s Legacy.- 7.10 Colophon.- 8. Structure and Dynamical Stability of Gibbsian States.- 8.1 Introduction.- 8.2 Preliminaries and Motivation.- 8.3 Gibbsian States: Necessary Conditions and Comparison Principles.- 8.4 Gibbsian States: Sufficient Conditions and Structure.- 8.5 Equivalent Problems of Thermostatics.- 8.6 Dynamical Stability.- References.- 9. Genericity and Gibbs’s Conjecture on the Maximum Number of Coexistent Phases.- 9.1 Gibbs’s Conjecture and the First Phase Rule.- 9.2 Counterexamples.- 9.3 Naive Reformulation of the Gibbs Conjecture for Substances.- 9.4 The Set of Gibbs Surfaces G.- 9.5 Strong and Weak Topologies on G.- 9.6 The Criterion.- 9.7 The Rule-Abiding Gibbs Surfaces. Denseness.- 9.8 Difficulty and Refinement of the Naive Reformulation.- 9.9 The Remaining Paradox.- References.- III Special Material Systems.- 10. Thermodynamics and the Constitutive Relations for Second Sound in Crystals.- Summary.- 10.1 Introduction.- 10.2 Derivation of Thermodynamical Relations.- References.- 11. Interstitial Working and a Nonclassical Continuum Thermodynamics.- 11.1 Introduction.- 11.2 Classical Continuum Thermodynamics: A Limitation.- 11.3 A Nonclassical Continuum Thermodynamics: Interstitial Work Flux.- 11.4 Forms and Effects of the Interstitial Work Flux.- 11.5 Materials of Korteweg Type.- 11.6 An Application: Rules Like Maxwell’s.- References.- 12. Phase Transformations and Non-Elliptic Free Energy Functions.- 12.1 Introduction.- 12.2 Kinematics of Co-Existence.- 12.3 Non-Elliptic Free Energy Functions for Materials Which Change Phase.- 12.4 Significance of Points of Convexity of the Free Energy.- 12.5 Geometry of the Domain of the Free Energy.- 12.6 Special Analysis for the Case of a Cubic Parent Phase.- References.- 13. Dynamic Changes of Phase in a van der Waals Fluid.- 13.1 Introduction.- 13.2 Basic Equations.- 13.3 The Hugoniot Curve.- 13.4 Existence of Compressive Shock Layers.- References.