Equidistribution of Dynamical Systems: Time-Quantitative Second Law

Description

We know very little about the time\-evolution of many\-particle dynamical systems, the subject of our book. Even the 3\-body problem has no explicit solution (we cannot solve the corresponding system of differential equations, and computer simulation indicates hopelessly chaotic behaviour). For example, what can we say about the typical time evolution of a large system starting from a stage far from equilibrium? What happens in a realistic time scale? The reader\x27s first reaction is probably: What about the famous Second Law (of thermodynamics)? Unfortunately, there are plenty of notorious mathematical problems surrounding the Second Law. (1) How to rigorously define entropy? How to convert the well known intuitions (like \x26#34;\x26#34;disorder\x26#34;\x26#34; and \x26#34;\x26#34;energy spreading\x26#34;\x26#34;) into precise mathematical definitions? (2) How to express the Second Law in forms of a rigorous mathematical theorem? (3) The Second Law is a \x26#34;\x26#34;soft\x26#34;\x26#34; qualitative statement about entropy increase, but does not say anything about the necessary time to reach equilibrium. The object of this book is to answer questions (1)\-(2)\-(3). We rigorously prove a Time\-Quantitative Second Law that works on a realistic time scale. As a by product, we clarify the Loschmidt\-paradox and the related reversibility\/irreversibility paradox. Contents Preface Chapter 1. Formulating a Time-Quantitative Second Law for Large Systems 1. Defining our implicit interaction models; Unrealistic versus realistic time scale 2. The first step is the hardest: How to define a quantitative form of microscopic equilibrium in large systems? 3. First surprise: Shockingly fast approach to micro-equilibrium in the Gaussian case 4. Second surprise: Shockingly slow approach to micro-equilibrium in the “photon-like” constant speed case 5. Estimating a crucial “variance”, and more on starting from Big Bang 6. Micro-Entropy and a time-quantitative Second Law: A preview Chapter 2. Starting the Proofs: Applying Fourier Analysis 7. Proof of Theorem 3.1 8. Proving Theorems 4.1–2 and Lemma 5.1 9. Proof of Theorem 5.1 10. Proving Theorem 3.2 and the Quick-Jump-Up Phenomenon Chapter 3. Proving Our Time-Quantitative Second Law 11. First step: Relative disparity — a Boltzmann entropy like quantity 12. Does the disparity decrease? 13. Proving a Second Law (I): First kind and second kind 14. Proving a Second Law (II): Second kind 15. Illustrations of the Second Law (I): Third kind 16. Illustrations of the Second Law (II): First kind and second kind 17. Illustrations of the Second Law (III): More on the second kind, and the Paradoxes Chapter 4. More on the Second Law 18. The case of general speed distribution (I) 19. The case of general speed distribution (II): A general Second Law 20. Returning to the torus-via-unfolding model 21. Billiards in other shapes 22. Equivalent dynamical systems (I) 23. Equivalent dynamical systems (II) 24. Extensions of the Second Law Chapter 5. Long-Term Stability of Equilibrium 25. Simultaneous box equilibrium 26. Starting the proof of Theorem 25.1 28. Proof of Theorem 26.1 (II) 29. Stability beyond the Gaussian case (I) 30. Stability beyond the Gaussian case (II) Appendix 1 The Second Law in Physics Appendix 2 Proving Lemmas 1.1–2 and Beyond References Index