Table of Contents
Preface v
Chapter 1 From Uniform Distribution to the Time-Evolution of Large Off-Equilibrium Systems 1
1 Traditional Uniform Distribution and Weyl's Criterion 1
2 Strong Uniformity 11
3 High-Dimensional Configuration Space of Large Systems and Unrealistic Time Scale 22
4 Dimension-Free Strong Uniformity on a Realistic Time Scale 39
5 Rapid Approach and Long-Term Stability of Square-Root Equilibrium 45
6 Non-ergodic Time-flow: Closed Orbit Spherical Systems 56
7 Closed Orbit Polar Systems 69
8 Snapshot Randomness (I): Poisson 84
9 Proofs of Theorems 4.2 and 4.3 96
Chapter 2 General Models 119
10 General Model: Unique Ergodicity via Typical Rotations 119
11 Asymptotic Time-Lapse Randomness 137
12 Short-Term Time-Lapse Randomness: Multiple Mixedupness (I) 149
13 Extensions of Theorem 4.2 beyond the Gaussian Case 170
14 Extensions of Theorem 4.2 to Nonlinear Curves on the Plane 188
Chapter 3 More Applications of Theorem 4.2 205
15 Snapshot Randomness (II): Central Limit Theorem 205
16 Snapshot Randomness (III) Case of Closed Orbits 216
17 Time-Lapse Randomness vs. Snapshot Randomness (I): A Fundamental Difference 231
18 Time-Lapse Randomness vs. Snapshot Randomness (II): A Fundamental Difference 240
19 CLT Time-Lapse Randomness: Upper Bound 246
Chapter 4 More Results about Randomness and Stability in Equilibrium 259
20 Simultaneous Square-Root Equilibrium Relative to Nice Sets (I) 259
21 Simultaneous Square-Root Equilibrium Relative to Nice Sets (II) 271
22 Simultaneous Square-Root Equilibrium Relative to Nice Sets (III) 283
23 On the Square-Root Logarithmic Threshold in the Gaussian Case 295
24 Beyond the Applications of Theorem 4.2 306
25 The Case of Singular Underlying Measure 316
Chapter 5 More Proofs 325
26 Proof of Theorem 4.1 325
27 Starting the Proofs of Theorems 13.1-13.4 335
28 Completing the Proof of Lemma 27.2 348
29 Finishing the Proofs of Theorems 13.1-13.4 359
30 Starting the Proof of Theorem 14.1 367
31 Finishing the Proof of Theorem 14.1 381
32 Proof of Theorem 14.2 392
33 Multiple Mixedupness (II): Proof of Lemma 12.2 405
34 Multiple Mixedupness (III): Proof of Theorem 12.2 420
References 437
Index 439
