A non-equilibrum statistical mechanics

✍ Scribed by Chen T.Q.

• Publisher: WS
• Year: 2003
• Tongue: English
• Leaves: 438
• Category: Library

✦ Synopsis

This work presents the construction of an asymptotic technique for solving the Liouville equation, which is an analogue of the Enskog-Chapman technique for the Boltzmann equation. Because the assumption of molecular chaos has not been introduced, the macroscopic variables defined by the arithmetic means of the corresponding microscopic variables are random in general. Therefore, it is convenient for describing the turbulence phenomena. The asymptotic technique for the Liouville equation reveals a term showing the interaction between the temperature and the velocity of the fluid flows, which will be lost under the assumption of molecular chaos.

✦ Table of Contents

Foreword......Page 8
Preface......Page 10
Contents......Page 14
1.1 Historical Background......Page 18
1.2 Outline of the Book......Page 28
2.1 Hydrodynamic Random Fields......Page 44
2.2 H-Functional......Page 47
3.1 Derivation of H-Functional Equation......Page 50
3.2 H-Functional Equation......Page 68
3.3 Balance Equations......Page 71
3.4 Reformulation......Page 81
4.1 Definition of K-Functional......Page 86
5.1 Some Useful Formulas......Page 92
5.2 A Remark on H-Functional Equation......Page 95
6.1 Asymptotic Analysis for Liouville Equation......Page 98
6.2 Turbulent Gibbs Distributions......Page 110
6.3 Gibbs Mean......Page 126
7.1 Expressions of B2 and B3......Page 136
7.2 Euler K-Functional Equation......Page 153
7.3 Reformulation......Page 157
7.4 Special Cases......Page 161
7.5 Case of Deterministic Flows......Page 165
8.1 K-Functionals and Turbulent Gibbs Distributions......Page 174
8.2 Turbulent Gibbs Measures......Page 181
8.3 Asymptotic Analysis......Page 186
9.1 Gross Determinism......Page 192
9.2 Temporal Part of Material Derivative of TN......Page 201
9.3 Spatial Part of Material Derivative of TN......Page 236
9.4 Stationary Local Liouville equation......Page 242
10.1 Case of Reynolds-Gibbs Distributions......Page 244
10.2 A Poly-spherical Coordinate System......Page 251
10.3 A Solution to the Equation (10.24)1......Page 255
10.4 A Solution to the Equation (10.24)2......Page 270
10.5 A Solution to the Equation (10.24)3......Page 271
10.6 A Solution to the Equation (10.24)4......Page 276
10.7 A Solution to the Equation (10.24)5......Page 277
10.8 A Solution to the Equation (10.24)6......Page 278
10.9 Equipartition of Energy......Page 280
11.1 The Expression of B2......Page 288
11.2 The Contribution of G1 to B2......Page 290
11.3 The Contribution of G2 to B2......Page 308
11.4 The Contribution of G6 to B2......Page 311
11.5 The Expression of B3......Page 313
11.6 The Contribution of G1 to B3......Page 315
11.7 The Contribution of G2 to B3......Page 318
11.8 The Contribution of G6 to B3......Page 320
11.9 The Contribution of G3 to B3......Page 322
11.10 The Contribution of G4 to B3......Page 335
11.11 The Contribution of G5 to B3......Page 345
11.12 A Finer K-Functional Equation......Page 353
12.1 A View on Turbulence......Page 356
12.2 Features of the Finer K-Functional Equation......Page 359
12.3 Justification of the Finer K-Functional Equation......Page 360
12.4 Open Problems......Page 362
A.1 Higher Dimensional Spherical Harmonics......Page 364
A.2 A List of Spherical Harmonics......Page 366
A.3 Products of Some Spherical Harmonics......Page 386
A.4 Derivatives of Some Spherical Harmonics......Page 419
Bibliography......Page 424
Index......Page 432