Preface ......................................................... 5
Introduction ................................................... 10
Chapter 1
System of Hard Spheres ......................................... 27
1.1. Introduction .............................................. 27
1.2. Hamiltonian Dynamics of a System of Hard Spheres .......... 27
1.2.1. Hamilton Equations ................................. 27
1.2.2. Existence of Trajectories .......................... 30
1.2.3. Liouville Theorem .................................. 32
1.3. Evolution Operator for a System of Hard Spheres ........... 33
1.3.1. Definition of the Evolution Operator ............... 33
1.3.2. Liouville Equation ................................. 36
1.3.3. Evolution Operator and Liouville Equation for
Negative Time ...................................... 39
1.4. BBGKY Hierarchy for Systems of Hard Spheres ............... 40
1.4.1. Definition of Correlation Functions ................ 40
1.4.2. Derivation of Hierarchy of Equations for
Correlation Functions .............................. 44
1.4.3. Solution ofthe BBGKY Hierarchy in the Space of
Summable Functions ................................. 46
1.4.4. Solution ofthe BBGKY Hierarchy ..................... 48
1.4.5. BBGKY Hierarchy with Nonstandard Normalization ..... 52
1.5. Justification ofthe Boltzmann-Grad Limit .................. 54
1.5.1. Definition ofthe Boltzmann-Grad Limit .............. 54
1.5.2. Auxiliary Lemmas ................................... 58
1.5.3. Convergence of Solutions of the BBGKY Hierarchy
of a System of Hard Spheres to Solutions of
the Ordinary Boltzmann Hierarchy in
the Boltzmann-Grad Limit ........................... 59
1.5.4. Convergence of Solutions of the BBGKY Hierarchy
of Systems of Hard Spheres to Solutions of
the Proper Stochastic Hierarchy in
the Boltzmann-Grad Limit ........................... 64
Chapter 2
Stochastic Dynamics as the Limit of the Hamiltonian Dynamics
of Hard Spheres ................................................ 67
2.1. Introduction .............................................. 67
2.2. Stochastic Trajectories as the Limit of the Hamiltonian
Trajectories of Hard Spheres .............................. 68
2.2.1. Hamiltonian Trajectories of Hard Spheres ........... 68
2.2.2. Stochastic Trajectories ............................ 70
2.2.3. Convergence of Hamiltonian Trajectories to
Stochastic Trajectories ............................ 73
2.3. New Representation of Hamiltonian and Stochastic
Trajectories .............................................. 77
2.3.1. Representation of Hamiltonian Trajectories ......... 77
2.3.2. Representation of Stochastic Trajectories .......... 79
2.4. Functional for a System of Two Hard Spheres ............... 82
2.4.1. Domain of Interaction and Functional ............... 82
2.4.2. Derivative of Functional ........................... 86
2.5. Functional for a System of Two Stochastic Particles ....... 88
2.5.1. Functional of Stochastic Particles as the Limit
of the Functional of Hard Spheres .................. 88
2.5.2. Derivative of Functional with Respect to Time ...... 94
2.6. General Case of Many-Particle System ...................... 97
2.6.1. Functional for Many Hard Spheres ................... 97
2.6.2. Derivative of Functional with Respect to Time ..... 100
2.6.3. Limit of the Average of the Functional for Hard
Spheres and the Functional of Stochastic
Particles ......................................... 101
2.7. Infinitesimal Operator of the Evolution Operator of
Stochastic Particles ..................................... 108
2.7.1. Dynamics of Finitely Many Particles ............... 108
2.7.2. Evolution Operator of Finitely Many Particles
and Its Infinitesimal Operator .................... 112
2.7.3. Evolution Operator for Negative Time .............. 119
2.7.4. Equivalence of the Infinitesimal Operators ........ 124
Chapter 3
Stochastic Boltzmann Hierarchy ................................ 129
3.1. Introduction ............................................. 129
3.2. Average of Observables over State ........................ 131
3.2.1. Stochastic Dynamics ............................... 131
3.2.2. Average for Infinitesimal Time .................... 132
3.2.3. Infinitesimal Operator ............................ 136
3.2.4. Infinitesimal Operator with Fixed Random
Vectors ........................................... 139
3.2.5. Duality Principle ................................. 143
3.2.6. Generalized Function .............................. 148
3.3. Hierarchy for Correlation Functions ...................... 150
3.3.1. Derivation of Hierarchy from Equation for
Distribution Function ............................. 150
3.3.2. Stochastic Hierarchy in Grand Canonical
Ensemble .......................................... 158
3.3.3. Duality Principle for Correlation Functions ....... 160
3.4. Derivation of Hierarchy from Functional Average .......... 162
3.4.1. Functional-Average for s-Particle Observable ...... 162
3.4.2. Derivation of the Stochastic Boltzmann Hierarchy
from the Ito-Liouville Equation ................... 167
3.4.3. Derivation of Ordinary Boltzmann Hierarchy from
Boltzmann Equation ................................ 170
3.5. Derivation of Stochastic Boltzmann Hierarchy from
the BBGKY Hierarchy for Hard Spheres ..................... 173
3.5.1. Stochastic Boltzmann Hierarchy .................... 173
3.5.2. Solutions of the Ordinary Boltzmann Hierarchy
and the Boltzmann-Grad Limit of Solutions of
the BBGKY Hierarchy ............................... 177
3.5.3. Derivation of the Stochastic Boltzmann Hierarchy
from the Evolution Operator of the BBGKY
Hierarchy for Hard Spheres ........................ 179
3.5.4. Functional for Correlation Functions .............. 183
3.5.5. Stochastic Boltzmann Hierarchy .................... 186
3.5.6. Different Representations of the Infinitesimal
Operator .......................................... 189
3.5.7. Different Equivalent Forms of the Stochastic
Boltzmann Hierarchy ............................... 192
3.6. Boltzmann Equation and Its Solutions in Terms of
Stochastic Dynamics ...................................... 193
3.6.1. Iterations of the Boltzmann Equation .............. 193
3.6.2. Iterations of the Boltzmann Hierarchies ........... 199
Chapter 4
Solutions of the Stochastic Boltzmann Hierarchy ............... 202
4.1. Introduction ............................................. 202
4.2. Solutions of the Stochastic Hierarchy in the Space of
Bounded Functions ........................................ 203
4.2.1. Abstract Form of the Stochastic Hierarchy ......... 203
4.2.2. Convergence of Series (4.2.8) in the Space Eξ,β .... 205
4.2.3. One Auxiliary Lemma ............................... 209
4.2.4. Convergence of Series (4.2.8) in the Space Eξ,β .... 211
4.3. Chaos Property of Solutions of the Stochastic
Hierarchy ................................................ 215
4.3.1. New Representation of the Series of Iterations .... 215
4.3.2. Chaos Property .................................... 218
4.3.3. Justification of the Thermodynamic Limit .......... 220
4.3.4. New Representation of the Series of Iterations .... 221
Chapter 5
Spatially Homogeneous Boltzmann Hierarchy ..................... 224
5.1. Introduction ............................................. 224
5.2. Stochastic Dynamics for the Spatially Homogeneous
Stochastic Boltzmann Hierarchy ........................... 228
5.2.1. System of N Particles ............................. 228
5.2.2. Equation for Spatially Homogeneous Distribution
Functions ......................................... 233
5.3. Derivation of the Spatially Homogeneous Hierarchy ........ 235
5.3.1. Spatially Homogeneous Hierarchy within
the Framework of Canonical and Grand Canonical
Ensemble .......................................... 235
5.3.2. Hierarchy with Fixed Random Vectors ............... 239
5.4. Representation of Solutions of the Spatially
Homogeneous Hierarchy ............................. 241
5.4.1. Representation of Solutions of the Spatially
Homogeneous Hierarchy through Series of
Iterations ........................................ 241
5.4.2. One-Particle Distribution Function as a Solution
of the Boltzmann Equation ......................... 247
Chapter 6
Stochastic Dynamics for the Boltzmann Equation with
Arbitrary Differential Scattering Cross Section ............... 253
6.1. Introduction ............................................. 253
6.2. Stochastic Dynamics ...................................... 256
6.2.1. Functional Average ................................ 256
6.2.2. Infinitesimal Operator ............................ 261
6.2.3. Infinitesimal Operator with Fixed Random
Vectors ........................................... 264
6.2.4. Duality Principle ................................. 267
6.3. Hierarchy for Correlation Functions ...................... 274
6.3.1. Derivation of Hierarchy from Equation for
Distribution Function ............................. 274
6.3.2. Derivation of Hierarchy from Functional Average ... 279
6.4. Solutions of the Stochastic Hierarchy .................... 284
6.4.1. Abstract Form of the Stochastic Hierarchy ......... 284
6.4.2. Chaos Property .................................... 286
6.4.3. Spatially Homogeneous Initial Data ................ 289
6.5. Stochastic Process in Momentum Space ..................... 290
6.5.1. Averaging Procedure in Spatially Homogeneous
Case .............................................. 290
6.5.2. Differential Equation for Spatially Homogeneous
Distribution Functions ............................ 293
6.5.3. Hierarchy for Correlation Functions in
Mean-Field Approximation .......................... 294
Chapter 7
Analog of Liouville Equation and BBGKY Hierarchy for
a System of Hard Spheres with Inelastic Collisions ............ 297
7.1. Introduction ............................................. 297
7.2. Trajectories of a System of Hard Spheres with Inelastic
Collisions ............................................... 299
7.2.1. Dynamics .......................................... 299
7.2.2. Trajectory ........................................ 302
7.3. Evolution Operator ....................................... 305
7.3.1. Definition of Evolution Operator .................. 305
7.3.2. Properties of Evolution Operator .................. 306
7.3.3. Differential Equation for Distribution Function ... 310
7.4. Equation for a Sequence of Correlation Functions ......... 318
7.4.1. Definition of Correlation Functions ............... 318
7.4.2. Equation for Correlation Functions ................ 319
7.4.3. Boundary Conditions for Correlation Functions ..... 325
7.4.4. Grand Canonical Ensemble .......................... 329
Appendix A .................................................... 330
Appendix B .................................................... 333
Chapter 8
Solution of the BBGKY Hierarchy for a System of Hard Spheres
with Inelastic Collisions ..................................... 335
8.1. Introduction ............................................. 335
8.2. Solution of Hierarchy for Correlation Functions .......... 338
8.2.1. Solution Formula .................................. 338
8.2.2. Convergence of Series ............................. 342
8.2.3. Group Property .................................... 343
8.2.4. Strong Continuity of the Group .................... 345
8.3. Infinitesimal Generator of the Group and a Solution of
the BBGKY Hierarchy ...................................... 346
8.3.1. Infinitesimal Generator ........................... 346
8.3.2. Existence of Solutions of the BBGKY Hierarchy ..... 348
8.3.3. States of Infinite Systems ........................ 349
8.4. Stochastic Boltzmann Hierarchy for Granular Flow ......... 350
8.4.1. Stochastic Dynamics for Hard Spheres with
Inelastic Collisions .............................. 350
8.4.2. Stochastic Trajectories and Operator of
Evolution ......................................... 351
8.4.3. Functional Average ................................ 353
8.4.4. Hierarchy for Correlation Functions ............... 355
8.4.5. Solution of the Stochastic Boltzmann Hierarchy .... 356
8.4.6. Ordinary Boltzmann Hierarchy ...................... 358
References .................................................... 362
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