Professor Boris V. Alexeev is Head of the Centre of the Theoretical Foundations of Nanotechnology and Head of the Physics Department at the Moscow Lomonosov University of Fine Chemical Technologies, Moscow, Russia. In the 1990s he was Visiting Professor at the University of Alabama, Huntsville, AL, USA, and Visiting Professor at the University of Provence, Marseille, France. Professor Alexeev has published over 290 articles in international scientific journals and 22 books. He has received several honors and awards, and is member of six societies.
Unified Non-Local Theory of Transport Processess, 2nd Edition provides a new theory of transport processes in gases, plasmas and liquids. It is shown that the well-known Boltzmann equation, which is the basis of the classical kinetic theory, is incorrect in the definite sense. Additional terms need to be added leading to a dramatic change in transport theory. The result is a strict theory of turbulence and the possibility to calculate turbulent flows from the first principles of physics. - Fully revised and expanded edition, providing applications in quantum non-local hydrodynamics, quantum solitons in solid matter, and plasmas- Uses generalized Boltzmann kinetic theory as an highly effective tool for solving many physical problems beyond classical physics- Addresses dark matter and energy- Presents non-local physics in many related problems of hydrodynamics, gravity, black holes, nonlinear optics, and applied mathematics
Front Cover 1
Unified Non-Local Theory of Transport Processes: Generalized Boltzmann Physical Kinetics 4
Copyright 5
Contents 6
Preface 10
Historical Introduction and the Problem Formulation 12
Chapter 1: Generalized Boltzmann Equation 28
1.1. Mathematical Introduction-Method of Many Scales 28
1.2. Hierarchy of Bogolubov Kinetic Equations 38
1.3. Derivation of the Generalized Boltzmann Equation 42
1.4. Generalized Boltzmann H-Theorem and the Problem of Irreversibility of Time 58
1.5. Generalized Boltzmann Equation and Iterative Construction of Higher-Order Equations in the Boltzmann Kinetic Theory 70
1.6. Generalized Boltzmann Equation and the Theory of Non-Local Kinetic Equations with Time Delay 73
Chapter 2: Theory of Generalized Hydrodynamic Equations 80
2.1. Transport of Molecular Characteristics 80
2.2. Hydrodynamic Enskog Equations 82
2.3. Transformations of the Generalized Boltzmann Equation 83
2.4. Generalized Continuity Equation 85
2.5. Generalized Momentum Equation for Component 87
2.6. Generalized Energy Equation for Component 90
2.7. Summary of the Generalized Enskog Equations and Derivation of the Generalized Hydrodynamic Euler Equations 94
Chapter 3: Quantum Non-Local Hydrodynamics 104
3.1. Generalized Hydrodynamic Equations and Quantum Mechanics 104
3.2. GHEs, Quantum Hydrodynamics. SE as the Consequence of GHE 109
3.3. SE and its Derivation from Liouville Equation 115
3.4. Direct Experimental Confirmations of the Non-Local Effects 116
Chapter 4: Application of Unified Non-Local Theory to the Calculation of the Electron and Proton Inner Structures 124
4.1. Generalized Quantum Hydrodynamic Equations 124
4.2. The Charge Internal Structure of Electron 127
4.3. The Derivation of the Angle Relaxation Equation 133
4.4. The Mathematical Modeling of the Charge Distribution in Electron and Proton 135
4.5. To the Theory of Proton and Electron as Ball-like Charged Objects 156
Chapter 5: Non-Local Quantum Hydrodynamics in the Theory of Plasmoids and the Atom Structure 158
5.1. The Stationary Single Spherical Plasmoid 158
5.2. Results of the Mathematical Modeling of the Rest Solitons 160
5.3. Nonstationary 1D Generalized Hydrodynamic Equations in the Self-Consistent Electrical Field. Quantization in the Gen ... 168
5.4. Moving Quantum Solitons in Self-Consistent Electric Field 171
5.5. Mathematical Modeling of Moving Solitons 173
5.6. Some Remarks Concerning CPT (Charge-Parity-Time) Principle 183
5.7. About Some Mysterious Events of the Last Hundred Years 189
5.7.1. Tunguska Event (TE) 189
5.7.2. Gagarin and Seryogin Air Crash 191
5.7.3. Accident with Malaysia Airlines Flight MH370 193
Chapter 6: Quantum Solitons in Solid Matter 196
6.1. Quantum Oscillators in the Unified Non-local Theory 196
6.2. Application of Non-Local Quantum Hydrodynamics to the Description of the Charged Density Waves in the Graphene Cryst ... 204
6.3. Generalized Quantum Hydrodynamic Equations Describing the Soliton Movement in the Crystal Lattice 206
6.4. Results of the Mathematical Modeling Without the External Electric Field 216
6.5. Results of the Mathematical Modeling With the External Electric Field 240
6.6. Spin Effects in the Generalized Quantum Hydrodynamic Equations 259
6.7. To the Theory of the SC 266
Chapter 7: Generalized Boltzmann Physical Kinetics in Physics of Plasma 270
7.1. Extension of Generalized Boltzmann Physical Kinetics for the Transport Processes Description in Plasma 270
7.2. Dispersion Equations of Plasma in Generalized Boltzmann Theory 277
7.3. The Generalized Theory of Landau Damping 281
7.4. Evaluation of Landau Integral 283
7.5. Estimation of the Accuracy of Landau Approximation 291
7.6. Alternative Analytical Solutions of the Vlasov-Landau Dispersion Equation 295
7.7. The Generalized Theory of Landau Damping in Collisional Media 302
Chapter 8: Physics of a Weakly Ionized Gas 312
8.1. Charged Particles Relaxation in ``Maxwellian´´ Gas and the Hydrodynamic Aspects of the Theory 312
8.2. Distribution Function (DF) of the Charged Particles in the ``Lorentz´´ Gas 315
8.3. Charged Particles in Alternating Electric Field 326
8.4. Conductivity of a Weakly Ionized Gas in the Crossed Electric and Magnetic Fields 328
8.5. Investigation of the GBE for Electron Energy Distribution in a Constant Electric Field with due Regard for Inelastic ... 333
Chapter 9: Generalized Boltzmann Equation in the Theory of the Rarefied Gases and Liquids 342
9.1. Kinetic Coefficients in the Theory of the Generalized Kinetic Equations. Linearization of the Generalized Boltzmann ... 342
9.2. Approximate Modified Chapman-Enskog Method 348
9.3. Kinetic Coefficient Calculation with Taking into Account the Statistical Fluctuations 356
9.4. Sound Propagation Studied with the Generalized Equations of Fluid Dynamics 360
9.5. Shock Wave Structure Examined with the Generalized Equations of Fluid Dynamics 372
9.6. Boundary Conditions in the Theory of the Generalized Hydrodynamic Equations 374
9.7. To the Kinetic and Hydrodynamic Theory of Liquids 379
Chapter 10: Strict Theory of Turbulence and Some Applications of the Generalized Hydrodynamic Theory 390
10.1. About Principles of Classical Theory of Turbulent Flows 390
10.2. Theory of Turbulence and Generalized Euler Equations 391
10.3. Theory of Turbulence and the Generalized Enskog Equations 400
10.4. Unsteady Flow of a Compressible Gas in a Cavity 404
10.5. Application of the GHE: To the Investigation of Gas Flows in Channels with a Step 414
10.6. Vortex and Turbulent Flow of Viscous Gas in Channel with Flat Plate 422
Chapter 11: Astrophysical Applications 440
11.1. Solution of the Dark Matter Problem in the Frame of the Non-Local Physics 440
11.2. Plasma-Gravitational Analogy in the Generalized Theory of Landau Damping 441
11.3. Disk Galaxy Rotation and the Problem of Dark Matter 452
11.4. Hubble Expansion and the Problem of Dark Energy 465
11.5. Propagation of Plane Gravitational Waves in Vacuum with Cosmic Microwave Background 472
11.6. Application of the Non-Local Physics in the Theory of the Matter Movement in Black Hole 488
11.7. Self-similar Solutions of the Non-local Equations 496
11.7.1. Preliminary Remarks 496
11.7.2. Self-similar Solutions of the Non-Local Equations in the Astrophysical Applications 499
Chapter 12: The Generalized Relativistic Kinetic Hydrodynamic Theory 520
12.1. Hydrodynamic Form of the Dirac Quantum Relativistic Equation 520
12.2. Generalized Relativistic Kinetic Equation 524
12.3. Generalized Enskog Relativistic Hydrodynamic Equations 529
12.3.1. Derivation of the Continuity Equation 529
12.3.2. Derivation of the Motion Equation 531
12.3.3. Derivation of the Energy Equation 533
12.4. Generalized System of the Relativistic Hydrodynamics and Transfer to the Generalized Relativistic non-Local Euler H ... 535
12.5. Generalized non-Local Relativistic Euler Equations 540
12.6. The Limit Transfer to the non-Relativistic Generalized non-Local Euler Equations 543
12.6.1. Some Auxiliary Expressions 543
12.6.2. Non-Relativistic Generalized Euler Equations as Asymptotic of the Relativistic Equations 544
12.7. Expansion of the Flat Harmonic Waves of Small Amplitudes in Ultra-relativistic Media 547
Some remarks to the conclusion of the monograph 558
Appendix 1: Perturbation Method of the Equation Solution Related to T[f] 560
Appendix 2: Using of Curvilinear Coordinates in the Generalized Hydrodynamic Theory 564
Appendix 3: Characteristic Scales in Plasma Physics 584
Appendix 4: Dispersion Relations in the Generalized Boltzmann Kinetic Theory Neglecting the Integral Collision Term 586
Appendix 5: Three-Diagonal Method of Gauss Elimination Techniques for the Differential Third- and Second-Order Equations 588
Appendix 6: Some Integral Calculations in the Generalized Navier-Stokes Approximation 594
Appendix 7: Derivation of Energy Equation for Invariant Ea=maVa22+ea 596
Appendix 8: To the Non-Local Theory of Cold Nuclear Fusion 602
Appendix 9: To the Non-Local Theory of Variable Stars 610
Appendix 10: To the Non-Local Theory of Levitation 620
References 628
Index 636
Generalized Boltzmann Equation
Abstract
In what follows, we intend to construct the generalized Boltzmann physical kinetics using the different methods of the kinetic equation derivations from the Bogolubov hierarchy.
Keywords
Generalized Boltzmann equation
Bogolubov hierarchy
Method of many scales
1.1 Mathematical Introduction—Method of Many Scales
In the sequel asymptotic methods will be used, but first of all, we will look at the method of many scales. The method of many scales is so popular that Nayfeh in his book [68] written more than 40 years ago said that method of many scales (MMS) is discovered by different authors every half a year. As a result, there exist many different variants of MMS. As a minimum four variants of MMS are considered in [68]. We are interested only in the main ideas of MMS, which are used further in theory of kinetic equations.
From this standpoint we demonstrate MMS possibilities using typical example of solution linear differential equation which also has the exact solution for comparing the results [68]. But in contrast with usual consideration, which can be found in literature, we intend to bring this example up to table and a graph.
Therefore let us consider the linear differential equation
x¨+ɛx˙+x=0.
(1.1.1)
We begin with the special case when δ = 1 and is a small parameter. Equation (1.1.1) has the exact solution
=ae−ɛt/2cost1−14ɛ2+ϕ,
(1.1.2)
where a and ϕ are arbitrary constants of integrating. In typical case of small parameter δ in front of senior derivative—in this case it would be ¨—the effects of boundary layer can be observed. Using the derivatives
x˙=−12ɛx−ae−ɛt/21−14ɛ2sint1−14ɛ2+ϕ,x¨=−12ɛx˙−x1−14ɛ2+12aɛe−ɛt/21−14ɛ2sint1−14ɛ2+ϕ,
for substitution in Eq. (1.1.1) we find the identical satisfaction of Eq. (1.1.1).
We begin with a direct expansion in small , using series
=x0+ɛx1+ɛ2x2+⋯,
(1.1.3)
and after differentiating
x˙=x˙0+ɛx˙1+ɛ2x˙2+⋯,x¨=x¨0+ɛx¨1+ɛ2x¨2+⋯.
Substitute series (1.1.3) into (1.1.1) and equalize coefficients in front of equal powers of , having
¨0+x0=0,
(1.1.4)
¨1+x1=−x˙0,
(1.1.5)
¨2+x2=−x˙1,
(1.1.6)
¨3+x3=−x˙2,
(1.1.7)
and so on. The general solution of homogeneous Eq. (1.1.4) has the form
0=acost+ϕ.
(1.1.8)
Substitute (1.1.8) in (1.1.5):
¨1+x1=asint+ϕ.
(1.1.9)
General solution (1.1.1) should contain only two arbitrary constants. In this case, both constants a and ϕ are contained in the main term of expansion defined by relation (1.1.8). Then we need find only particular solution of Eq. (1.1.9); which can be found as follows
1=−12atcost+ϕ.
(1.1.10)
Really,
x˙1=−12acost+ϕ+12atsin(t+ϕ),x¨1=12asint+ϕ−x1+12asin(t+ϕ).
After substitution in (1.1.9), we find identity. Equation (1.1.6) can be rewritten as
¨2+x2=12acost+ϕ−12atsint+ϕ,
(1.1.11)
and its solution
2=18at2cost+ϕ+18atsint+ϕ.
(1.1.12)
Really,
˙2=14atcost+ϕ−18at2sint+ϕ+18asint+ϕ+18atcost+ϕ,
¨2=12acost+ϕ−58atsint+ϕ−18at2cost+ϕ.
Substitution into left-hand side of Eq. (1.1.11), lead to result
2acost+ϕ−58atsin(t+ϕ)−18at2cos(t+ϕ)+18at2cost+ϕ+18atsin(t+ϕ)=12acost+ϕ−12atsin(t+ϕ).
Then we state the identical satisfaction of Eq. (1.1.11) by solution (1.1.12). In analogous way the solution of Eq. (1.1.7) is written as cubic polynomial in t. For the first three terms of Eq. (1.1.3) series the solution is
=acost+ϕ−12ɛatcos(t+ϕ)+18ɛ2at2cost+ϕ+tsint+ϕ+Oɛ3.
(1.1.13)
At our desire the variable t can be considered as dimensionless time. Suppose, of course, that we wish to have a solution for arbitrary time moments. But it is not possible in the developed procedure, because the series construction regards the successive terms of the series to be smaller than the forgoing terms; in other case, it is impossible to speak about series convergence. But for fixed , the time moment can be found when successive term of expansion is no smaller than the forgoing term. Figure 1.1 contains comparison of the exact solution (1.1.2) for concrete parameters of calculations a = 1, ϕ = 0, = 0.2 with approximate solutions
0=costx1=cost−0.1tcostx2=cost−0.1tcost+0.005t2cost+tsintxex=e−0.1tcost0.99.
As we could expect, the divergence of solution 2x and exact solution exx appears when t is of order 10; or, in the common case, if t ~ − 1. But as it follows from Fig. 1.1, the situation is much worse, because, for example, for t = 4π approximate solution 1x gives wrong sign in comparison with the exact solution exx. For the mathematical model of an oscillator with damping—which is reflecting by Eq. (1.1.1), it means that approximate solution 1x forecasts a deviation in the opposite direction for the mentioned oscillator. By the way, solution 1x is also worse in comparison with solution 0x, in this sense the minor approximation is better than senior ones.
This poses the question how to improve the situation remaining in the frame of asymptotic methods. To answer this question, let us consider the exact solution (1.1.2). Exponential and cosine terms containing in this solution, can be expand in the following series for small and fixed t:
−ɛt/2=1−12ɛt+18ɛt2−164ɛt3+⋯
(1.1.14)
cost1−14ɛ2+ϕ=cost1−18ɛ2−1128ɛ4−11024ɛ6−⋯+ϕ=cost+ϕ−18tɛ2−1128tɛ4−11024tɛ6−⋯≅cost+ϕ+18tɛ2+1128tɛ4+11024tɛ6+⋯sin(t+ϕ)=cost+ϕ+18tɛ2sin(t+ϕ)+1128tɛ4sin(t+ϕ)+⋯.
(1.1.15)
Obviously, product of the first terms in expansions (1.1.14) and (1.1.15) gives 0x, and retaining of terms of O(3) lead to result
≅a1−12ɛt+18ɛ2t2cost+ϕ+18tɛ2sint+ϕ≅acost+ϕ−12aɛtcos(t+ϕ)+18atɛ2sin(t+ϕ)+18aɛ2t2cos(t+ϕ).
Then we state that the used construction of asymptotic solution is based in deed on the assumption that combination ɛt is small. If it is not so (for t having the order − 1), then expansions (1.1.14) and (1.1.15) are wrong or need to take into account all terms of expansions. But asymptotic expansion can be organized by another way, using additional variables:
1=ɛt
(1.1.16)
and
2=ɛ2t.
(1.1.17)
In this case
−ɛt/2=e−T1/2,
(1.1.18)
and expansion (1.1.15) is replaced by other...
| Erscheint lt. Verlag | 10.2.2015 |
|---|---|
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Physik / Astronomie ► Mechanik |
| Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik | |
| Technik ► Maschinenbau | |
| ISBN-10 | 0-444-63487-8 / 0444634878 |
| ISBN-13 | 978-0-444-63487-0 / 9780444634870 |
| Haben Sie eine Frage zum Produkt? |
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