Kinetic Theory

by Liboff, Richard L.

• ISBN13: 9780387955513
• ISBN10: 0387955518
• Edition: 3rd
• Format: Hardcover
• Copyright: 2003-09-01
• Publisher: Springer Verlag

Summary

Kinetic Theory: Classical, Quantum, and Relativistic Descriptions goes beyond the scope of other works in the field with its thorough treatment of applications in a wide variety of disciplines. Its clear exposition and emphasis on concrete examples will make it not only an excellent graduate text but also a valuable resource for researchers in such disciplines as aerospace, mechanical, and chemical engineering; astrophysics, solid state and laser physics and devices, plasma physics, and controlled and thermonuclear fusion.Among the topics covered are:- The Liouville equation and analyses of the Liouville equation, including two independent derivations - The Boltzmann equation and Boltzmann's H-theorem - Analysis of the linearized collision operator- Fluid dynamics and irreversibility- Assorted kinetic equations with applications to plasmas and neutral fluids- Elements of quantum kinetic theory, including the Green's-function formalism and the Wigner-Moyal equation- Relativistic kinetic theory and Lorentz invariants- Kinetic properties of metals and amorphous media- Monte-Carlo analysis in kinetic theory- Kinetic study of shock wavesThis third revised edition features a new section on constants of motion and symmetry and a new appendix on the Lorentz-Legendre expansion.Each chapter concludes with a variety of problems, many of which provide self-contained descriptions of related topics; lists of such "topical problems" are included in the Table of Contents. Numerous appendices supply vector formulas and tensor notation, properties of special functions, physical constants, references, and a historical time chart.

Table of Contents

Preface to the Third Edition	ix
1 The Liouville Equation	1	(76)
1.1 Elements of Classical Mechanics	2	(9)
1.1.1 Generalized Coordinates and the Lagrangian	2	(2)
1.1.2 Hamilton's Equations	4	(2)
1.1.3 Constants of the Motion	6	(1)
1.1.4 F-Space	7	(2)
1.1.5 Dynamic Reversibility	9	(1)
1.1.6 Equation of Motion for Dynamical Variables	10	(1)
1.2 Canonical Transformations	11	(6)
1.2.1 Generating Functions	11	(3)
1.2.2 Generating Other Transformations	14	(1)
1.2.3 Canonical Invariants	15	(1)
1.2.4 Group Property of Canonical Transformations	15	(1)
1.2.5 Constants of Motion and Symmetry	16	(1)
1.3 Liouville Theorem	17	(3)
1.3.1 Proof	17	(1)
1.3.2 Geometric Significance	18	(1)
1.3.3 Action Generates the Motion	19	(1)
1.4 Liouville Equation	20	(9)
1.4.1 The Ensemble: Density in Phase Space	20	(1)
1.4.2 First Interpretation of D(q, p, t)	20	(2)
1.4.3 Most General Solution: Second Interpretation of D(q, p, t)	22	(1)
1.4.4 Incompressible Ensemble	23	(1)
1.4.5 Method of Characteristics	24	(1)
1.4.6 Solutions to the Initial-Value Problem	25	(2)
1.4.7 Liouville Operator	27	(2)
1.5 Eigenfunction Expansions and the Resolvent	29	(6)
1.5.1 Liouville Equation Integrating Factor	29	(1)
1.5.2 Example: The Ideal Gas	30	(2)
1.5.3 Free-Particle Propagator	32	(1)
1.5.4 The Resolvent	33	(2)
1.6 Distribution Functions	35	(6)
1.6.1 Third Interpretation of D(q, p, t)	35	(1)
1.6.2 Joint-Probability Distribution	36	(1)
1.6.3 Reduced Distributions	37	(1)
1.6.4 Conditional Distribution	37	(1)
1.6.5 s-Tuple Distribution	38	(1)
1.6.6 Symmetric Properties of Distributions	39	(2)
1.7 Markov Process	41	(6)
1.7.1 Two-Time Distributions	41	(1)
1.7.2 Chapman-Kolmogorov Equation	41	(2)
1.7.3 Homogeneous Processes in Time	43	(1)
1.7.4 Master Equation	43	(1)
1.7.5 Application to Random Walk	44	(3)
1.8 Central-Limit Theorem	47	(13)
1.8.1 Random Variables and the Characteristic Function	47	(1)
1.8.2 Expectation, Variance, and the Characteristic Function	47	(2)
1.8.3 Sums of Random Variables	49	(1)
1.8.4 Application to Random Walk	50	(4)
1.8.5 Large n Limit: Central-Limit Theorem	54	(1)
1.8.6 Random Walk in Large n Limit	55	(2)
1.8.7 Poisson and Gaussian Distributions	57	(2)
1.8.8 Covariance and Autocorrelation Function	59	(1)
Problems	60	(17)
2 Analyses of the Liouville Equation	77	(60)
2.1 BBKGY Hierarchy	78	(8)
2.1.1 Liouville, Kinetic Energy, and Remainder Operators	78	(2)
2.1.2 Reduction of the Liouville Equation	80	(1)
2.1.3 Further Symmetry Reductions	81	(2)
2.1.4 Conservation of Energy from BY1 and BY2	83	(3)
2.2 Correlation Expansions: The Vlasov Limit	86	(11)
2.2.1 Nondimensionalization	86	(3)
2.2.2 Correlation Functions	89	(1)
2.2.3 The Vlasov Limit	90	(2)
2.2.4 The Vlasov Equation: Self-Consistent Solution	92	(3)
2.2.5 Debye Distance and the Vlasov Limit	95	(1)
2.2.6 Radial Distribution Function	96	(1)
2.3 Diagrams: Prigogine Analysis	97	(18)
2.3.1 Perturbation Liouville Operator	98	(1)
2.3.2 Generalized Fourier Series	99	(1)
2.3.3 Interpretation of al Coefficients	99	(3)
2.3.4 Equations of Motion for a(k) Coefficients	102	(1)
2.3.5 Selection Rules for Matrix Elements	103	(1)
2.3.6 Properties of Diagrams	104	(3)
2.3.7 Long-Time Diagrams: The Boltzmann Equation	107	(1)
2.3.8 Reduction of Time Integrals	108	(5)
2.3.9 Application of Diagrams to Plasmas	113	(2)
2.4 Bogoliubov Hypothesis	115	(9)
2.4.1 Time and Length Intervals	115	(1)
2.4.2 The Three Temporal Stages	115	(2)
2.4.3 Bogoliubov Distributions	117	(1)
2.4.4 Density Expansion	117	(2)
2.4.5 Construction of F2(0)	119	(2)
2.4.6 Derivation of the Boltzmann Equations	121	(3)
2.5 Klimontovich Picture	124	(3)
2.5.1 Phase Densities	125	(1)
2.5.2 Phase Density Averages	125	(1)
2.5.3 Relation to Correlation Functions	126	(1)
2.5.4 Equation of Motion	127	(1)
2.6 Grad's Analysis	127	(5)
2.6.1 Liouville Equation Revisited	127	(1)
2.6.2 Truncated Distributions	128	(1)
2.6.3 Grand's First and Second Equations	129	(1)
2.6.4 The Boltzmann Equation	130	(2)
Problems	132	(5)
3 The Boltzmann Equation, Fluid Dynamics, and Irreversibility	137	(141)
3.1 Scattering Concepts	138	(11)
3.1.1 Separation of the Hamiltonian	138	(2)
3.1.2 Scattering Angle	140	(3)
3.1.3 Cross Section	143	(3)
3.1.4 Kinematics	146	(3)
3.2 The Boltzmann Equation	149	(6)
3.2.1 Collisional Parameters and Derivation	149	(4)
3.2.2 Multicomponent Gas	153	(1)
3.2.3 Representation of Collision Integral for Rigid Spheres	153	(2)
3.3 Fluid Dynamic Equations and the Boltzmann H Theorem	155	(19)
3.3.1 Collisional Invariants	155	(2)
3.3.2 Macroscopic Variables and Conservative Equations	157	(2)
3.3.3 Conservation Equations and the Boltzmann Equation	159	(2)
3.3.4 Temperance: Variance of the Velocity Distribution	161	(1)
3.3.5 Irreversibility	162	(1)
3.3.6 Poincaré Recurrence Theorem	163	(1)
3.3.7 Boltzmann and Gibbs Entropies	164	(2)
3.3.8 Boltzmann's H Theorem	166	(1)
3.3.9 Statistical Balance	167	(1)
3.3.10 The Maxwellian	168	(2)
3.3.11 The Barometer Formula	170	(2)
3.3.12 Central-Limit Theorem Revisited	172	(2)
3.4 Transport Coefficients	174	(20)
3.4.1 Response to Gradient Perturbations	174	(4)
3.4.2 Elementary Mean-Free-Path Estimates	178	(8)
3.4.3 Diffusion and Random Walk	186	(2)
3.4.4 Autocorrelation Functions, Transport Coefficients, and Kubo Formula	188	(6)
3.5 The Chapman-Enskog Expansion	194	(23)
3.5.1 Collision Frequency	194	(1)
3.5.2 The Expansion	195	(4)
3.5.3 Second-Order Solution	199	(3)
3.5.4 Thermal Conductivity and Stress Tensor	202	(1)
3.5.5 Sonine Polynomials	203	(4)
3.5.6 Application to Rigid Spheres	207	(1)
3.5.7 Diffusion and Electrical Conductivity	208	(1)
3.5.8 Expressions of Q(l,q) for Inverse Power Interaction Forces	209	(2)
3.5.9 Interaction Models and Experimental Values	211	(2)
3.5.10 The Method of Moments	213	(4)
3.6 The Linear Boltzmann Collision Operator	217	(8)
3.6.1 Symmetry of the Kernel	217	(1)
3.6.2 Negative Eigenvalues	218	(1)
3.6.3 Comparison of Boltzmann and Liouville Operators	219	(1)
3.6.4 Hard and Soft Potentials	220	(1)
3.6.5 Maxwell Molecule Spectrum	220	(2)
3.6.6 Further Spectral Properties	222	(3)
3.7 The Druyvesteyn Distribution	225	(15)
3.7.1 Basic Parameters and Starting Equations	225	(2)
3.7.2 Legendre Polynomial Expansion	227	(5)
3.7.3 Reduction of J^0(f0 | F)	232	(1)
3.7.4 Evaluation of I0 and I1 Integrals	233	(3)
3.7.5 Druyvesteyn Equation	236	(1)
3.7.6 Normalization, Velocity Shift, and Electrical Conductivity	237	(3)
3.8 Further Remarks on Irreversibility	240	(18)
3.8.1 Ergodic Flow	240	(3)
3.8.2 Mixing Flow and Coarse Graining	243	(2)
3.8.3 Action-Angle Variables	245	(3)
3.8.4 Hamilton-Jacobi Equation	248	(1)
3.8.5 Conditionally Periodic Motion and Classical Degeneracy	249	(4)
3.8.6 Bruns's Theorem	253	(1)
3.8.7 Anharmonic Oscillator	253	(2)
3.8.8 Resonant Domains and the KAM Theorem	255	(3)
Problems	258	(20)
4 Assorted Kinetic Equations with Applications to Plasmas and Neutral Fluids	278	(51)
4.1 Application of the Vlasov Equation to a Plasma	279	(15)
4.1.1 Debye Potential and Dielectric Constant	279	(6)
4.1.2 Waves, Instabilities, and Damping	285	(3)
4.1.3 Landau Damping	288	(4)
4.1.4 Nyquist Criterion	292	(2)
4.2 Further Kinetic Equations of Plasmas and Neutral Fluids	294	(25)
4.2.1 Krook-Bhatnager-Gross Equation	295	(3)
4.2.2 KBG Analysis of Shock Waves	298	(3)
4.2.3 The Fokker-Planck Equation	301	(6)
4.2.4 The Landau Equation	307	(2)
4.2.5 The Balescu-Lenard Equation	309	(6)
4.2.6 Convergent Kinetic Equation	315	(2)
4.2.7 Fokker-Planck Equation Revisited	317	(2)
4.3 Monte Carlo Analysis in Kinetic Theory	319	(6)
4.3.1 Master Equation	319	(1)
4.3.2 Equivalence of Master and Integral Equations	320	(1)
4.3.3 Interpretation of Terms	321	(2)
4.3.4 Application of Random Numbers	323	(1)
4.3.5 Program for Evaluation of Distribution Function	324	(1)
Problems	325	(4)
5 Elements of Quantum Kinetic Theory	329	(120)
5.1 Basic Principles	330	(11)
5.1.1 The Wave Function and Its Properties	330	(2)
5.1.2 Commutators and Measurement	332	(2)
5.1.3 Representations	334	(1)
5.1.4 Coordinate and Momentum Representations	335	(2)
5.1.5 Superposition Principle	337	(2)
5.1.6 Statistics and the Pauli Principle	339	(1)
5.1.7 Heisenberg Picture	339	(2)
5.2 The Density Matrix	341	(22)
5.2.1 The Density Operator	342	(4)
5.2.2 The Pauli Equation	346	(5)
5.2.3 The Wigner Distribution	351	(3)
5.2.4 Weyl Correspondence	354	(3)
5.2.5 Wigner-Moya1 Equation	357	(2)
5.2.6 Homogeneous Limit: Pauli Equation Revisited	359	(4)
5.3 Application of the KBG Equation to Quantum Systems	363	(22)
5.3.1 Equilibrium Distributions	363	(3)
5.3.2 Photon Kinetic Equation	366	(3)
5.3.3 Electron Transport in Metals	369	(7)
5.3.4 Thomas-Fermi Screening	376	(2)
5.3.5 Mott Transition	378	(1)
5.3.6 Relaxation Time for Charge-Carrier Phonon Interaction	379	(6)
5.4 Quantum Modifications of the Boltzmann Equation	385	(9)
5.4.1 Quasi-Classical Boltzmann Equation	385	(2)
5.4.2 Kinetic Theory for Excitations in a Fermi Liquid	387	(5)
5.4.3 H Theorem for Quasi-Classical Distribution	392	(2)
5.5 Overview of Classical and Quantum Hierarchies	394	(9)
5.5.1 Second Quantization and Fock Space	394	(1)
5.5.2 Classical and Quantum Distribution Functions	395	(4)
5.5.3 Equations of Motion	399	(1)
5.5.4 Generalized Hierarchies	400	(3)
5.6 Kubo Formula Revisited	403	(5)
5.6.1 Charge Density and Current	403	(1)
5.6.2 Identifications for Kubo Formula	404	(1)
5.6.3 Electrical Conductivity	405	(2)
5.6.4 Reduction to Drude Conductivity	407	(1)
5.7 Elements of the Green's Function Formalism	408	(13)
5.7.1 Schrödinger Equation Green's Function	408	(2)
5.7.2 The s-Body Green's Function	410	(1)
5.7.3 Averages and the Green's Function	410	(3)
5.7.4 The Quasi-Free Particle	413	(1)
5.7.5 One-Body Green's Function	414	(1)
5.7.6 Retarded and Advanced Green's Functions	415	(1)
5.7.7 Coupled Green's Function Equations	415	(1)
5.7.8 Diagrams and Expansion Techniques	416	(5)
5.8 Spectral Function for Electron-Phonon Interactions	421	(8)
5.8.1 Hamiltonian	421	(2)
5.8.2 Green's Function Equations of Motion	423	(2)
5.8.3 The Spectral Function	425	(1)
5.8.4 Lorentzian Form	426	(2)
5.8.5 Lifetime and Energy of a Quasi-Free Particle	428	(1)
Problems	429	(20)
6 Relativistic Kinetic Theory	449	(31)
6.1 Preliminaries	450	(6)
6.1.1 Postulates	450	(1)
6.1.2 Events, World Lines, and the Light Cone	450	(1)
6.1.3 Four-Vectors	451	(1)
6.1.4 Lorentz Transformation	452	(2)
6.1.5 Length Contraction, Time Dilation, and Proper Time	454	(1)
6.1.6 Covariance, Hamiltonian, and Hamilton's Equations	454	(1)
6.1.7 Criterion for Relativistic Analysis	455	(1)
6.2 Covariant Kinetic Formulation	456	(11)
6.2.1 Distribution Function	456	(1)
6.2.2 One-Particle Liouville Equation	457	(1)
6.2.3 Covariant Electrodynamics	458	(1)
6.2.4 Vlasov Equation	459	(1)
6.2.5 Covariant Drude Formulation of Ohm's Law	460	(1)
6.2.6 Lorentz Invariants in Kinetic Theory	461	(3)
6.2.7 Relativistic Electron Gas and Darwin Lagrangian	464	(3)
6.3 The Relativistic Maxwellian	467	(3)
6.3.1 Normalization	467	(2)
6.3.2 The Nonrelativistic Domain	469	(1)
6.4 Non-Cartesian Coordinates	470	(4)
6.4.1 Covariant and Contravariant Vectors	470	(1)
6.4.2 Metric Tensor	471	(1)
6.4.3 Lagrange's Equations	472	(1)
6.4.4 The Christoffel Symbol	473	(1)
6.4.5 Liouville Equation	474	(1)
Problems	474	(6)
7 Kinetic Properties of Metals and Amorphous Media	480	(47)
7.1 Metallic Electrical and Thermal Conduction	482	(23)
7.1.1 Background	482	(2)
7.1.2 Thermopower	484	(3)
7.1.3 Electron-Phonon Scattering Matrix Elements	487	(6)
7.1.4 Quantum Boltzmann Equation	493	(3)
7.1.5 Perturbation Distribution	496	(2)
7.1.6 Electrical Resistivity	498	(5)
7.1.7 Scale Parameters of Po	503	(1)
7.1.8 Electron Distribution Function	504	(1)
7.1.9 Thermal Conductivity	504	(1)
7.2 Amorphous Media	505	(19)
7.2.1 Background	505	(3)
7.2.2 Localization	508	(3)
7.2.3 Conduction Mechanisms and Hopping	511	(3)
7.2.4 Percolation Phenomena	514	(3)
7.2.5 Pair-Connectedness Function and Scaling	517	(3)
7.2.6 Localization in Second Quantization	520	(4)
Problems	524	(3)
Location of Key Equations	527	(2)
List of Symbols	529	(8)
Appendices
A Vector Formulas and Tensor Notation	537	(4)
A.1 Definitions	537	(2)
A.2 Vector Formulas and Tensor Equivalents	539	(2)
B Mathematical Formulas	541	(9)
B.1 Tensor Integrals and Unit Vector Products	541	(1)
B.2 Exponential Integral, T Function, ζ Function, and Error Function	542	(3)
B.2.1 Exponential Integral	542	(3)
B.3 Other Useful Integrals	545	(1)
B.4 Hermite and Laguerre Polynomials and the Hypergeometric Function	546	(4)
C Physical Constants	550	(2)
D Lorentz-Legendre Expansion	552	(2)
E Additional References	554	(9)
E.1 Early Works	554	(4)
E.2 Recent Contributions to Kinetic Theory and Allied Topics	558	(5)
F Science and Society in the Classical Greek and Roman Eras	563	(2)
Index	565