Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, Third Edition Richard L Liboff Springer Graduate Texts in Contemporary Physics Series Editors : R Stephen Berry Joseph L Birman Mark P Silverman H Eugene Stanley Mikhail Voloshin Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo This page intentionally left blank Richard L Liboff Kinetic Theory Classical, Quantum, and Relativistic Descriptions Third Edition With 112 Illustrations 13 Richard L Liboff School of Electrical Engineering Cornell University Ithaca, NY 14853 richard@ee.cornell.edu Series Editors R Stephen Berry Department of Chemistry University of Chicago Chicago, IL 60637 USA Joseph L Birman Department of Physics City College of CUNY New York, NY 10031 USA H Eugene Stanley Center For Polymer Studies Physics Department Boston University Boston, MA 02215 USA Mikhail Voloshin Theoretical Physics Institute Tate Laboratory of Physics 424 The University of Minnesota Minneapolis, MN 55455 USA Mark P Silverman Department of Physics Trinity College Hartford, CT 06106 USA Library of Congress Cataloging-in-Publication Data Liboff, Richard L., 1931– Kinetic theory: classical, quantum, and relativistic descriptions / Richard L Liboff 3rd ed p cm.—(Graduate texts in contemporary physics) Includes bibliographical references and index ISBN 0-387-95551-8 (alk paper) Kinetic theory of matter I Title II Series QC174.9 L54 2003 530.13 6—dc21 2002026658 ISBN 0-387-95551-8 Printed on acid-free paper The second edition of this book was published © 1998 by John Wiley & Sons © 2003 Springer-Verlag New York, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed in the United States of America SPIN 10887268 www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH To the Memory of Harold Grad Teacher and Friend This page intentionally left blank I learned much from my teachers, more from my colleagues and most of all from my students Rabbi Judah ha-Nasi (ca 135–C.E.) Babylonian Talmud (Makkot, 10a) Ludwig Boltzmann (University of Vienna, courtesy AIP Emilio Segr` Visual e Archives) Preface to the Third Edition Since the first edition of this work, kinetic theory has maintained its position as a cornerstone of a number of disciplines in science and mathematics In physics, such is the case for quantum and relativistic kinetic theory Quantum kinetic theory finds application in the transport of particles and radiation through material media, as well as the non-stationary quantum–many-body problem Relativistic kinetic theory is relevant to controlled thermonuclear fusion and to a number of problems in astrophysics In applied mathematics, kinetic theory relates to the phenomena of localization, percolation, and hopping, relevant to transport properties in porous media Classical kinetic theory is the foundation of fluid dynamics and thus is important to aerospace, mechanical, and chemical engineering Important to the study of transport in metals is the Lorentz– Legendre expansion, which in this new edition appears in an appendix A new section in Chapter was included in this new edition that addresses constants of motion and symmetry A number of small but important revisions were likewise made in this new edition A more complete description of the contents of the text follows The text comprises seven chapters In Chapter 1, the transformation theory of classical mechanics is developed for the purpose of deriving Liouville’s theorem and the Liouville equation Four distinct interpretations of the solution to this equation are presented The fourth interpretation addresses Gibbs’s notion of a distribution function that is the connecting link between the Liouville equation and experimental observation The notion of a Markov process is discussed, and the central-limit theorem is derived and applied to the random walk problem In Chapter 2, the very significant BBKGY hierarchy is obtained from the Liouville equation, and the first two equations of this sequence are applied in the derivation of conservation of energy for a gas of interacting particles In E.1 Early Works 557 (f) Svensk Akad (Arkiv f Mat., Ast och Fys.), 21A, No 13 (1928) Hasse, H R Phil Mag., 1, 139 (1926) Hasse, H R., and Cook, W R (a) Phil Mag., 3, 977 (1927) (b) Proc Roy Soc., 196 (1929) (c) Phil Mag., 12, 554 (1931) Hilbert, D Math Ann., 72, 562 (1912) James, C G F Proc Camb Phil Soc., 20, 477 (1921) Jeans, J (a) An Introduction to the Kinetic Theory of Gases, Cambridge University Press, New York, 1952 (b) Kinetic Theory of Gases, Cambridge University Press, New York, 1946 (c) Phil Trans Roy Soc A, 196, 399 (1901) (d) Quart J Math., 25, 224 (1904) Kennard, E H Kinetic Theory of Gases, McGraw-Hill, New York, 1938 Loeb, L B The Kinetic Theory of Gases, 2d ed., McGraw-Hill, New York, 1934 Lorentz, H A (a) “On the Equilibrium of Kinetic Energy Among Gas-Molecules,” Wien Ber., 95, 115 (1887) (b) “The Motions of Electrons in Metallic Bodies,” Proc Amsterdam Acad., 7, 438, 585, 684 (1905) (c) The Theory of Electrons, B G Teubner, Leipzig, 1909 Massey, H S W., and Mohr, C B O (a) “On the Rigid Sphere Model,” Proc Roy Soc A, 141, 434 (1933) (b) “On the Determination of the Laws of Force between Atoms and Molecules,” Proc Roy Soc., 144, 188 (1934) Pidduck, F B (a) “The Kinetic Theory of the Motions of Ions in Gases,” Proc Lond Math Soc., 15, 89 (1916) (b) “The Kinetic Theory of a Special Type of Rigid Molecule,” Proc Roy Soc A., 101, 101 (1922) 558 E Additional References E.2 Recent Contributions to Kinetic Theory and Allied Topics Abrikosov, A A., Gorkov, L P., and Dzyaloshinskii, I E., Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, N.J (1963) Reprinted by Dover, New York (1975) Akhiezer, A I., and Peletminskii, S V., Methods of Statistical Physics, Pergamon, Elmsford, N.Y (1981) Arnold, V I., Mathematical Methods of Classical Mechanics, Springer-Verlag, New York (1978) Allis, W P., “Motion of Ions and Electrons,” Hand Physik, vol XXI, SpringerVerlag, Berlin (1956) Balescu, R., Statistical Mechanics of Charged Particles, Wiley, New York (1967) Barut, A O., Electrodynamics and Classical Theory of Fields and Particles, Dover, New York (1980) Bellman, R., G Birkhoff and Abu-Shumays, I., eds., Transport Theory, American Mathematical Society, Providence, R.I (1969) Bernstein, J., Kinetic Theory in the Expanding Universe, Cambridge, New York (1988) Brenig, W., Statistical Theory of Heat: Non-Equilibrium Phenomena, Springer, New York (1989) Burgers, J M., The Nonlinear Diffusion Equation, D Reidel, Dordrecht, The Netherlands (1974) Burshtein, A I., Introduction to Thermodynamics and Kinetic Theory of Matter, Wiley, New York (1996) Carruthers, P., and Zachariasen, F., “Quantum Collision Theory with PhaseSpace Distributions,” Rev Mod Phys 55, 245 (1983) Cercignani, C., Mathematical Methods in Kinetic Theory, Plenum, New York (1969) ——, Theory and Application of the Boltzmann Equation, Elsevier, New York (1975) Case, K M., and Zweifel, P F., Linear Transport Theory, Addison-Wesley, Reading, Mass (1967) Chapman, S., and Cowling, T G., The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge University Press, New York (1974) Cohen, E D G., Fundamental Problems in Statistical Mechanics, Wiley, New York (1962) Cohen, E G D., and Thiring, W (eds.), The Boltzmann Equation: Theory and Application, Springer-Verlag, New York (1973) Curtiss, C F., Hirschfelder, J O., and Bird, R B., The Molecular Theory of Gases and Liquids, Wiley, New York (1969) Davidson, R C., Methods in Nonlinear Plasma Theory, Academic Press, New York (1972) E.2 Recent Contributions to Kinetic Theory and Allied Topics 559 de Boer, J., and Uhlenbeck, G E., Studies in Statistical Mechanics, NorthHolland, Amsterdam (1970) de Groot, S R., and Mazur, P., Non-Equilibrium Thermodynamics, North Holland, Amsterdam (1962) Doniach, S., and Sondheimer, E H., Green’s Functions for Solid State Physicists, Benjamin-Cummings, Menlo Park, Calif (1974) Dresden, M., “Recent Developments in the Quantum Theory of Transport and Galvonomagnetic Phenomena” Rev Mod Phys 33, 265 (1961) Duderstadt, J J., and Martin, W R., Transport Theory, Wiley, New York (1979) Ebeling, W., Transport Properties of Dense Plasma, Birkhauser, Boston, (1984) Ecker, G., Theory of Fully Ionized Plasmas, Academic Press, New York (1972) Eu, B C., Kinetic Theory and Irreversible Thermodynamics, Wiley, New York (1992) Family, F., and Landau, D P., eds Kinetics of Aggregation and Gelation, North Holland, Amsterdam (1984) Farquhar, I E., Ergodic Theory in Statistical Mechanics, Wiley-Interscience, New York (1964) Ferziger, J H., and Kaper, H G., Mathematical Theory of Transport Processes in Gases, North Holland, Amsterdam (1962) Fetter, A L., and Walecka, J D., Quantum Theory of Many-Particle Systems, McGraw-Hill, New York (1971) Finkelstein, R J., Nonrelativistic Mechanics, Benjamin, Menlo Park, Calif (1973) Frenkel, J., Kinetic Theory of Liquids, Oxford, New York (1946) Fujita, S., Introduction to Non-Equilibrium Statistical Mechanics, W B Saunders, Philadelphia (1966) Goldstein, H., Classical Mechanics, 2nd ed., Addison-Wesley, Reading, Mass (1973) Gombosi, T I., Gas Kinetic Theory, Cambridge, Now York (1994) Grad, H., “On the Kinetic Theory of Rarefied Gases,” Comm Pure and Appl Math 2, 331 (1949) ——, “Principles of the Kinetic Theory of Gases,” Hand Physik, vol XII, Springer-Verlag, Berlin (1958) ——, “The Many Faces of Entropy,” Comm Pure Appl Math 14, 323 (1961) Grandy, W T., Foundations of Statistical Mechanics, Vol 2, Nonequilibrium Phenomena, D Reidel, Boston (1988) Groot, S R de, and Leeuwen, W A Van., Relativistic Kinetic Theory: Principles and Applications, North Holland, New York (1980) Haken, H., Quantum Field Theory of Solids, North-Holland, Amsterdam (1976) Harris, S., Introduction to the Theory of the Boltzmann Equation, Holt, Rinehart and Winston, New York (1971) Harrison, L G., Kinetic Theory of Living Patterns, Cambridge, New York (1993) 560 E Additional References Haug, A., Theoretical Solid State Physics, H S H Massey (trans.), Pergamon, Elmsford, N.Y (1972) Hillery, M., O’Connell, R F., Scully, M O., and Wigner, E P., “Distribution Functions in Physics: Fundamentals,” Phys Repts 106, 122 (1984) Kac, M., Probability and Related Topics in Physical Sciences, Wiley Interscience, New York (1959) Kadanoff, L P., and Baym, G., Quantum Statistical Mechanics, Benjamin, Menlo Park, Calif (1962) Keldysh, L V., “Diagram Technique for Nonequilibrium Processes,” Sov Phys JETP 20, 1018 (1965) Klimontovich, Y., Statistical Theory of Non-Equilibrium Processes in Plasma, MIT Press, Cambridge, Mass (1969) Koga, T., Introduction to Kinetic Theory Stochastic Processes in Gaseous Systems, Pergamon Press, Elmsford, N.Y (1970) Kogan, M N., Rarefied Gas Dynamics, Plenum, New York (1969) Kohn, W., and Luttinger, J M., “Quantum Theory of Electron Transport Phenomena,” Phys Rev 108, 570 (1975) Kubo, R., Statistical Mechanics, Wiley, New York (1965) Liboff, R L., Introduction to the Theory of Kinetic Equations, Wiley, New York (1969); Mir, Moscow (1974), Krieger, Melbourne, Fla (1979) ——, Introductory Quantum Mechanics, 4th ed., Addison-Wesley, San Francisco (2002) ——, and Rostoker, N (eds.), Kinetic Equations, Gordon and Breach, New York (1971) Lie, T-J., and Liboff, R L., “Consideration of Particle Exchange in Quantum Kinetic Theory,” Ann Phys 67, 349 (1971) Lifshitz, E M., and Pitaevskii, L P., Physical Kinetics, Pergamon, Elmsford, N.Y (1981) ——, and ——, Statistical Physics, Part Two, Pergamon, Elmsford, N.Y (1980) Mahan, G D., “Quantum Transport Equation for Electric and Magnetic Fields,” Phys Repts 145, 251 (1987) ——, Many-Particle Physics, Plenum, New York (1981) Mitchner, M., and Kruger, C H., Jr., Partially Ionized Gases, Wiley, New York (1973) McQuarrie, D A., Statistical Mechanics, Harper & Row, New York (1976) Montgomery, D C., and Tidman, D A., Plasma Kinetic Theory, McGraw-Hill, New York (1964) Negele, J W., and Orland, H., Quantum Many-Particle Systems, AddisonWesley, Reading, Mass (1988) Nicholson, D R., Introduction to Plasma Theory, Wiley, New York (1983) O’Raifeartaigh, L., General Relativity Papers in Honor of J L Synge, Oxford University Press, New York (1972) Pathria, R K., Statistical Mechanics, Pergamon, Elmsford, N Y (1972) E.2 Recent Contributions to Kinetic Theory and Allied Topics 561 Peletminskii, S., and Vatsenko, A., “Contribution to the Quantum Theory of Kinetic and Relaxation Processes,” Sov Phys JETP, 26, 773 (1968) Pines, D., The Many-Body Problem, Benjamin-Cummings, Menlo Park, Calif., (1962) Pomraning, G C., The Equations of Radiation Hydrodynamics, Pergamon, Elmsford, N.Y (1973) Pozhar, L.A., Transport of Inhomogeneous Fluids, World Scientific, Singapore (1994) Present, R D., Kinetic Theory of Gases, McGraw-Hill, New York (1958) Prigogine, I., Non-Equilibrium Statistical Mechanics, Wiley, New York (1962) ——, From Being to Becoming, W A Freeman, San Francisco (1980) Rammer, J., and Smith, H., “Quantum Field-Theory Methods in Transport Theory of Metals,” Rev Mod Phys., 58, 323 (1986) Reed, T M., and Gubbins, K E., Applied Statistical Mechanics, McGraw-Hill, New York (1973) Reichl, L., A Modem Course in Statistical Physics, University of Texas Press, Austin (1986) Resibois, P., and de Leener, M., Classical Kinetic Theory of Fluids, Wiley, New York (1977) Rickayzen, G., Green’s Functions and Condensed Matter, Academic Press, New York (1980) Riskin, H., The Fokker-Planck Equation, 2nd ed., Springer-Verlag, New York (1989) Roos, B W., Analytic Functions and Distributions in Physics and Engineering, Wiley, New York (1969) Sampson, D H., Radiative Contributions to Energy and Momentum Transport in a Gas, Wiley-Interscience, New York (1965) Schenter, K G., A Unified View of Classical and Quantum Kinetic Theory with Application to Charge-Carrier Transport in Semiconductors, Thesis, Cornell (1988) Sinai, Ya G., Introduction to Ergodic Theory, Princeton University Press, Princeton, N.J (1976) Stewart, J M., Non-Equilibrium Relativistic Kinetic Theory, Springer-Verlag, New York (1971) Sudarshan, E C G., and Mukunda, N., Classical Dynamics, Wiley, New York (1974) Synge, J L., Relativity, the Special Theory, 2nd ed., North-Holland, Amsterdam (1965) ——, Relativity, the General Theory, North-Holland, Amsterdam (1966) Thornber, K K., and Feynman, R P., “Velocity Acquired by an Electron in a Finite Crystal in a Polar Field,” Phys Rev B1, 4900 (1970) Thouless, D J., Quantum Mechanics of Many Body Systems, Academic Press, New York (1961) Uehling, E A., and Uhlenbeck, G E., “Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases I,” Phys Rev., 43, 552 (1933) 562 E Additional References Uhlenbeck, G E., and Ford, G W., Lectures in Statistical Mechanics, American Mathematical Society, Providence, R.I (1963) van Leewen, W A., van Weert, Ch G., and de Groot, S R., Relativistic Kinetic Theory, North-Holland, Amsterdam (1980) Vincenti, W G., and Kruger, C H., Introduction to Physical Gas Dynamics, Wiley, New York (1965) Wing G M, An Introduction to Transport Theory, Wiley, New York (1962) Wolfram, S., “Cellular Automaton Fluids 1: Basic Theory,” J Stat Phys., 4, 471 (1986) Wu, T Y., Kinetic Equations of Gases and Plasmas, Addison-Wesley Reading, Mass (1966) Zaslavsky, G M., Chaos in Dynamical Systems, Harwood Academic, New York (1984) Ziff, R M., New Class of Solvable-Model Boltzmann Equations, Phys Rev Letts, 45 306 (1980) Zimon, J M., Electrons and Phonons, Oxford University Press, New York (1960) ——, Elements of Advanced Quantum Theory, Cambridge University Press, New York (1969) Zubarev, D N., Nonequilibrium Statistical Thermodynamics, P J Shepard (trans.), Consultants Bureau, New York (1974) ——, and Kalashnikov, V P., “Derivation of the Nonequilibrium Statistical Operator from the Extremum of the Information Entropy,” Physica, 46, 550 (1970) APPENDIX F This page intentionally left blank Index Acoustic phonons, 379 Action-angle variables, 245 Action integral, 2, 478 Activation energy, 512 Adiabatic law, 198, 265 Alkali metals, 487 Alloy, 480 binary, 480 ordered and disordered, 480, 520 Alumina, 480 Amorphous media, 506ff Anderson, P W., 508 Apsidal vector, 142, 302 Atmospheres, law of (See Barometer formula) Autocorrelation function, 59, 189, 312, 327 and Green’s function, 441 Averages (See also Expectation) ensemble, 343 phase, 37, 241 phase density, 125 time, 242 Balazs, N L., 354 Balescu-Lenard equation, 313, 316 Barometer formula, 171 Barut, A O., 455 BBKGY hierarchy, 78 Bergman, P G., 449 Bethe lattice, 517 Birkhoff’s theorem, 242 Black hole, 478 Bloch condition, 492 waves, 508 Bogoliubov distribution, 117 Bogoliubov hypothesis, 115 Boltzmann equation, 112, 124, 131, 152 assumptions in derivation, 152 quasi-classical, 385, 493 Bose-Einstein distribution, 382, 387 (See also Planck distribution) Boson, composite, 439 β-Brass, 480 Brass, 480 Brillouin zone, 488 Bronze, 480 Bruns’s theorem, 253 Burnett equations, 199 Canonical distribution, 133, 344 Canonical invariants, 15 Canonical transformation, 11, 247 Carbon dioxide, viscosity, 212 Central limit theorem, 54, 172, 270 Chapman-Enskog expansion, 194, 265 566 Index Chapman-Kolmogorov equation, 41, 317 Characteristic curves, 25 Characteristic function, 48, 69 Characteristics, method of, 24, 94, 304 Chebyshev’s inequality, 71 Christoffel symbol, 473 Coarse graining, 243 Collision: inverse, 147 reverse, 148 Collisional invariants, 155 Collision frequency, 194, 297 Collision operator, linear, 217 negative eigenvalues, 218 spectrum for Maxwell molecules, 220 symmetry, 217 Completely continuous operator, 224 Conductivity, 177, 209, 238 Conservation equations: absolute, 159 from BY and BY, 83 relative, 160 Constants of motion, 4, 243, 254 Contravariant vectors, 470 Convergent kinetic equation, 315 Convolution integral, 191 Coordinates, generalized, Correlation expansion, 89, 310 Correlation function, 59, 89 total, 96, 134 Coulomb gauge, 464 Covariance, 59, 452 Covariant vectors, 470 Cross section, scattering: Coulomb, 145, 234 differential, 143 rigid sphere, 145 total, 144 Current density entropy, 486 particle, 486 thermal, 486 Darwin Lagrangian, 467 deBroglie wavelength, 364 Debye distance, 95, 113, 326 frequency, 493 shielding potential, 284 temperature, 482, 498 wave number, 283, 314, 495 Deformation potential, 379 Degeneracy: classical, 250 quantum, 377 Degenerate plasma, 377 Degrees of freedom, de Groot, S R., 449 Density expansion, 117 Density matrix, 395, 430 equation of motion, 399 Density operator, 331 Deuterium, liquid, 259 Diagrammatic analysis (See Prigogine analysis) Dielectric function, plasma (See Plasma) Dielectric time, 113 Diffusion, 175 coefficient, 483 mutual, 181, 208 self, 180, 188 Diffusion coefficient, 304, 327 Diffusion equation, 67, 187 Dispersion relation: degenerate plasma, 377 warm plasma, 286, 377 Distribution functions, classical (See also Quantum distributions) conditional, 37 joint probability, 36 reduced, 37 s-tuple, 38 Doppler effect, 477 Drift-diffusion equation, 177 Drude conductivity, 177, 239, 407 covariant, 460 Druyvesteyn distribution, 236 absolute, 237 Edwards, J T., 510 Electrical conductivity, 487 temperature domains, 512 Electrochemical potential, 178, 484 Electrodynamics, covariant, 458 Index 567 Electron gas, 464 Electron mobility (See Mobility, electrical) Electron-phonon scattering rate (See Scattering rate) Elliptic integrals, complete, 221 Elyutin, P V., 510 Energy bands, 369 Energy equation, 97 Energy shell, 35, 240 Ensemble, 20 average, 189, 343 Entropy: Boltzmann, 165 change, 275 Gibbs, 164, 275 Equal a priori probabilities, 345 Equation of state, 97, 134 Equipartition of energy, 179, 382 Ergodic hypothesis, 241 Ergodic motion, 240, 264 Error function, 545 Euler equations, 198 Euler’s constant, 542 Event, relativistic, 450 Exchange transformation, 13 Expectation: of a classical function, 37 of a random variable, 47 and wave function, 330 and Wigner distribution, 353 Exponential integral, 542 Exponents, critical, 518 Extended states, 506, 521 Fokker-Planck equation, 304, 308, 315, 317, 327 Four-current, 453 Four-vector, 451 covariant and contravariant, 471 Four-vector potential, 453, 459 Four-wave vector, 453 Free energy, 518 Friction coefficient, 304, 327 Friedel oscillations, 377 Fermi-Dirac distribution, 371, 387, 483, 492 Fermi energy, 371, 4343 Fermi liquid, 387 Fermi momentum, 388 Fermi sphere, 374 Field tensor, electromagnetic, 458 Flow chart, 333 Fluid dynamical variables (See Macroscopic variables) Fock space, 395, 433 Hamiltonian, 433 operator, 396, 438 Hamiltonian, 4, 61, 66, 139, 254, 262, 272, 274, 396, 399 relativistic, 5, 455, 467 Hamilton-Jacobi equation, 248 Hamilton’s equations, covariant, 455 Hamilton’s principle, 2, 76 Hard potential (See Potential) Heisenberg picture, 339, 396, 422, 429 H´ non-Heiles Hamiltonian, 254 e anti, 274 Hermite polynomials, tensor, 213, 546 Gamma function, 543 Gauss’ equation, 280, 281 Gaussian distribution, 56, 59, 173 Generalized coordinates, Generalized potential (See Potential) Generating function, 11 Goldstein, H., 455 Grad’s first and second equations, 129, 262 Grad’s method of moments, 213 Grains, 480 Grazing collisions, 301, 303 Green’s function equations, coupled, 415, 422 Green’s functions: and averages, 410 diagrammatic representation, 416, 440 retarded and advanced, 415 s-body, 410, 438 and Schroedinger equation, 408 Group property: canonical transformations, 15 Lorentz transformations, 475 568 Index Hermiticity, 27 Hessian, 257 Heteroclinic orbit, 74 Hierarchies, quantum and classical, table of, 402 Hillery, R F., 354 Histogram, bin storage, 323 Hole mobility, 369 Homoclinic orbit, 74 Honeycomb lattice, 517 Hopping, 506 thermally assisted, 513 variable range, 513 H-theorem, 166, 392, 446 Hydrofracturing of a rock, 515 Hypergeometric function, 548 confluent, 548 Hypersonic flow, 298 Ideal gas, 30, 270 Impact parameter, 142 Incompressible phase density, 24, 136 Inelastic scattering, 319 Insulators, 480 Integrable motion, 254 Integral invariants of Poincar´ , 68 e Integral of motion (See Constants of motion) Inverse scattering, 147 Ioffe-Regel minimum, 512 Irreversibility, 162, 167, 240 Jacobian, 17 Jacobi’s identity, 11 Jennings, B K., 354 Joint probability distribution (See Distribution functions) Kagom´ lattice, 517 e KAM theorem, 257, 274 KBG equation (See KrookBhatnager-Gross equation) Kinetic equation: convergent, 315 definition, 82 photon, 366 table of, 296 Klimontovich picture, 124, 136 Kronicker-delta symbol, four dimensional, 472 Krook-Bhatnager-Gross equation, 96, 194, 295, 328 Kubo formula: classical, 192 quantum, 404 Kursunoglu, B., 456 Lagrange’s equations, covariant, 472 Lagrangian, 2, 60, 62, 132 relativistic, 466, 472 Laguerre polynomials, 204, 547 Landau damping, 288, 326, 377 Landau equation, 309, 315 Laplace-Runge-Lenz vector Large-mass consistency limit, 503 Lasing criterion, 368 Lattice: Bethe, 517 Honeycomb, 517 Kagom´ , 517 e one-dimensional, 522 Legendre expansion, 227, 261 Legendre polynomials, 229 Legendre transformation, 14 Length contraction, 454 Levi-Civita symbol, 538 Liapunov exponent, 257 Lie derivative, 135 Light cone, 450 Liouville equation, 22, 78, 344, 399 in non-Cartesian coordinates, 474 one-particle, 93 relativistic, 457 Liouville operator, 27, 98 resolvent, 33 Liouville theorem, 17, 276 Localization, 508 length, 510 in second quantization, 520ff Localized states, 506 Logarithmic singularity, 295, 303 Long-time limit, 110 Long-time tails, 189, 277 Lorentz expansion, 494 force, 5, 62 Index gauge, 66, 464 invariants, 453 in kinetic theory, table of, 465 transformation, 452, 475 Lorentzian form (See Spectral function) Lorenz number, 373, 376 relation, 483 Mach number, 298 Macroscopic variables: absolute, 159 relative, 160 Magnetic alloys, dilute, 515 Mandelstam variables, 465, 478 Markov process, 41 Master equation, 43, 319, 446 Maxwellian, 96, 225, 275, 306, 327 absolute, 169 local, 169 relativistic, 467, 475 Maxwell interaction, 144, 219, 221, 271 Maxwell molecules (See Maxwell interaction) Maxwell’s equations, 66, 459 Mean-free-path, 179 acoustic phonon, 384 estimates, 178 Measurement: and commutators, 332 and operators, 331 Measure of a set, 241 Metal-insulator transition, 506, 510 Metals, 480 electron transport in, 369 Method of moments, Grad’s, 213 Metrically indecomposable sets, 242 Metrically transitive transformations, 242 Metric space, 133 Metric tensor, 471 Microcanonical distribution, 344 Mixed states, 341, 434 Mixing flow, 243 Mobility edge, 506, 511 electrical, 177, 192 569 electron, 369 gap, 511 hole, 369 Monte Carlo analysis, 319, 328 Mott, N F., 510 Mott transition, 378 Navier-Stokes equations, 199 Neural networks, 515 Neutron, magnetic moment, 430 Nitrogen, viscosity, 212 Noble metals, 487 Non-Cartesian coordinates, 474 Nyquist criterion, 292, 327 Occupation numbers, 395 O’Connell, M O., 354 Optical phonons, 379 Pair connectedness, 518 Partition function, 432 grand, 433 Pauli principle, 339, 394 Percolation, 506, 514ff bond, 515 cluster, 515 site, 515 Periodic motion, conditional, 249 Phase average (See Averages) Phase density averages, 125 Phonon polarization, 490 Phonon scattering (See Relaxation time) Planck distribution, 364, 379 Plasma: dielectric function, 282 frequency, 113, 282, 326 parameter, 113, 304, 314 Plasma waves, 285 stable modes, 286 unstable modes, 291 Plemelj’s formula, 362 Poincar´ map, 254 e Poincar´ recurrence theorem, 163 e Poisson brackets, 10, 64 Poisson distribution, 52, 58, 69 Polycrystalline material, 480 Polymer gelatin, 515 Potential: 570 Index Potential: (continued) generalized, 66 hard, 220 soft, 220 Prandtl number, 185 Pratt, W P., 488 Pressure tensor, 175 Prigogine analysis, 97 Principal part, 326 Projection representation, 350, 438 Proper frequency, 461 Proper time, 454 Proper volume, 463 Pure state, 341, 397, 398, 432 Quantum distributions (See Bose-Einstein distribution; Fermi-Dirac distribution; Planck distribution) Quantum-modified Boltzmann equation (See Boltzmann equation) Quantum statistics, 386 Quartz, 480 Quasi-classical kinetic equation, 433 (See also Boltzmann equation) Quasi-classical limit, 386, 392 Quasi-free particle, 413 energy, 428 lifetime, 428 Radial distribution function, 96, 135, 448, 480 Radiation field, 366 Random numbers, 323 Random phases, 345, 510 Random variables: definition, 47 sums of, 49 Random walk, 44, 50, 55, 187 Range of interaction, 86 Rayleigh dissipation, 304 Rayleigh’s formula, 365 Reichl, L E., 394 Relativistic Maxwellian (See Maxwellian) Relativistic Vlasov equation (See Vlasov equation) Relativity, postulates, 450 Relaxation time approximation, 485 phonon scattering, 379 Representations: coordinate, 335 momentum, 335 Resistivity, metallic, 481 Bloch component of, 501 electrical, 498, 501 low-temperature, 483 reduced, 504 residual, 488, 501 Resolvent of Liouville operator (See Liouville operator) Riemann zeta function, 544 Rigid spheres: collision integral, 153 transport coefficients, 210 Rosenbluth-Rostoker limit, 96 Rough spheres, 273 Sample space, 47 Saxon-Hunter theorem, 523 Scattering: angle, 140 inelastic, 319 matrix, 142, 258, 488, 493 rate, 362 Scattering cross section (See Cross section) Schenter, G K., 394, 488 Schmidt number, 185 Schroeder, P A., 488 Schwartzchild metric, 478 Scully, M O., 354 Second quantization, 395 Self-consistent solution, 92 Self-scattering mechanism, 320 Semiconductor, 369, 480 Semimetals, 480 Shock front, 298 Shock waves, 298 Sine-Gordon equation, 70, 71 Single-crystal material, 480 Small shot noise, 53 Smooth spheres, 273 Sodium, band structure, 370 Soft potential (See Potential) Solids: categories of, 480 Index 571 Sonine polynomials, 204, 221, 224, 259, 266 Sound speed, 268 Spectral function, 409, 425 Lorentzian form, 426 Spectral theorem, 224 Spherical harmonics, 220 Spin, 338 and density matrix, 348, 430 and density of states, 371 Spinor, 430 Spontaneous decay, 366 Stable modes (See Plasma) Statistical balance, 167, 381, 387 Stefan-Boltzmann law, 366 Stimulated decay, 366 Stirling’s approximation, 57, 544 Stosszahlansatz, 152 Strain, 176 Strain-acoustic interaction, 379 Strain-optical interaction, 379 Strain tensor, symmetric, 176 Stress tensor, 175, 203, 265, 266 Structure factor, 97, 135 Subsonic flow, 298 Superposition principle, 41, 337 Sutherland model, 211 Symmetry properties of distribution functions, 39, 66, 73, 75 Synge, J L., 467 Time dilation, 454 Tori, invariant, 252, 257 Transfer integral, 510 Transfer matrix method, 522 Transonic flow, 298 Transport coefficients, 174 Chapman-Enskog estimates, 210 mean-free-path estimates, 186 Temperature, 161 Tensor: covariant and contravariant, 472 equivalents, 539 integrals, 471 metric, 471 Thermal conductivity, 175, 185, 202 coefficient of, 483, 487, 505 Thermal speed, 169, 179, 275 Thermopower, 484 Thomas-Fermi potential, 377, 378 Thomas-Fermi screening: distance, 376, 377 wave number, 376, 377 Thouless, D., 510 Three-body problem, 253 Tight-binding approximation, 508, 520 Walker, C H., 257 Wang Chang, C S., 204, 220 Wave number of closest approach, 314 Weidemann-Franz Law, 483 Weyl correspondence, 354, 435, 438 Wigner distribution, 351, 396, 435, 436, 437, 438 equation of motion, 353, 396 Wigner-Moyal equation, 358, 399, 431 World line, 450 Uehling-Uhlenbeck equation, 388 Uhlenbeck, G E., 204, 220 Umklapp process, 482 Uncertainty, in quantum mechanics, 333, 334 Unstable modes (See Plasma waves) Van Leewen, W A., 449 van Weert, Ch G., 449 Variance, 48, 161 Variational technique, 442 Vector potential, 66, 452, 466, 475 Velocity autocorrelation function, 441 Viscosity coefficient, 175, 184, 204, 211 observed values, 212 Vlasov equation, 92 for a plasma, 279 relativistic, 459, 475 Vlasov fluid, 92 Zallen, R., 510 Zero-point energy, ion, 492 Zero sound, 377, 391 Ziman, J M., 508 Zwanzig, R., 188 ... intentionally left blank Richard L Liboff Kinetic Theory Classical, Quantum, and Relativistic Descriptions Third Edition With 112 Illustrations 13 Richard L Liboff School of Electrical Engineering... of Congress Cataloging-in-Publication Data Liboff, Richard L., 1931– Kinetic theory: classical, quantum, and relativistic descriptions / Richard L Liboff 3rd ed p cm.—(Graduate texts in contemporary... this work, kinetic theory has maintained its position as a cornerstone of a number of disciplines in science and mathematics In physics, such is the case for quantum and relativistic kinetic theory