General Bibliography

Classic Literatures: G.l L.Boltzmann: Vorlesungen iiber Gastheorie, 2 Bde. (J.A.Barth 1912) Lectures on Gas

Theory [English trans!. by S.B.Brush] (Univ. of California Press 1964) G.2 J.W.Gibbs: Elementary Principles in Statistical Mechanics (Yale Univ. 1902 reprint

Dover) G.3 P. and T.Ehrenfest: The Conceptional Foundation of the Statistical Approach in Me­

chanics [trans!. by M.J.Moravcsik] (Cornell Univ. Press 1959). This was originally published in Enzyklopiidie der mathematischen Wissenschaften Bd. 4 Art 32 (1911)

G.4 R.H.Fowler: Statistical Mechanics, 2nd ed. (Cambridge Univ. Press 1936) G.5 C.Tolman: The Principles of Statistical Mechanics (Oxford Univ. Press 1938)

General Textbooks: G.6 R.Balescu: Equilibrium and Nonequilibrium Statistical Mechanics (John Wiley & Sons

1975) G.7 R.Becker: Theorie der Wcirme (Springer, Berlin, Gottingen, Heidelberg 1955) G.8 N.N.Bogolyubov: Lectures on Quantum Statistics (Gordon and Breach 1967) G.9 R.P.Feynman: Statistical Mechancis(Benjamin 1972) G.I0 S.Flugge (ed.): Principles of Thermodynamics and Statistical Mechanics, Encyclopedia

of Physics, Vo!. 3 Part 2 (Springer, Berlin, Gottingen, Heidelberg 1959) G.11 R.H.Fowler, E.A.Guggenheim: Statistical Thermodynamics, 2nd ed. (Cambridge

Univ. Press 1965) G.12 J.I.Frenkel: Statistische Physik (Akademie-Verlag 1957) G.13 T.L.Hill: Statistical Mechanics (McGraw-Hill 1956) G.14 K.Huang: Satistical Mechancis (John Wiley & Sons 1963) G.15 A.lshihara: Statistical Physics (Academic Press, New York 1971) G.16 C.Kittel: Elementary Statistical Physics (John Wiley & Sons 1958) G.17 R.Kubo (ed.): Statistical Mechanics (North-Holland 1965) G.18 L.D.Landau, E.M.Lifshitz: Statistical Physics (Pergamon 1958) There are a series

of revised and enlarged editions G.19 J.E.Mayer, M.G.Mayer: Statistical Mechanics (John Wiley & Sons 1940) G.20 A.MUnster: Statistische Thermodynamik (Springer, Berlin, Gottingen, Heidelberg

1956) G.21 F.Reif: Statistical and Thermal Physics (McGraw-Hill 1965) G.22 A.Sommerfeld: Thermodynamik und Statistik (Dietrich 1952) Thermodynamics and

Statistical Mechanics [trans!. by J. Kestin] (Academic Press 1956) G.23 D.terHarr: Elements of Statistical Mechanics (Holt, Rinehart & Winston 1961) G.24 G.H.Wannier: Statistical Physics (John Wiley & Sons 1966)

Mathematical Aspects of Fnndamental Problems: G.25 A.I.Khinchin: Mathematical Foundations of Statistical Mechanics [trans!. by G. Ga­

mow] (Dover 1949)

240 General Bibliography

G.26 R.Kurth: Axiomatics of Classical Statistical Mechanics (Pergamon 1960) G.27 O.Penrose: Foundations of Statistical Mechanics (Pergamon 1970) G.28 R.Ruelle: Statistical Mechanics (Benjamin 1969)

References

Chapter 1

1.1 L.Boltzmann: Vorlesungen iiber Gastheorie, 2 Bde. (J.A.Barth, Leipzig 1912), English trans!' by S. G .Brush : Lectures on Gas Theory (U niv. of California Press 1964)

1.2 J.W.Gibbs: Elementary Principles in Statistical Mechanics (Yale Univ. Press 1902, reprint Dover)

1.3 P.Ehrenfest, T.Ehrenfest: The Conceptional Foundations 0/ the Statistical Approach in Mechanics [trans!. by M.J.Moravcsik] (Cornell Univ. Press 1959), Enzyklopiidie der mathematischen Wissenscha/ten, Bd. 4 Art 32 (1911)

1.4 A.Einstein: Investigations on The Theory 0/ the Brownian Motion, ed. by R.Furth (trans!. by A.D.Cowper, Dover 1956)

1.5 P.Jordan: Statistische Mechanik au/ Quantenmechanischer Grundlage (Vieweg Braun­schweig 1933)

1.6 R.C.Tolman: The Principles 0/ Statistical Mechanics (Oxford Univ. Press 1938) 1.9 A.I.Khinchin: Mathematical Foundations 0/ Statistical Mechanics [trans!. by G.Gamow]

(Dover 1949) 1.1 0 R.Kurth: Axiomatics 0/ Classical Statistical Mechanics (Pergamon 1960) 1.11 O.Penrose: Foundations 0/ Statistical Mechanics (Pergamon 1970) 1.12 R.Ruelle: Statistical Mechanics (Benjamin 1969) 1.13 E.H.Kennard: Kinetic Theory o/Gases (McGraw-Hill, New York, London 1938) 1.14 The method employed here and in Sect.3.3.2 has been presented before only in Japanese

by M.Toda in Recent Problems in Physics [in Japanese] (Iwanami Shoten Pub!. 1948) 1.15 R.Kubo, M.Toda, N.Hashitsume: Statistical Physics II, Springer Ser. Solid-State Sci,

Vo!' 31 (Springer, Berlin, Heidelberg, New York 1983)

Chapter 2

2.1 L.D.Landau, E.M.Lifshitz: Statistical Physics (trans!. by D.Shoenberg, Clarendon Press, Oxford 1938, Pergamon 1958)

2.2 J.E.Mayer, M.G.Mayer: Statistical Mechanics (John Wiley & Sons 1940) 2.3 D.terHarr: Elements 0/ Statistical Mechanics (Holt, Rinehart & Winston 1961) 2.4 D.terHaar: Elements o/Thermostatics (Holt, Rinehart & Winston 1966) 2.5 R.Kubo, H.lchimura, T.Usui, N.Hashitsume: Statistical Mechanics (North-Holland,

Amsterdam 1965) 2.6 C.Kittel: Elementary Statistical Mechanics (John Wiley & Sons 1958) 2.7 R.W.Gurney: Introduction to Statistical Mechanics (McGraw-Hill 1949) 2.8 G.S.Rushbrooke: Introduction to Statistical Mechanics (Oxford 1951) 2.9 A.Sommerfeld: Thermodynamik und Statistik (Dietrich 1952) English trans!' by

J.Kestin: Thermodynamics and Statistical Mechanics (Academic Press 1956) 2.10 R.Becker: Theorie der Wiirme (Springer, Berlin, G6ttingen, Heidelberg 1955) 2.11 A.Mlinster: Statistische Thermodynamik (Springer, Berlin, G6ttingen, Heidelberg

1956, English trans!.: Statistical Thermodynamics, Springer, Berlin, Heidelberg, New York 1969)

242 References

2.12 T.L.HiIl: Statistical Mechanics (McGraw-Hill 1956) 2.13 S.Fliigge (ed.): Principles of Thermodynamics and Statics, Encyclopedia of Physics,

Vol. 3, Part 2 (Springer, Berlin, Gottingen, Heidelberg 1959) 2.14 K.Huang: Statistical Mechancis (John Wiley & Sons 1963) 2.15 F.Reif: Statistical and Thermal Physics (McGraw-Hill 1965) 2.16 G.H.Wannier: Statistical Physics (John Wiley & Sons 1966) 2.17 A.Isihara: Statistical Mechanics (Academic Press, New York 1971) 2.18 K.Husimi: Proc. Phys.-Math. Soc. Jpn. 22, 246 (1940)

Chapter 3

3.1 M.Toda: J. Phys. Soc. Japan 7, 230 (1952) 3.2 The method of steepest descent (the saddle point method) was applied to statistical

mechanics by C.G.Darwin and R.H.Fowler with wide applications: F.G.Fowler: Statistical Mechanics, 2nd ed. (Cambridge Univ. Press 1936)

3.3 R.H.Fowler, E.A.Guggenheim: Statistical Thermodynamics (Cambridge Univ. Press 1939, 1965)

3.4 J.O.Hirschfelder, C.F.Curtiss, R.B.Bird: Molecular Theory of Gases and Liquids (John Wiley & Sons, New York 1954)

3.5 J.deBoer: Physica 14,139 (1948) 3.6 J.G.Kirkwood, F.P.Buff: J.Chem. Phys. 17, 338 (1949) 3.7 A.Harasima: J.Phy. Soc. Jpn. 8, 343 (1953) 3.8 M.Toda: J. Phys. Soc. Jpn. 10, 512 (1955) 3.9 S.Ono, S.Kondo: Molecular Theory of Surface Tension in Liquids, Handbuch der Physik,

Bd. 10, Struktur der Fliissigkeiten (Springer, Berlin, Gottingen, Heidelberg 1960) 3.10 J.E.Mayer, M.G.Mayer: Statistical Mechanics (John Wiley & Sons 1940) 3.11 R.H.Fowler, E.A.Guggenheim: Statistical Thermodynamics (Cambridge Univ. Press

1939,1965) 3.12 K.Husimi: J.Chem. Phys. 18,682 (1950) 3.13 J.deBoer, G.E.Uhlenbeck (ed.): Studies in Statistical Mechanics, Vol. I, continuing

series (North-Holland, Amsterdam 1962) 3.14 P.Debye, E.Hiikel: Physik. Z. 24,185 (1923)

Chapter 4 [4.1-4.13] may serve as general references to Chap. 4. 4.1 R.Brout: Phase Transitions (Benjamin 1965) 4.2 M.Fisher: The Theory of Equilibrium Critical Phenomena, Reports on Progress in

Physics 30, Part 2, 615 (1967) 4.3 M.Fisher: The Nature of Critical Points, ed. by W.E.Britten, Lectures on Theoretical

Physics (U niv. of Colorado Press 1965) 4.4 H.S.Green, C.A.Hurst: Order-Disorder Phenomena (lnterscience 1964) 4.5 G.F.Newell, E.W.Montroll: Revs. Mod. Phys. 25, 353 (1953) 4.6 H.E.Stanley: Introduction to Critical Phenomena (Oxford Univ. Press 1971) 4.7 L.P.Kadanoff, W.Gotze, D.Hamblen, R.Hecht, E.A.S.Lewis, V.V.Palciauskas, M.

Rayl, J.Swift: Revs. Mod. Phys. 39, 395 (1967) 4.8 Shang-Keng Ma: Modern Theory of Critical Phenomena (Benjamin 1976) 4.9 P.Pfeuty, G.Toulouse: Introduction to the Renormalization Group and Critical Phenomena

[transl. by G.Barton] (John Wiley & Sons 1977) 4.10 K.G.Wilson, G.Kogut: The Renormalization Group and the e Expansion. Phys. Report

C. 12, No.2, 75 (1974) 4.11 K.G.Wilson: The Renormalization Group: Critical Phenomena and the Kondo Problem.

References 243

Revs. Mod. Phys. 47, 773 (1975) The above two articles were written by Wilson himself who introduced the renormalization method to the Critical Phenomena. 4.12 C.Domb, M.E.Green (ed.): Phase Transitions and Critical Phenomena, 1 (to) 4, 5A,

5B (Academic Press 1972-76) This series contains many review articles on experimental and theoretical approaches to the studies of phase transition.

4.13 H.E.Stanley (ed.): Cooperative Phenomena near Phase Transitions, a Bibliography with Selected Readings (MIT Press 1973)

4.14 E.Leib, T.Schultz, D.Mattis: Ann. Phys. 16, 407 (1961); S.Katsura: Phys. Rev. 127, 1508 (1962), 129, 2835 (1963)

4.15 C.N.Yang: Phys. Rev. 85, 808 (1952) 4.16 C.N.Yang, T.D.Lee: Phys. Rev. 87, 404 (1952) 4.17 T.D.Lee, C.N.Yang: Phys. Rev. 87,410(1952) 4.18 T.Asano: J. Phys. Soc. Jpn. 29, 350(1970) 4.19 E.H.Lieb, D:C.Mattis (ed.): Mathematical Physics in One Dimension (Academic Press,

New York 1966); H. Takahashi: Proc. Phys. Math. Soc. Jpn. 24, 60 (1942) 4.20 R.Ruelle: Commun. Math. Phys. 9, 267 (1968) 4.21 F.J.Dyson: Commun, Math. Phys.12, 91, 212(1969) 4.22 G.A.Baker: Phys. Rev. 126, 2071 (1962) 4.23 M.Kac, G.E.Uhlenbeck, P.C.Hemmer: J.Math. Phys. 4, 216 (1963) 4.24 N.G.vanKampen: Phys. Rev. 13SA, 362 (1964); Physica 48, 313 (1970) 4.25 O.Penrose, J.L.Lebowitz: J.Stat. Phys. 3, 211 (1971); see also P.C.Hemmer, J.L.

Lebowitz in [4.12]: SB, 108 (1976) 4.26 D.Poland, H.A.Scherega: Theory of Helix-coil Transitions in Biopolymers (Academic

Press 1970) 4.27 H.A.Kramers, G.H.Wannier: Phys. Rev. 60, 252 (1941) 4.28 L.Onsager: Phys. Rev. 65,117 (1944) 4.29 R.J.Baxter: Ann. Physics 70,193 (1972)

Chapter 5

[5.1-5.7] may serve as general references to Chap. 5. 5.1 D.terHaar: Foundations of Statistical Mechanics, Revs. Mod. Phys. 27, 289 (1955) 5.2 Proceedings of the International School of Physics "Enrico Fermi" XIV, Ergodic The­

ories (Academic Press 1961) 5.3 I.E.Farquhar: Ergodic Theory in Statistical Mechanics (lnterscience 1964) 5.4 R.Jancel: Foundations of Classical and Quantum Statistical Mechanics (Pergamon

1963) 5.5 E.Hopf: Ergodentheorie (Chelsea 1948) 5.6 A.I.Khinchin: Mathematical Foundations of Statistical Mechanics (Dover 1949) 5.7 V.I.Arnold, A.Avez: Ergodic Problems of Classical Mechanics (Benjamin 1968) 5.8 P. and T.Ehrenfest: The Conceptual Foundations of Statistical Approach in Mechanics

[Trans!. by M.J.Moravcsik] (Cornell Univ. Press 1959) 5.9 G.D.Birkhoff: Proc. Nat. Acad. Sci. 17, 656 (1931) 5.10 B.O.Koopmann: Proc. Nat. Acad. Sci. 17, 315 (1931) 5.11 G.D.Birkhoff, B.O.Koopmann: Proc. Nat. Acad. Sci. 18, 279 (1932) 5.12 P.A.Smith,: J. Math. 7, 365 (1928) 5.13 L.VanHove: Math. Rev. 17,812 (1956) 5.14 D.S.Ornstein: Adv. in Math. 4, 337 (1970) 5.15 Ja.G.Sinai: Ergodicity of Boltzmann's Gas Model. In Statistical Mechanics, Founda­

tions and Applications, Proceedings of the I.U.P.A.P. Meeting, Copenhagen, 1966,

244 References

p. 559 [T.A.Bak, ed.] (Benjamin 1967) 5.16 Y.Aizawa: J.Phys. Soc. Jpn. 33,1693 (1972) [5.17-5.26] may serve as reviews of materials on Sects. 5.4-5.7. 5.17 S.Joma (ed.): Topics in Nonlinear Dynamics, AlP Conference Proceedings, No. 46.

Amer. Inst. Phys. (New York 1978). Especially, see papers by J.Moser, M.V.Berry, J.Ford, Y.M.Treve

5.18 J.Ford: Fundamental Problems in Statistical Mechanics 1 [B.D.G.Cohen, ed.] (North­Holland 1975)

5.19 G.E.O.Giacaglia: Perturbation Methods in Nonlinear Systems (Springer, Berlin, Heidel­berg, New York 1972)

5.20 C.L.Siegel, J.K.Moser: Lectures on Celestial Mechanics (Springer, Berlin, Heidelberg, New York 1971)

5.21 J.Moser: Stable and Random Motions in Dynamical Systems (Princeton 1973) 5.22 Y.Hagiwara: Celestial Mechanics, Vol. 4, Part 2 (Japan Soc. for the Promotion of

Science, Tokyo 1975) 5.23 G.Casati, J.Ford (eds.): Stochastic Behavior in Classical and Quantum Hamiltonian

Systems, Como, 1977, Lecture Notes in Phys. 93 (Springer, Berlin, Heidelberg, New York 1979)

5.24 I.C.Percival: Semiclassical Theory of Bound States, (I.Prigogine, S.A.Rice, eds.) Advances in Chemical Physics XXXVI (John Wiley & Sons 1977)

5.25 B.V.Chirikov: Phys. Repts. 52, 263 (1979) 5.26 R.H.G.Helleman: In Fundamental Problems in Statistical Mechanics V [E.G.D.Cohen

ed.] (North-Holland 1980) 5.27 H.Bruns: Acta Math. 11, 25 (1887) 5.28 H.Poincare: Les Methodes Nouvelles de la Mecanique Celeste (Dover 1957) 5.29 E.Fermi: Physik. Z. 24, 261 (1923) 5.30 E.Fermi, J .Pasta, S. Ulam: Los Alamos Rpt. LA-1940 (1955); Collected Papers of Enrico

Fermi (Univ. of Chicago Press, Chicago 1965) Vol. II, p. 977 5.31 P.C.Hemmer: Thesis, Dynamic and Stochastic Types of Motion in the Linear Chain

(Trondheim 1959) 5.32 J.Ford, J.Waters: J.Math. Phys. 4,1293 (1963) ~.33 N.Saito, N.Hirotomi, A.Ichimura: J. Phys. Soc. Jpn. 39,1931 (1975) 5.34 M.Henon, C.Heiles: Astron. J. 69, 73 (1964) 5.35 J.N.Greene: J.Math. Phys. 9, 760 (1968) 5.36 A.N.Kolmogorov: Intern. Congress Mathematician, Amsterdam I, 315 (1954) 5.37 J.Moser: Comm. Pure and Appl. Math. 9, 673 (1956) 5.38 N.Saito, H.Hirooka, J.Ford, F.Vivaldi, G.H.Walker: Physica D, 273 (1982) 5.39 G.Ludwig: In Ref. [5.2] and Die Grundlagen der Quantenmechanik (Springer, Berlin,

Gottingen, Heidelberg 1954) 5.40 N.G.vanKampen: Physica 20, 603 (1954) 5.41 M.Fierz: Helv. Phys. Acta 28, 705 (1955) 5.42 J.E.Farquhar, P.T.Landsberg: Proc. Roy. Soc. London, 239, 134 (1957) 5.43 P.Bocchieri, A.Loinger, Phys. Rev 111, 668 (1958): Phys. Rev. 114, 948 (1959);

G.M.Prosperi, A.Scotti: Nuovo Cimento 13,1007 (1959); 17,267 (1960); P.Bocchieri, G.M.Prosperi: Ref. [5.15, p. 17]

5.44 S.Golden, H.C.Longuet-Higgins: J.Chem. Phys. 33, 1479 (1960) 5.45 N.Pomphrey: J. Phys. B. 7,1909 (1974) 5.46 A.Wright: Phys. Rev. 76,1826 (1949) 5.47 M.J.K1ein: Phys. Rev. 86, 807 (1952)

Subject Index

Absolute temperature 36,38,40, 51,53

Absolute zero of temperature 69 Abstract dynamical system 195 Action integral 180 Action variable 25,47,179,180,

181 Adiabatic

change 43,91 demagnetization 70 process 235 susceptibility 236 theorem 43 theorem in classical mechanics

48 theorem in statistical mechanics

45 Angle variable 175,180,181 Approach to equilibrium 3 a priori probability 29 Automorphism 195 Average 4,53 Average convergence 190

Baker's transformation 195 Bernoulli scheme 195,196,198,

200,201 Bernoulli's equation 90 Bernoulli's system 225 Bethe approximation 155 Binomial distribution 5 Birkhoff's first theorem 186, 188 Birkhoff's second theorem 187 Birkhoff's theorem 188,227 Bloch equation 59 Boltzmann constant 31 Boltzmann distribution 114 Boltzmann factor 52

Boltzmann's principle 31 Boltzmann's relation 85 Boltzmann-Planck's method 54 Bose

condensation 94, 118 distribution 75,78 gas 93 particle 10, 73 system 74, 82, 86, 90

Bose-Einstein distribution 78 Boson 73 Boyle-Charles'law 10,13,92,

118 Bragg-Williams approximation

153 Brownian motion 6, 10 Bruns'theorem 204

C-system 202,203,215 Canonical distribution 52 Canonical ensemble 51 Chemical potential 63,64, 65 Classical

distribution 79 gas 100 limit 59 mechanics 1 particle 88 statistics 100, 102 sum over states 61 systems 100

Cluster integral 108 Combined system 31,37 Compressibility equation 10 Computer experiment 212,217,

226,227 Condensation 39,129, 130 Correlation 6

246 Subject Index

Correlation coefficient 193 Correspondence between classical

and quantum mechanics 25 Coulomb force 112 Critical

exponent 149,159,160,175 phenomenon 160, 164 point 68 temperature 102, 123

Crystal lattice 76 Curie point 68,141 Curie temperature 118,148,154 Curie's law 12, 118, 143 Curve of section 218

de Boer parameter 102 de Boer's corresponding state 102 Debye-Hiickel theory 116 Debye length 113 Debye temperature 77 Debye's specific heat formula 77 Debye's T3-law 76 Degenerate 96 Density

fluctuation 8, 92 matrix 18,29,58,60,228,

236 of states 30 operator 58

Detailed balancing 82,83,84 Diffeomorphism 195 Discernibility 232 Dual lattice 140 Dulong-Petit's law 78

Eight-vertex model 151 Einstein-Brillouin-Keller quantiza­

tion 233 Einstein's specific heat formula

77 Electrolyte 114 Electrolyte solutions Electromagnetic wave Electron 73

gas 112 gas in metal 92

Electronic specific heat

117 75,90

96,98

Elliptic fixed point 220,225,227 Energy 85

by mixing 34 change 51

Ensemble average 28 Entropy 31, 33, 43, 51, 53, 56,

85 and fluctuation 84 of the partition 200

Equal a priori probability 177, 178

Equilibrium condition Equilibrium distribution Equipartition of energy Ergodic

component 227

38 82

104

hypothesis 28,45, 177, 187 problem 29

Ergodicity 187 Expectation value 6 Extensive variable 65 External force 41

Fermi-Dirac distribution 79 Fermi

distribution 79,95 energy 95 gas 95,99 level 95 particle 73 system 82, 86, 90 temperature 96

Fermi-Pasta-Ulam's experiment 216

Fermi's problem 206 Fermion 73 First-order phase transition 39 Fixed point 174, 220 Fluctuation 4, 67, 87

of energy 67 of light intensity 9 of magnetization 11

Free energy 63, 85 Free particle 32 Frequency spectrum 76 Fundamental laws of thermo-

dynamics 51

G-path 186-189, 192 Gaussian curvature 184,203 Gaussian-Markovian process 136 Geodesic 183, 203 Gibbs'

distribution 52 ensemble 28 free energy 63 paradox 34

Ginzburg-Landau Hamiltonian 166,174

Grand canonical ensemble 65 Grand partition function 87

Harmonic oscillator 26, 180 Heat radiation 76 Heisenberg model 119 Helix -coil transition 137 Helmholtz's free energy 57 Henon-Heils system 217, 234 Highly relativistic case 98 High temperature expansion 91 Homoclinic point 225,226 Hopf's theorem 191, 230 Hyperbolic fixed point 220,

225-227

Imperfect gas 107 Index of the fixed point 221 Individual ergodic theorem 189 Inner virial 14 Integrable system 181 Intensity of light 9 Invariant curve 218,221, 225 Invariant function 189 Ion 114 Ion atmosphere 114 Irreducible integral 110 Ising

magnet 121 model 119,130,152,160 system 123,137

Isolated system 28 Isolated susceptibility 237 Isolating integral 216-218, 233 Isomorphic dynamical system

197

Subject Index 247

Isomorphism 195,201 Isothermal distribution Isothermal susceptibility

52 236

Joule-Thomson effect 70

K-system 199-201,203 Kadanoff transformation 172 KAM theorem 221,225,233

(Kolmogorov, Arnol'd and Moser) theorem 222

Khinchin's theorem 193 Kolmogoroventropy 198, 201 Kolmogorov transformation 199

Lagrange's indeterminate multiplier 56

A-transition 94 Laser 40 Lattice gas 120,121,123,124,

133 Law of the equipartition of energy

13, 103 Lennard -J ones potential 100 Level density 90

of a free particle Liouville equation Liouville operator Liouville's theorem

185

88 19 21,189,191

19,28,178,

in classical mechanics 20 Liquid helium 94 Lissajous' figure 21 Ljapunov number 219 Long-range interaction 135 Long-time average 230, 227 Long-time behavior 208

Macro-observable 230 Macroscopic 1 Magnetic moment II Magnetic susceptibility II, 68 Magnetization 67 Many-particle system 72 Mathieu's equation 214 Maxwell-Boltzmann distribution

79

248 Subject Index

Mayer function 107 Mean ergodic theorem 189 Measure preserving transformation

178,202,219 Mechanical pressure 42 Melting 42 Metallic electron 97 Metric indecomposability 192 Metrical transitivity 191, 219 Microcanonical ensemble 29 Microscopic 1 Mixing 192,201 Molecular field approximation

153 Molecular field theory 149

Nature of wave 88 Neel temperature 126, 150 Negative curvature 184, 203 Negative temperature 36,40 Nernst-Plank's theorem 68 Nernst's theorem 68 Newton's method 225 Nonlinear lattice vibration 205,

206 Normal mode 208, 212 Nuclear demagnetization 70 Number-density 6 Number of microscopic state 30,

32

One-body density 6 Ornstein-Zernike relation 10 Oscillator system 74 Osmotic pressure 117 Outer virial 14

Pair distribution function 7 Paramagnetic substance 70 Particles in a one-dimensional

parabolic potential 80 Partition 199 Partition function 57, 102, 103 Pauli's (exclusion) principle 74,

95 Perfect gas 33, 72, 88, 90 Permanent 74

Permutation 34 Phase

average 186, 187 integral 61, 103 space 15,19

Phonon 74 Photon 9,74,89 Photon gas 99 Planck's

characteristic function 57 law of radiation 75 radiation formula 84

Poisson equation 114 Poisson's adiabatic equation of

state 91 Poisson's distribution 5 Poincare mapping 217 J 221 Poincare-Birkhoff fixed-point

theorem 218,221,222 Poincare-Fermi's theorem 204 Pressure 41,50,53,62, 104,

105 Pressure equilibrium 41 Principle of equal (a priori) prob­

ability 3,28,29 Principles of statistical mechanics

28 Probability 4

distribution 4 of a state 53

Proton 73

qp space 19 Quantity of heat 51, 53 Quantum

effect 92, 100, 102 mechanics 2 statistics 72

Quasi-ergodic 206 Quasi-ergodic hypothesis 195

Radial distribution function 7 Radiation 83 Radiation field 75 Random phase assumption 228, 230

Relative fluctuation 5

Relativistic classical gas 1 00 gas 98 particle 89

Renormalization group 171 Representative point 20 Residual entropy 69 Residue of the fixed

point 220 Resonance condition 209 Rest energy 98

Saddle point method 84 Scaling 168 SchrOdinger equation 88, 101 Semi-conductor 92 Simple molecule 100 Slater determinant 74 Small denominator 210,224 Solute ion 117 Sommerfeld constant 96 Specific heat 67

of a metal 97 of solid 76

Spectral invariant 191 Spectral isomorphism 197 Spherical model 150 Spin 73 Spin system 35,40 Spin-alignment 70 Spin-coordinate 73 Spin-function 73 Spontaneous magnetization 68 State density 89 Statistical thermodynamics Statistical weight 31 Stefan-Boltzmann's law of radiation

75 Stirling's formula 87 Stochastic region 219 Stochastic region (or sea) 218 Strong convergence 190 Subsystems

with a given chemical potential 63

with a given pressure 61

Subject Index 249

with a given temperature 51 Sum over states 57, 84, 103 Surface tension 106, 109 Susceptibility 235

Temperature equilibrium 37 Theorem of Poincare and Fermi

225 Thermal equilibrium 39 Thermodynamics 53

limit 37, 126, 127, 129 potential 63,116 relation 50 weight 31

Third integral 216 Third law of thermodynamics

68 Time average 186, 187 Total differential 53 T-P distribution 62 T-Il distribution 65 Trajectory 20 Transition from quantum mechan-

ics to classical mechanics 60 Transition probability 3,82 Two-body density 6 Two-dimensional perfect gas 81

van der Waals attraction 100 van der Waals' equation of state

11,137 Virial

coefficient 112 expansion 112 theorem 12,14,105

von Neumann's theorem 190

Wave 8 Weak convergence 191,232 Weiss approximation 153 Weyl's billiard 22 Wigner's representation 22 Work done upon the system 51

XY model 120, 137

Zero-point energy 95

The Monte Carlo Methods in Atmospheric Optics By G.I.Marchuk, G.A.Mikhailov, M.A.Nazaraliev, R.A.Darbinjan, B.A. Kargin, B.S.Elepov

1980. 44 figures, 40 tables. VIII, 208 pages (Springer Series in Optical Sciences, Volume 12) ISBN 3-540-09402-4

Contents: Introduction. - Elements of Radia­tive-Transfer Theory Used in Monte Carlo Methods. - General Questions About the Monte Carlo Technique for Solving Integral Equations of Transfer. - Monte Carlo Methods for Solving Direct and Inverse Problems ofthe Theory of Radiative Transfer in a Spherical Atmosphere. - Monte Carlo Algorithms for Solving Nonstationary Problems of the Theory of Narrow-Beam Pro­pagation in the Atmosphere and Ocean. -Monte Carlo Algorithms for Estimating the Correlation Function of Strong Light F1uctua­tions in a Turbulent Medium. - References. -Subject Index.

Monte Carlo Methods in Statistical Physics Editor: K.Binder

1979.91 figures, 10 tables. XV, 376 pages (Topics in Current Physics, Volume 7) ISBN 3-540-09018-5

Contents: K Binder: Introduction: Theory and "Tech­nical" Aspects of Monte Carlo Simulations. -D.Levesque, l.l. Weis, l. P. Hansen: Simulation of Classical F1uids. - D. P. Landau: Phase Diagrams of Mixtures and Magnetic Systems. - D. M Ceperiey, M H. Kalos: Quan­tum Many-Body Problems. - H. Milller­Krumbhaar: Simulation of Small Systems. -KBinder, MH.Kalos: Monte Carlo Studies of Relaxation Phenomena: Kinetics of Phase Changes and Critical Slowing Down. -H.Milller-Krumbhaar: Monte Carlo Simula­tion of Crystal Growth. - K Binder, D. Stauffer: Monte Carlo Studies of Systems with Dis­orders. - D. P. Landau: Applications in Surface Physics.

Real-Space Renormalization Editors: T. W.Burkhardt, J.M.J. van Leeuwen

1982.60 figures. XIII, 214 pages (Topics in Current Physics, Volume 30) ISBN 3-540-11459-9

Contents: T. WBurkhardt, l.M.l. van Leeuwen: Progress and Problems in Real-Space Renormaliza­tion. - T. W Burkhardt: Bond-Moving and Variational Methods in Real-Space Renorma­lization. - R. H.Swendsen: Monte Carlo Renor­malization. - G.F.Mazenko, O. T. Valls:The Real-Space Dynamic Renormalization Group. - P.l1euty, R.Jullien, KA.Penson: Renormalization for Quantum Systems. -M Schick: Application of the Real-Space Renormalization to Adsorbed Systems. -H.E.Stanley, P.l.Reynolds, S.Redner, F. Family: Position-Space Renormalization Group for Models of Linear Polymers, Branched Polymers, and Gels. - Subject Index.

Structural Phase Transitions I Editors: K.A.Miiller, H.Thomas

1981. 61 figures. IX, 190 pages (Topics in Current Physics, Volume 23) ISBN 3-540-10329-5

Contents: K A. Milller: Introduction. - P. A. Fleury, K Lyons: Optical Studies of Structural Phase Transitions. - B. Domer: Investigation of Structural Phase Transformations by Inelastic Neutron Scattering. - B.Lilthi, WRehwald: Ultrasonic Studies Near Structural Phase Transitions.

Springer-Verlag Berlin Heidelberg New York

Springer Series in

Synergetics Series Editor: H.Haken

Volume 1: H.Haken Synergetics An Introduction. Nonequilibrium Phase Transi­tions and Self-Organization in Physics, Chemistry and Biology. 2nd enlarged edition. 1978. 152 figures, 4 tables. XII, 355 pages. ISBN 3-540-08866-0

Volume 2: Synergetics A Workshop. Proceedings ofthe International Workshop on Synergetics at SchloB Elmau, Bavaria, May 2-7, 1977 Editor: H.Haken 1977. 136 figures. VIII, 274 pages. ISBN 3-540-08483-5

Volume 3: Synergetics Far from Equilibrium Proceedings of the Conference Far from Equi­librium: Instabilities and Structures, Bordeaux, France, September 27-29, 1978 Editors: A. Pacault, C. Vidal 1979. 109 figures, 3 tables. IX, 175 pages. ISBN 3-540-093044

Volume 4: Structural Stability in Physics Proceedings of Two International Symposia on Applications of Catastrophe Theory and Topo­logical Concepts in Physics, Tiibingen, Federal .RepublicofGermany, May 2-6 and December 11-14,1978 Editors: W.Giittinger, H.Eikemeier 1979. 108 figures, 8 tables. VIII, 311 pages. ISBN 3-540-09463-6 Volume 5: Pattern Formation by Dynamic Systems and Pattern Recognition Proceedings of the International Symposium on Synergetics at SchloB Elmau, Bavaria, April 30-May 5, 1979 Editor: H.Haken 1979. 156 figures, 16 tables. VIII, 305 pages. ISBN 3-540-09770-8 Volume 6: Dynamics of Synergetic Systems Proceedings of the International Symposium on Synergetics, Bielefeld, Federal Republic of Germany, September 24-29, 1979 Editor: H.Haken 1980. 146 figures, some in color, 5 tables. VIII, 271 pages. ISBN 3-540-09918-2

Volume 7: L. A. Blumenfeld Problems of Biological Physics 1981. 38 figures. IX, 224 pages. ISBN 3-540-10401-1

Volume 8: Stochastic Nonlinear Systems in Physics, Che!I1istry, and Biology Proceedings of the Workshop, Bielefeld, Federal Republic of Germany, October 5-11, 1980 Editors: L.Amold, R.Lefever 1981. 48 figures. VIII, 237 pages. ISBN 3-540-107134

Volume 9: Numerical Methods in the Study of Critical Phenomena Proceedings of a Colloquium, Carry-Ie-Rouet, France, June 2-4, 1980 Editors: J.DelIa Dora, J.Demongeot, B.LacolIe 1981. 83 figures. IX, 269 pages. ISBN 3-540-11009-7

Volume 10: Yu. L.KIimontovich The Kinetic TheolY of Electromagnetic Processes Translated from the Russian by A Dobroslavsky 1982. Approx. 320 pages ISBN 3-540-11458-0. In preparation

Volume 11: Chaos and Order in Nature Proceedings of the International Symposium on Synergetics at SchloB Elmau, Bavaria, April 27-May 2, 1981 Editor: H.Haken 1981. 134 figures. VIII, 275 pages. ISBN 3-540-11101-8

Volume 12: Nonlinear Phenomena in Chemical Dynamics Proceedings of an International Conference, Bordeaux, France, September 7-11,1981 Editors: C. Vidal, A.Pacault 1981. 124 figures. X, 280 pages. ISBN 3-540-112944

Volume 13: C. W.Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences 1982. 29 figures. Approx. 450 pages ISBN 3-540-11357-6. In preparation

Volume 14: W.Weidlich, G.Haag Dynamics of Interacting Populations in Society ISBN 3-540-113584. In preparation

Volume 15: W.Horsthemke, R.Lefever Nonequilibrium Transitions Induced by External Noise ISBN 3-540-11359-2. In preparation

Volume 16: L.A.Blumenfeld Physics of Bioenergetic Processes ISBN 3-540-11417-3. In preparation

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