Bibliography

[AA68] Arnold, V.I., Avez, A.: Ergodic problems of classical mechanics, Ben­jamin, New York, 1968.

[ACG96] Artuso, R., Casati, G., Guarneri, 1.: Numerical experiments on billiards, Journal of Statistical Physics, 83, 145-166, 1996.

[AGM71] Abraham, D., Gallavotti, G., Martin-Lof, A.: Surface tension in the Ising model, Lettere al Nuovo Cimento, 2,143-146,1971. And Surface tension in the two-dimensional Ising model, Physica, 65, 73-88, 1973.

[AL95] Abraham, D.B., Latremoliere, F.T.: Corner spontaneous magnetization, Journal of Statistical Physics, 81, 539-559, 1995.

[AM73] Abraham, D.B., Martin-Lof, A.: The transfer matrix for a pure phase in the two-dimensional Ising model, Communications in Mathematical Physics, 31, 245-268, 1973.

[AR76] Abraham, D.B., Reed, P.: Interface profile in the two-dimensional Ising model, Communications in Mathematical Physics, 49, 35-46, 1976.

[AW68] Adler, R., Weiss, B.: Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society, 98, 1968.

[Ab71] Abraham, D.B.: On the transfer matrix for the two-dimensional Ising model, Studies in Applied Mathematics, 50, 71-88, 1971.

[Ab78a] Abraham, D.B.: n-point functions for the rectangular Ising ferromagnet, Communications in Mathematical Physics, 60, 205-213, 1978.

[Ab78b] Abraham, D.W.: Block spins in the edge of an Ising ferromagnet in a half plane, Journal of Statistical Physics, 19, 553-556, 1978.

[Ab84] Abraham, D.W.: Intrinsic structure and long ranged correlations in interfaces, Physical Review, 29B, 526-526, 1984.

[Ai80] Aizenman, M.: Translation invariance and instability of phase coexis­tence in the two-dimensional Ising system, Communications in Mathe­matical Physics, 73, 83-94, 1980.

[An82] Andrej, 1.: The rate of entropy change in non-Hamiltonian systems, Physics Letters, lIlA, 45-46, 1982. And Ideal gas in empty space, Nuovo Cimento, B69, 136-144, 1982. See also The relation between entropy production and K -entropy, Progress in Theoretical Physics, 75, 1258-1260, 1986.

[An84] Anderson, P.W.: Basic notions of condensed matter physics, Benjamin­Cummings, Menlo Park, California, 1984.

[As70] Asano, T.: Theorems on the partition function of the Heisenberg ferro­magnet, Journal of the Physical Society of Japan, 29, 350-359, 1970.

[BC94] Borgs, C., Chayes, J.: The covariance matrix in the Potts model: a random cluster analysis, Journal of Statistical Physics, 82, 1235-1297, 1995.

[BCS57] Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Theory of superconductivity, Physical Review, 108, 1175-1204, 1957.

314 Bibliography

[BF67]

[BG95]

[BG97]

[BGG84]

[BGG93]

[BGG97]

[BGJS73]

[BH93]

[BJS72]

[BK52]

[BLP079]

[BR75]

[BS67]

[BaOO]

[Ba82] [Ba90]

[Be31]

[Be69]

[Be84]

[Be87]

[Be94]

Burford, R.J., Fisher, M.E.: Theory of critical scattering and correla­tions. I. The Ising model, Physical Review, 156, 583-622, 1967. Benfatto, G., Gallavotti, G.: Renormalization group, Princeton Univer­sity Press, 1995. Bonetto, F., Gallavotti, G.: Reversibility, coarse graining and the cha­oticity principle, Communications in Mathematical Physics, 189, 263-276, 1997. Benettin, G., Galgani, L., Giorgilli, A.: Boltzmann's ultraviolet cut-off and Nekhoroshev's theorem on Arnold diffusion, Nature, 311, 444-445, 1984. Benettin, G., Galgani, L., Giorgilli, A.: The dynamical foundations of classical statistical mechanics and the Boltzmann-Jeans conjecture, Edited by S. Kuksin, V.F. Lazutkin, J. Poschel, Birkhauser, 1993. Bonetto, F., Gallavotti, G., Garrido, P.: Chaotic principle: an experi­mental test, Physica D, 105, 226-252, 1997. Benettin, G., Gallavotti, G., Jona-Lasinio, G., Stella, A.: On the Onsager-Yang value of the spontaneous magnetization, Communica­tions in Mathematical Physics, 30, 45-54, 1973. Bohm, D., Hiley, B.J.: The undivided universe: an ontological interpre­tation of quantum mechanics, Routledge & Kegan Paul, London, 1993. Benettin, G., Jona-Lasinio, G., Stella, A.: Duality and correlation func­tions in the Ising model, Lettere Nuovo Cimento, 4, 443-447, 1972. Berlin, T.H., Kac, M.: The spherical model of a ferromagnet, Physical Review, 86, 821-835, 1952. Bricmont, J., Lebowitz, J.L., Pfister, C., Olivieri, E.: Nontranslation­invariant Gibbs states with coexisting phases, Communications in Math­ematical Physics, 66, 1-20, 1979; and the first three authors Nontrans­lation-invariant Gibbs states with coexisting phases, Communications in Mathematical Physics, 66, 21-36,1979 and Absence of symmetry break­down and uniqueness of the vacuum for multicomponent field theories, Communications in Mathematical Physics, 69, 267-291, 1979. Bowen, R., Ruelle, D.: The ergodic theory of axiom A flows, Inventiones Mathematicae, 29, 181-205, 1975. Berezin, F.A., Sinai, Y.G.: The existence of phase transition for a lattice gas with attraction between the particles, Transactions of the Moscow Mathematical Society, 19, 219-234, 1967. Bachelier, L.: Theorie de la speculation, Annales Scientifiques de l'Ecole Normale Superieure, 17, 21-36, 1900. Baxter, R.: Exactly solved models, Academic Press, London, 1982. Bach, A.: Boltzmann's probability distribution of 1877, Archive for the History of Exact Sciences, 41, 1-40, 1990. Bethe, H.A.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktio­nen der linearen Atomkette, Zeitschrift fur Physik, 71, 205-226, 1931. Berezin, F.A.: The plain Ising model, Russian Mathematical Surveys, 24, 2, 1-22, 1969. Benettin, G.: On the Boltzmann conjecture for a classical ''freezing'' of fast degrees of freedom, in "Boltzmann's legacy 150 years after his birth, Atti dei Convegni Lincei, 131, Accademia dei Lincei, 89-105, 1997. Bell, J.S.: Speakable and unspeakable in quantum mechanics, Cambridge University Press, 1987. Benettin, G.: Ergodicity and time-scales for statistical equilbrium in classical dynamical systems, Progress of Theoretical Physics, Supple­ment, 116, 207-234, 1994.

[B064]

[B066]

[B068]

[B070]

[Bo7la]

[Bo7lb]

[Bo7lc]

[B072]

[B074]

[B077]

[B084]

[B096]

[B097]

[B097b] [Br65] [Br66] [Br69]

[Br76]

Bibliography 315

Boltzmann, L.: Lectures on gas theory, English edition annotated by S. Brush, University of California Press, Berkeley, 1964 (reprint). Boltzmann, L.: Uber die mechanische Bedeutung des zweiten Haupt­satzes der Wiirmetheorie, in Wissenschaftliche Abhandlungen, ed. F. Hasenohrl, vol. I, pp. 9-33, reprinted by Chelsea, New York. Boltzmann, L.: Studien iiber das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punk ten, in Wissenschaftliche Abhand­lungen, ed. F. Hasenohrl, vol. I, pp. 49-96, reprinted by Chelsea, New York. Bowen, R. : Markov partitions for Axiom A difJeomorphisms, American Journal of Mathematics, 92, 725-747, 1970. And: Markov partitions and minimal sets for Axiom A difJeomorphisms, American Journal of Mathematics, 92, 907-918, 1970. Boltzmann, L.: Analytischer Beweis des zweiten Hauptsatzes der mecha­nischen Wiirmetheorie aus den Siitzen iiber das Gleichgewicht der lebendigen Kraft, in Wissenschaftliche Abhandlungen, ed. F. Hasenohrl, vol. I, pp. 288-308, reprinted by Chelsea, New York. Boltzmann, L.: Einige allgemeine Siitze iiber Wiirmegleichgewicht, in Wissenschaftliche Abhandlungen, ed. F. Hasenohrl, vol. I, pp. 259-287, reprinted by Chelsea, New York. Boltzmann, L.: Zusammenhang zwischen den Siitzen iiber das Verhalten mehratomiger Gasmolekiile mit Jacobi's Princip des letzten Multiplica­tors. This is sometimes quoted as a paper but it is sec. 1 of [Bo7lb], see the introduction of [Gi81]. Boltzmann, L.: Weitere Studien iiber das Wiirmegleichgewicht unter Gasmolekiilen, English translation in S. Brush, Kinetic theory, vol. 2, p. 88. Original in Wissenschaftliche Abhandlungen, ed. F. Hasenohrl, vol. I, pp. 316-402, reprinted by Chelsea, New York. Boltzmann, L.: Theoretical physics and philosophical writings, ed. B. McGuinness, Reidel, Dordrecht, 1974. Boltzmann, L.: Uber die Beziehung zwischen dem zweiten Hauptsatze der mechanischen W iirmetheorie und der Wahrscheinlichkeitsrechnung, r'espektive den Siitzen iiber das Wiirmegleichgewicht, in Wissenschaftli­che Abhandlungen, vol. II, pp. 164-223, F. Hasenohrl, Chelsea, New York, 1968 (reprint).

Boltzmann, L.: Uber die Eigenschaften monzyklischer und anderer damit verwandter Systeme, in Wissenschaftliche Abhandlungen, ed. F.P. Hasenohrl, vol. III, p. 122-152, Chelsea, New York, 1968 (reprint). Boltzmann, L.: Entgegnung auf die wiirmetheoretischen Betrachtungen des Hrn. E. Zermelo, Engl. transl.: S. Brush, Kinetic Theory, vol. 2, p. 218, Pergamon Press, Oxford, 1965-1972. Boltzmann, L.: Zu Hrn. Zermelo's Abhandlung "Ueber die mechanische Erkliirung irreversibler Vorgiinge", Engl. trans. in S. Brush, Kinetic Theory, vol. 2, p. 238, Pergamon Press, Oxford, 1965-1972. Bonetto, F.: private communication. Brout, R.H.: Phase transitions, Benjamin, New York, 1965. Brush, S.G.: Kinetic theory, Pergamon, Oxford, 1966. Brush, S.G.: For a history of the Ising model, Reviews of Modern Physics, 39, 883 (1969). Brush, S.: The kind of motion we call heat, North Holland, Amsterdam, 1976 (vol. II), 1986 (vol. I).

316 Bibliography

[Br86] Brydges, D.C.: A short course on cluster expansions, in Critical phe­nomena, random systems, gauge theories, ed. K. Osterwalder, R Stora, Les Houches XLIII, North Holland, Amsterdam, 1986.

[CC53] Chapman, S., Cowling, T.: The mathematical theory of nonuniform gases, Cambridge University Press, 1953.

[CC074] Capocaccia, D., Cassandro, M., Olivieri, E.: A study of metastability in the Ising model, Communications in Mathematical Physics, 39, 185-205, 1974.

[CELS93] Chernov, N. I., Eyink, G. L., Lebowitz, J.L., Sinai, Y.: Steady state electric conductivity in the periodic Lorentz gas, Communications in Mathematical Physics, 154, 569-601, 1993.

[CF71] Camp, W.J., Fisher, M.E.: Behavior of two point correlation functions at high temperature, Physical Review Letters, 26, 73-77 and 565-568, 1971.

[CG99] Cohen, E.G.D., Gallavotti, G.: A note on two theorems in nonequilib­rium statistical mechanics, in cond-mat 9903148, in print in Journal of Statistical Physics.

[CL98] Ciliberto, S., Laroche, C.: An experimental verification of the Galla­votti-Cohen fluctuation theorem, Journal de Physique, 8, 215-222, 1998.

[Ca67] Cantoni, G.: Su alcune condizioni fisiche dell' affinitiL e sui moto brow­niano, Il Nuovo Cimento, 27, 156-167, 1867.

[Ca83] Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems, Communications in Mathematical Physics, 85, 517-528, 1983.

[Ce69] Cercignani, C.: Mathematical methods in kinetic theory, Plenum, New York,1969.

[C165] Clausius, R: The nature of the motion which we call heat, translated in Kinetic theory, ed. S. Brush, vol. 1, pp. 111-134, Pergamon Press, Oxford, 1965.

[Co69] Cohen, E.G.D.: The kinetic theory of dilute gases, in Transport phe­nomena in fluids, H. Hanley ed., Ch. VI, pp. 119-156, Dekker, New York,1969.

[Co93] Cohen, E.G.D.: Fifty years of kinetic theory, Physica A, 194, 229-257, 1993.

[DKS92] Dobrushin, RL., Kotecky, R, Shlosman, S.: The Wulff construction, Translations of Mathematical Monographs, 104, American Mathemat­ical Society, Providence, 1992. See also: Dobrushin, RL., Kotecky, R, Shlosman, S.: A microscopic justification of the Wulff construction, Journal of Statistical Physics, 72, 1-14, 1993; and Dobrushin, RL., Hryniv, 0.: Fluctuations of the phase boundary in the 2D Ising model, Communications in Mathematical Physics, 189, 395-445, 1997.

[DL67] Dyson, F., Lenard, A.: Stability of matter, I, Journal of Mathemati­cal Physics, 8, 423-434, 1967. And Stability of matter, II, Journal of Mathematical Physics, 9, 698-711, 1968.

[DLS78] Dyson, F.J., Lieb, E.H., Simon, B.: Phase transitions in quantum spin systems with isotropic and anisotropic interactions, Journal of Statisti­cal Physics, 18, 335-383, 1978.

[DM96] Dettman, C.P., Morris, G.P.: Proof of conjugate pairing for an isokinetic thermostat, Physical Review 53 E, 5545-5549, 1996.

[DM67] Dobrushin, R, Minlos, R: Existence and continuity of the pressure in classical statistical mechanics, Theory of Probability and its Applica­tions, 12, 535-559, 1967.

Bibliography 317

[DPH96] Dellago, C., Posch, H., Hoover, W.: Lyapunov 'instability in ens.tex­system of hard disks in equilibrium and non-equilibrium steady ens.tex­states, Physical Review, 53E, 1485~~1501, 1996.

[Do68a] Dobrushin, RL.: The description of a random field by means of condi­tional probabilities and conditions of its regularity, Theory of Probabil­ity and its Applications, 13, 197-224, 1968.

[Do68b] Dobrushin, RL.: Gibbsian random fields for lattice systems with pair­wise interactions, Functional Analysis and Applications, 2, 31-43, 1968; and Problem of uniqueness of a Gibbs random field and phase transi­tion, Functional Analysis and Applications, 2, 44-57, 1968.

[Do68c] Dobrushin, RL.: Gibbs field: the general case, Functional Analysis and Applications, 3, 27-35, 1969.

[Do72] Dobrushin, RL.: Gibbs state describing coexistence of phases for a three-dimensional Ising model, Theory of Probability and its Appli­cations, 17, 582-600, 1972. And Investigation of Gibbsian states for three-dimensional lattice systems, Theory of Probability and its Appli­cations, 18, 253-271, 1973. Both reprinted in [Si91].

[Do80] Dobrushin, R.L.: Automodel generated random fields and their renor­malization group, in Multicomponent random fields, Advances in Prob­ability and Related Topics, 6, 153-198, 1980.

[Do98] Dorfman, J.R: An introduction to chaos in non-equilibrium statistical mechanics, pp. 1-287, University of Maryland, preprint, 1998.

[Dr88] Dressler, U.: Symmetry property of the Lyapunov exponents of a class of dissipative dynamical systems with viscous damping, Physical Review, 38A, 2103~ 2109, 1988.

[Du59] Dugas, R: La tMorie physique au sens de Boltzmann, Griffon, Neu­chatel, 1959.

[Dy69] Dyson, F.: Existence of a phase transition in a one-dimensional Ising ferromagnet, Communications in Mathematical Physics, 12, 91-107, 1969.

[ECM90] Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Viscosity of a simple fluid from its maximal Lyapunov exponents, Physical Review, 42A, 5990-5997, 1990.

[ECM93] Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady flows, Physical Review Letters, 71, 2401-2404, 1993.

[EEll] Ehrenfest, P., Ehrenfest, T.: The conceptual foundations of the statis­tical approach in Mechanics, Dover, New York, 1990 (reprint).

[EG75] Esposito, R, Gallavotti, G.: Approximate simmetries and their spon­taneous breakdown, Annales de l'Institut Henri Poincare: 22, 159-172, 1975.

[EGM98] Esposito, R, Gallavotti, G., Marra, R.: Boltzmann's equation and ther­mostats, preprint, Rome, 1998.

[EM90] Evans, D.J., Morriss, G.P.: Statistical mechanics of nonequilibrium flu­ids, Academic Press, New York, 1990.

[ER85] Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attrac­tors, Reviews of Modern Physics, 57, 617-656, 1985.

[ES93] Evans, D.J., Sarman, S.: Equivalence of thermostatted nonlinear re­sponse, Physical Review E, 48 E, 65-70, 1993.

[ES94] Evans, D., Searles, D.: Equilibrium microstates which generate the sec­ond law violating steady states, Physical Review E, 50E, 1645-1648, 1994.

318 Bibliography

[Ei56]

[Ev93]

[FKG71]

[FPU55]

[FR66]

[FS62]

[Fe63]

[Fe72]

[Fe85]

[Fi64]

[Fi65] [Fi67a]

[Fi67b] [Fi67c]

[Fi98]

[Fr81]

[Fr86]

[Fr97] [GC95]

[GG94]

[GH64]

[GJ75]

[GHM77]

[GL71]

Einstein, A.: Investigations on the theory of the Brownian movement, edited by R Furth, A. Cowper, Dover, New York, 1956. Evans, D.J.: The equivalence of Norton and Thevenin ensembles, Molec­ular Physics, 80, 221-224, 1993. Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets, Communications in Mathematical Physics, 22, 89-104, 1971. Fermi, E., Pasta, J., Ulam, S.: Los Alamos report LA-1940, 1955, printed in E. Fermi, Collected papers, 1965. Fisher, M.E., Ruelle, D.: The stability of many particles systems, Jour­nal of Mathematical Physics, 7, 260-270, 1966. Fisher, M.E., Sykes, M.F.: Antiferromagnetic susceptibility of the simple cubic and body centered cubic Ising lattice, Physica, 28, 939-956, 1962. Feynman, RP., Leighton, RB., Sands, M.: Lectures in Physics., Vols. I-III, Addison Wesley, Reading, Massachusetts, 1963. Feynman, R.P.: Statistical mechanics, Benjamin, Reading, Massachu­setts, 1972. Fefferman, C.: The atomic and molecular nature of matter, Revista Matematica Iberoamericana, 1, 1-44, 1985. Fisher, M.E.: The free energy of a macroscopic system, Archive for Rational Mechanics and Analysis, 17, 377-410, 1964. Fisher, M.E.: Lectures in physics, vol. 7C, Boulder, Colorado, 1965. Fisher, M.E.: Theory of equilibrium critical phenomena, Reports on Progress in Theoretical Physics, 30, 615, 1967. Fisher, M.E.: Physical Review, 162, 475, 1967. Fisher, M.E.: Theory of condensation and the critical point, Physics­Physica-Fyzika, 3, 255-283, 1967. Fisher, M.E.: Renormalization group theory: its basis and formulation in statistical physics, Reviews of Modern Physics, 653-681, 1998. Frolich, J.: On the triviality of AcP~ theories and the approach to the critical point in (d ~ 4) dimensions, Nuclear Physics, 200 B, 281-296, 1981. Frolich, J.: Mathematical aspects of the physics of disordered systems, in Critical phenomena, random systems, gauge theories, ed. K. Oster­walder, R Stora, pp. 725-893, North Holland, Amsterdam, 1986. Frisch, U.: Turbulence, Cambridge University Press, 1997. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in non-equilibrium statistical mechanics, Physical Review Letters, 74, 2694-2697, 1995. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states, Journal of Statistical Physics, 80, 931-970, 1995. Gallavotti, G., Garrido, P.: Billiards correlation functions, Journal of Statistical Physics, 76, 549-586, 1994. Green, H.S., Hurst, C.H.: Order disorder phenomena, Wiley, New York, 1964. Gallavotti, G., Jona, G.: Limit theorems for multidimensional Markov processes, Communications in Mathematical Physics, 41, 301-307, 1975. Gruber, C., Hintermann, A., Messager, A., Miracle-Sole, S.: Uniqueness of the invariant equilibrium state and surface tension, Communications in Mathematical Physics, 56, 147-159, 1977. Gallavotti, G., Lebowitz, J.L.: Phase transitions in binary lattice gases, Journal of Mathematical Physics, 12, 1129-1133, 1971.

Bibliography 319

[GLL 70] Gallavotti, G., Lanford, 0., Lebowitz, J.: Thermodynamic limit of time­dependent correlation functions for one-dimensional systems, Journal of Mathematical Physics, 11, 2898-2905, 1970.

[GM72a] Gallavotti, G., Miracle-Sole, S.: Equilibrium states of the Ising model in the two-phases region, Physical Review, 5B, 2555-2559, 1972.

[GM72b] Gallavotti, G., Martin-LM A.: Surface tension in the Ising model, Com­munications in Mathematical Physics, 25, 87-126, 1972.

[GMM72] Gallavotti, G., Martin-Lof, A., Miracle-Sole, S.: Some problems con­nected with the description of coexisting phases at low temperatures in the Ising model, in Springer Lecture Notes in Physics, edited by A. Lenard, vol. 20, pp. 159-204, Berlin, 1972.

[GMR69] Gallavotti, G., Miracle-Sole, S., Ruelle, D.: Absence of phase transi­tions in one-dimensional systems with hard core, Physics Letters, 26A, 350-351, 1968. And Gallavotti, G., Miracle-Sole, S.: Absence of phase transitions in hard core one-dimensional systems with long range inter­actions, Journal of Mathematical Physics, 11, 147-154, 1970.

[GR97] Gallavotti, G., Ruelle, D.: SRB states and non-equilibrium statisti­cal mechanics close to equilibrium, Communications in Mathematical Physics, 190, 279·285, 1997.

[GS72] Galgani, 1., Scotti, A.: Planck-like distributions in classical nonlinear mechanics, Physical Review Letters, 28, 1173-1176, 1972.

[GV72] Gallavotti, G., Van Beijeren, H.: The phase separation line in the two­dimensional Ising model and its relationships with the non-free random walk problem, Lettere al Nuovo Cimento, 4, 699--703, 1972.

[Ga69] G. Gallavotti, Time evolution problems in classical statistical mechan­ics and the wind-tree-model, in Cargese Lectures in Physics, vol. IV, ed. D. Kastler, Gordon Breach, Paris, 1970, pp. 257-275; see pp. 271-272, formula (6.2). And G. Gallavotti, Divergences and approach to equilib­r'ium in the Lorentz and the wind-tree models, Physical Review, 185, 308-322, 1969.

[Ga72a] Gallavotti, G.: Phase separation line in the two-dimensional Ising model, Communications in Mathematical Physics, 27, 103-136, 1972.

[Ga72b] Gallavotti, G.: Instabilities and phase transitions in the Ising model, La Rivista del Nuovo Cimento, 2, 133-169, 1972.

[Ga76] Gallavotti, G.: Probabilistic aspects of critical fluctuations, in Critical Phenomena, ed. J. Brey, R.B. Jones, Lecture Notes in Physics Series, 54, pp. 250273, Springer, Berlin, 1976.

[Ga81] Gallavotti, G.: Aspetti della teoria ergodica qualitativa e statistica del moto, Quaderno Unione Matematica Italiana, no. 21, pp. 1-300, ed. Pitagora, Bologna, 1981.

[Ga89] Gallavotti, G.: L' hypothese ergodique et Boltzmann, in Dictionnaire Phylosophique, Presses Universitaires de France, pp. 1081-1086, Paris, 1989.

[Ga95a] Gallavotti, G.: Ergodicity, ensembles, irreversibility in Boltzmann and beyond, Journal of Statistical Physics, 78, 1571--1589, 1995.

[Ga95b] Gallavotti, G.: Trattatello di meccanica statistica, Quaderni CNR­GNFM, 50, pp. 1-350, 1995, Firenze. Free copies can be obtained by requesting them to CNR-GNFM, via S. Marta 13/a, 50139 Firenze, Italy, or by fax at CNR-GNFM +39-55-475915.

[Ga95c] Gallavotti, G.: Reversible Anosov mapsand large deviations, Mathemat­ical Physics Electronic Journal, MPEJ, (http:// mpej.unige.ch), 1, 12 pp., 1995.

320

[Ga95d]

[Ga95e]

[Ga96a]

[Ga96b]

[Ga96c]

[Ga97a]

[Ga97b]

[Ga98a] [Ga98b]

[Ga98c]

[Ga98d]

[Ga98e]

[Ge93]

[Ge98]

[Gi67]

[Gi69a]

[Gi69b]

[Gi70]

[Gi81]

Bibliography

Gallavotti, G: Topics on chaotic dynamics, in Third Granada Lectures in Computational Physics, ed. P. Garrido, J. Marro, in Lecture Notes in Physics, Springer Verlag, 448, pp. 271-311, 1995. Gallavotti, G.: Meccanica statistica, in Quaderni del CNR-GNFM, vol. 50, pp. 1-350, Consiglio Nazionale delle Richerche Gruppo Nazionale di Fisica Matematica, Firenze, 1995. Gallavotti, G.: Chaotic hypothesis: Onsager reciprocity and fluctuation­dissipation theorem, Journal of Statistical Physics, 84, 899-926, 1996. Gallavotti, G.: Extension of Onsager's reciprocity to large fields and the chaotic hypothesis, Physical Review Letters, 78, 4334-4337, 1996. Gallavotti, G.: Equivalence of dynamical ensembles and Navier-Stokes equations, Physics Letters, 223, 91-95, 1996. Gallavotti, G.: Dynamical ensembles equivalence in fluid mechanics, Physica D, 105, 163-184, 1997. Gallavotti, G.: Chaotic principle: some applications to developed turbu­lence, Journal of Statistical Physics, 86, 907-934, 1997. Gallavotti, G.: A local fluctuation theorem, Physica A, 263, 39-50, 1999. Gallavotti, G.: Fluctuation patterns and conditional reversibility in non­equilibrium systems, Annales de l'Institut H. Poincare, 70, 429-443, 1999. Gallavotti, G.: New methods in nonequilibrium gases and fluids, Pro­ceedings of the conference Let's face chaos through nonlinear dynamics, University of Maribor, 24 June-5 July 1996, ed. M. Robnik, in press, Open Systems and Information Dynamics, Vol. 6,1999 (also in mp_arc@ math. utexas.edu #96-533 and chao-dyn #9610018). Gallavotti, G.: Chaotic dynamics, fluctuations, non-equilibrium ensem­bles, Chaos, 8, 384-392, 1998. Gallavotti, G.: Chaotic hypothesis and universal large deviations prop­erties, chao-dyn 9808004 and Documenta Mathematica, extra volume ICM98, vol. I, 1998. Georgii, H.O.: The equivalence of ensembles for classical systems of particles, Journal of Statistical Physics, 80, 1341-1378, 1995. Gentile, G: Large deviation rule for Anosov flows, Forum Mathe­maticum, 10, 89-118, 1998. Ginibre, J.: Rigorous lower bound on the compressibility of a classical system, Physics Letters, 24A, 223-224, 1967. See also: Continuity of the pressure as a function of the density for some quantum systems, Physical Review Letters, 24, 1473-1475, 1973. Ginibre, J.: Simple proof and generalization of Griffiths second inequal­ity, Physical Review Letters, 23, 828-830, 1969. See also: General for­mulation of Griffiths inequalities, Communications in Mathematical Physics, 16, 310-328, 1970. And GKS to plane rotators, Cargese Lec­ture Notes, vol. 4, ed. D. Kastler, Gordon & Breach, Paris, 1970. Ginibre, J.: Existence of phase transitions in quantum lattice systems, Communications in Mathematical Physics, 14, 205-234, 1969. See also: dilute quantum systems, in Phase transitions and critical points, edited by C. Domb and M.S. Green, pp. 112-137, Wiley, New York, 1972. Ginibre, J.: Systemes a un nombre infini de degres de liberte, Colloques du CNRS, ed. L. Michel, D. Ruelle, Paris, 1970. Gibbs, J.W.: Elementary principles in statistical mechanics, Ox Bow Press, Woodbridge (Connecticut), 1981 (reprint of the 1902 edition).

[Gr58]

[Gr62]

[Gr67]

[Gr70]

[Gr72]

[HHP87]

[HKU63]

[HL76]

[HS92]

[He95a]

[He95b]

[Hi81]

[Hu38]

[Hu63] [IM65]

[Is84]

[Ja84]

[Jo95]

[KM86]

[KM87]

Bibliography 321

Grad, H.: Principles of the kinetic theory of gases, in Handbuch der Physik, vol. XII, pp. 205-294, ed. S. Fliigge, Springer-Verlag, Heidel­berg, 1958. Groeneveld, J.: Two theorems on classical many particles systems, Physics Letters, 3, 50-51, 1962. Griffiths, RB.: Correlations in Ising ferromagnets, I, Journal of Math­ematical Physics, 8, 478-483, 1967. Griffiths, RB.: Concavity of magnetization, in A critical review of ther­modynamics, ed. E.B. Stuart, B. Gal-or, A.J. Brainard, Mono Book Corporations, Baltimore, 1970. Griffiths, RB.: Rigorous results and theorems, in Phase transitions and critical points, edited by C. Domb and M.S. Green, pp. 7-109, Wiley, New York, 1972. Holian, B.L., Hoover, W.G., Posch. H.A.: Resolution of Loschmidt's paradox: the origin of irreversible behavior in reversible atomistic dy­namics, Physical Review Letters, 59, 10-13, 1987. Hemmer, P., Kac, M., Uhlenbeck, G.E.: On the van der Waals theory of the liquid-vapor equilibrium, I-III, Journal of Mathematical Physics, 4, 216-228, 1963; 4, 229-247, 1963; 5, 60-84, 1964. Hemmer, P.C., Lebowitz, J.L.: Systems with weak long range potentials, in Phase transitions and critical points, vol. 4, edited by C. Domb and M.S. Green, pp. 108-198, Wiley, New York, 1976. Hara, T., Slade, G.: The lace expansion for self-avoiding walk in five or more dimensions, Reviews in Mathematical Physics, 4, 235-327, 1992. Helmholtz, H.: Principien der Statik monocyklischer Systeme, in Wis­senschajtliche Abhandlungen, vol. III, pp. 142-162 and pp. 179-202, Leipzig, 1895. Helmholtz, H.: Studien zur Statik monocyklischer Systeme, in Wis­senschajtliche Abhandlungen, vol. III, pp. 163-172 and pp. 173- 178, Leipzig, 1895. Higuchi, Y: On the absence of non translationally invariant Gibbs states for the two-dimensional Ising system, in Random fields, editors J. Fritz, J.L. Lebowitz and D. Szaz, North Holland, Amsterdam, 1981. Hulthen, L.: Uber das A ustauschproblem eines K ristalles, Arkiv for Matematik, Astronomi och Fysik, 26A, 1-106, 1938. Huang, K: Statistical mechanics, Wiley, New York, 1963. Ito, K., McKean, H.: Diffusion processes and their sample paths, Sprin­ger-Verlag, New York, 1965. Isakov, S.N.: Non analytic features of the first or·der phase transition in the Ising model, Communications in Mathematical Physics, 95, 427-443, 1984. Jacobs, K: Ergodic theory and combinatorics, in Proceedings of the con­ference on modern analysis ansd probability, June 1982. Contemporary Mathematics, 26, 171-187, 1984. Johansson, K: On the separation of phase on one-dimensional gases, Communications in Mathematical Physics, 169,521-561, 1995. Kotecky, R, Miracle-Sole, S.: A model with roughening transition at low temperature, Physical Review, 34B, 2049-2051, 1986. Kotecky, R, Miracle-Sole, S.: Roughening transition for the Ising model on a BCC lattice. A case in the theory of ground states, Journal of Statistical Physics, 47, 773-799, 1987. And Kotecky, R, Miracle-Sole, S., Ruiz, .1.: Roughening transition for the Ising model on a BCC lattice, Journal of Statistical Physics, 47, 773-799, 1987.

322 Bibliography

[K049J

[K093J

[KP86J

[KSiS95J

[KW41J

[Ka49J

[Ka61J

[Ka68J

[Ka69J

[Ka74J

[K162J

[KI72J

[K173J

[Kr79J

[Ku83]

[Ku87]

[Ku97]

[LL63],

[LL67J [LL72J

Kaufman, B., Onsager, L.: Comment to the "theory of regular solutions" following the paper by G.S. Rushbrooke, Supplemento Nuovo Cimento, 6, 261, 1949. Kotecky, R., Olivieri, E.: Droplet dynamics for asymmetric Ising model, Journal of Statistical Physics, 70, 1121-1148, 1993. Kotecky, R., Preiss, D.: Cluster expansion for abstract polymers, Com­munications in Mathematical Physics, 103, 491-498, 1986. Knimli, A., Simanyi, N., Szasz, D.: Ergodic properties of semi-dispersing billiards, Nonlinearity, 2, 311-326,1995. See also Szasz, D.: Boltzmann's ergodic hypothesis, a conjecture for centuries?, Studia Sci. Math. Hun­garicae, 31, 299-322, 1996. Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferro­magnetic Ising model, Physical Review, 60, 252-276, 1941. Kaufman, B.: Crystal statistics. II. Partition function evaluation by spinor analysis, Physical Review, 76, 1232-1243, 1949. See also Kauf­man, B., Onsager, L.: Crystal Statistics. III. Short range order in a binary Ising lattice, Physical Review, 76, 1244-1252, 1949. Kasteleyn, P.W.: The statistics of dimers on a lattice, Physica, 27, 1209-1225, 1961. Kasteleyn, P.W.: Phase transitions, in Fundamental Problems in Statis­tical Mechanics. II, edited by E.G.D. Cohen, pp. 30-70, North Holland, Amsterdam, 1968. Kadanoff, L.: Correlations along a line in the two-dimensional Ising model, Physical Review, 188, 859-863, 1969. See also Kadanoff, L., Ceva, H.: Determination of an operator algebra for the two-dimension­al Ising model, Physical Review, 3 B, 3918-3939, 1971. Kasteleyn, P.W.: Combinatorial lattice problems in statistical mechan­ics, Proceedings of the International School of Mathematical Physics, pp. 101-136, ed. G. Gallavotti, Camerino, 1974, University of Camerino Press, 1976. Klein, M.: Maxwell and the beginning of the quantum theory, Archive for the History of Exact Sciences, 1, 459-479, 1962. Klein, M.: Mechanical explanations at the end of the nineteenth century, Centaurus, 17, 58-82, 1972. Klein, M.: The development of Boltzmann statistical ideas, in The Boltz­mann equation, ed. E.G.D. Cohen, W. Thirring, Acta Physica Austri­aca, suppl. X, Vienna, pp. 53-106, 1973. Krylov, N.S.: Works on the foundations in statistical physics, Princeton University Press, 1979 (English translations of selected papers). Kurota, K.: Phase separations in Ising model with free boundary condi­tions, Journal of Statistical Physics, 36, 1-13, 1983. Kuhn, T.: Black body theory and the quantum discontinuity, 1814-1912, University of Chicago Press, 1987. Kurchan, J.: Fluctuation theorem for stochastic dynamics, Journal of Physics, A, 31, 3719--3729, 1998. Lieb, E., Lininger, W.: Exact analysis of an interacting Bose gas. 1. The general solution and the ground state, Physical Review, 130, 1605-1616, 1963; reprinted in ([LM66]). Landau, L., Lifschitz, L.E.: Physique statistique, MIR, Moscow, 1967. Lieb, E.H., Lebowitz, J.L.: Lectures on the thermodynamic limit for Coulomb systems, in Springer Lecture Notes in Physics, edited by A. Lenard, vol. 20, pp. 135-161, Berlin, 1972.

[LLP97]

[LM66]

Bibliography 323

Lepri. S., Livi, R., Politi, A.: Energy transport in anharmonic lat­tices close and far from equilibrium, preprint, cond-mat@xyz. lanl. gov #9709156, to appear on Physical Review E. Lieb, E., Mattis, D.C.: Mathematical physics in one dimension, Aca­demic Press, New York, 1966.

[LMMRS91]Laanait, L., Messager, A., Miracle-Sole, S., Ruiz, J., Shlosman, S.: In­terfaces in the Potts model. (I) Pirogov-Sinai theory of the Fortuin­Kasteleyn representation. Communications of Mathematical Physics, 140, 81-91, 1991.

[LMP98] Lebowitz, J.L., Mazel, A., Presutti, E.: Rigorous proof of a liquid-vapor

[LP66]

[LP76]

[LP79]

[LR69]

[LS82]

[LS94] [LS98]

[LT75]

[LV93]

[LW72]

[LY52]

[LY73]

[La68]

[La69]

[La70],

phase transition in a continuum particle system, in print on Physical Review Letters, 1998. Lebowitz, J., Percus, J.: Kinetic equations and exactly solvable one­dimensional systems, Physical Review, 155, 122--138, 1966; see also J. Lebowitz, J. Percus, J. Sykes, Time evolution of the total distribution function of a one-dimensional systems of hard rods, Physical Review, 171, 224-235, 1968. Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of un­bounded spins, 50, 195-218, 1976. Lebowitz, J.L., Penrose, 0.: Towards a rigorous molecular theory of metastab'ility, in Fluctuation phenomena, ed. E.W. Montroll, J.L. Lebo­witz, North Holland, Amsterdam, 1979. Lanford, 0., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics, Communications in Mathe­matical Physics, 13, 194-215, 1969. Lebowitz, J., Spohn, H.: Steady state self diffusion at low density, Jour­nal of Statistical Physics, 29, 39-55, 1982. Liddell, H., Scott, R.: Greek-English lexicon, Oxford, 1994. Lebowitz, J.L., Spohn, H.: The Gallavotti-Cohen fluctuation theorem for stochastic dynamics, preprint, Rutgers University, June 1998. Lieb, E., Thirring, W.: Bound for the kinetic energy of fermions which proves the stability of matter, Physical Review Letters, 11, 687-689, 1975. Levesque, D., Verlet, L.: Molecular dynamics and time reversibility, Journal of Statistical Physics, 72, 519-537, 1993. Lieb, E., F.Y. Wu: Two-dimensional ferroelectric models, in Phase tran­sitions and critical phenomena, edited by C. Domb and M. Green, vol. 1, pp. 332·-490, Academic Press, London, 1972. Lee, T.D., Yang C.N.: Statistical theory of equations of state and phase transitions, II. Lattice gas and Ising model, Physical Review, 87, 410-419, 1952. Lasota, A., Yorke, J.A.: On the existence ungheria.tex-of invariant measures for piecewise monotonic transformations, ungheria.tex-Trans­actions American Mathematical Society, 183, 481, 1973. Lanford, 0.: The classical mechanics of one-dimensional systems of infinitely many particles. I. An existence theorem, Communications in Mathematical Physics, 9, 176-191, 1968. Lanford, 0.: The classical mechanics of one-dimensional systems of infinitely many particles. II. Kinetic theory, Communications in Math­ematical Physics, 11, 257-292, 1969. Lanford, 0.: Quantum spin systems, in Cargese Lectures in Physics, vol. IV, ed. D. Kastler, pp, 113-115, Gordon & Breach, Paris, 1970.

324 Bibliography

[La72J, Lanford, 0.: Entropy and equilibrium states in classical statistical me­chanics, in Springer Lecture Notes in Physics, edited by A. Lenard, vol. 20, pp. 1-113, Berlin, 1972.

[La74] Lanford, 0.: Time evolution of large classical systems, in Dynamical systems, theory and applications, ed. J. Moser, Lecture Notes in Physics, vol. 38, pp. 1-111, Springer-Verlag, Berlin, 1974.

[La85] Lawler, G.F.: Intersections of random walks in four dimensions, Com­munications in Mathematical Physics, 97, 583-594, 1985. See also In­tersections of simple random walks in Particle systems, random me­dia and large deviations, Series 'Contemporary Mathematics', vol. 41, American Mathematical Society, Providence, 1985.

[Le74] Lebowitz, J.L.: GHS and other inequalities, Communications in Math­ematical Physics, 28, 313-321, 1974.

[Le93] Lebowitz, J.L.: Boltzmann's entropy and time's arrow, Physics Today, 32-38, 1993. And Microscopic reversibility and macroscopic behavior: physical explanations and mathematical derivations, in 25 years of non­equilibrium statistical mechanics, ed. J. Brey, J. Marro, J. Rubi, M. San Miguel, Lecture Notes in Physics, vol. 445, Springer, Berlin, 1995.

[Li67a] Lieb, E.: Exact Solution of the Problem of the Entropy of two-dimen­sional Ice: Physical Review Letters, 18,692-694, 1967. And The Resid­ual Entropy of Square Ice, Physical Review, 162, 162-172, 1967.

[Li67b] Lieb, E.: Exact solution of the F model of an antiferroelectric, Physical Review Letters, 18, 1046-1048, 1967.

[Li67c] Lieb, E.: Exact solution of the two-dimensional slater KDP model of a ferroelectric, Physical Review Letters, 19, 108-110, 1967.

[Li81] Lieb, E: Thomas-Fermi and related theories of atoms and molecules, Reviews of Modern Physics, 53, 603-641, 1981.

[Li96] Liverani, C.: Deacy of correlations, Annals of Mathematics, 142, 239-301, 1995.

[MK86] Miracle-Sole, S., Kotecky, R.: A model with roughening transition at low temperatures, Physical Review, 34 B, 2049-2051, 1986. And Roughen­ing transition for the Ising model on a B. C. C. lattice. A case in the theory of ground states, Journal of Statistical Physics, 47, 773-799, 1987.

[ML72] Martin-Lof, A., Lebowitz, J.L.: On the uniqueness of the equilibrium state for Ising spin systems, Communications in Mathematical Physics, 25, 276-282, 1972.

[MM77] Messager, A., Miracle-Sole, S.: Correlation functions and boundary con­ditions in the Ising ferromagnet, Journal of Statistical Physics, 17, 245-262, 1977.

[MMR92] Messager, A., Miracle-Sole, S., Ruiz, J.: Convexity properties of surface tension and equilibrium crystals, Journal of Statistical Physics, 67, 449-470, 1992.

[MMRS91] Messager, A., Miracle-Sole, S., Ruiz, J., Shlosman, S.: Interfaces in the Potts model. (II) Antonov rule and rigidity of the order-disorder inter­face. Communications in Mathematical Physics, 140, 275-290, 1991.

[MOS90] Martinelli, F., Olivieri, E., Scoppola, E.: Metastability and exponential approach to equilibrium for low temperature stochastic Ising models, J. Stat. Physics, 61,1105-1119,1990.

[MP72] Marchioro, C., Presutti, E.: Thermodynamics of particle systems in presence of external macroscopic fields, Communications in Mathemat­ical Physics, 27, 146-154, 1972. And 29, 265-284, 1973.

[MR94]

[MR96]

[MR97]

[MS67]

[MS70]

[MTW77]

[MW66]

[MW73]

[MW86]

[Ma65] [Ma72]

[Ma79]

[Ma8S]

[Ma98]

[Me67]

[Me71]

[Mi68]

[Mi95]

[M056]

Bibliography 325

Miracle-Soh§, S., Ruiz, J.: On the Wulff construction as a problem of equivalence of ensembles, in Micro, Meso and Macroscopic Approaches in Physics, ed. M. Fannes, A. Verbeure, Plenum, New York, 1994. Morriss, G.P., Rondoni, L.: Equivalence of "nonequilibrium" ensembles for simple maps, Physica A, 233, 767-784, 1996. Morriss, G.P., Rondoni, L.: Applications of periodic orbit theory to N­particle systems, Journal of Statistical Physics, 86, 991-1009, 1997. Minlos, R.A., Sinai, J.G.: The phenomenon of phase separation at low temperatures in some lattice models of a gas, I, Math. USSR Sbornik, 2, 335-395, 1967. And: The phenomenon of phase separation at low temperatures in some lattice models of a gas, II, Transactions of the Moscow Mathematical Society, 19, 121-196, 1968. Both reprinted in [Si91]. Minlos, R.A., Sinai, J.G.: Investigation of the spectra of stochastic oper­ators arising in lattice gas models (Russian), Teoreticheskaia y Matem­aticheskaia Fyzika, 2, No.2, 230-243, 1970. McCoy, B.M., Tracy, C., Wu, T.T.: Two-dimensional Ising model as an explicitly solvable relativistic field theory: explict formulas for n-point functions, Physical Review Letters, 18, 1253-1258, 1978. Mermin, N.D., Wagner, H. Absence of ferromagnetism or antiferro­magnetism in 1 or 2 dimensional isotropic Heisenberg models, Physical Review Letters, 17, 1183-1186, 1966. McCoy, B.M., Wu, T.T.: The two-dimensional Ising model, Harvard University Press, Cambridge, 1973. McCoy, B.M., Wu, T.T.: Some exact results for the electric correlation functions for the eight vertex model, Physical Review Letters, 56, 2645-2648, 1986. Mattis, D.C.: Theory of magnetism, Harper Row, New York, 1965. Martin-Lof, A.: On the spontaneous magnetization in the Ising model, Communications in Mathematical Physics, 24, 253-259, 1971. Maxwell, J.: On Boltzmann's theorem on the average distribution of energy in a system of material points, in The scientific papers of J. C. Maxwell, ed. W. Niven, vol. II, pp. 713-741, Cambridge University Press, 1890. Mathieu, M.: On the origin of the notion 'ergodic theory', Expositiones Mathematicae, 6, 373-377, 1988. Mastropietro, V.: Renormalization group and correlation functions for the XYZ model, preprint, Rome, 1998. Mermin, D.: Absence of ordering in certain classical systems, Journal of Mathematical Physics, 8, 1061-1064, 1967. Mermin, D.: Solvable model of a vapor-liquid transition with singu­lar coexistence-curve diameter, Physical Review Letters, 26, 168-172, 1971. Minlos, R.A.: Lectures in statistical mechanics, Russian Mathematical Surveys, 23, 137-196, 1968. Miracle-Sole, S.: On the microscopic theory of phase coexistence, in 25 Years of N on-Equilibrium Statistical Mechanics, ed. J.J. Brey, Lecture Notes in Physics, vol. 445, pp. 312-322. Springer, Berlin, 1995. And Surface tension, step free energy and facets in the equilibrium crystal shape, Journal of Statistical Physics, 79, 183-214, 1995. Morrey, C.B.: On the derivation of the equations of hydrodynamics from statistical mechanics, Communications in Pure and Applied Mathemat­ics, 8, 279-326, 1955.

326

[NM53]

[Ne67]

[On44]

[Or58]

[Or74]

[PS76]

[PY96]

[PY99]

[Pa82]

[Pe36]

[Pe63]

[Pe70] [PoOO]

[RM75]

[RT71]

[RT98]

[Ru64]

[Ru68]

[Ru69] [Ru70]

Bibliography

Newell, G.F., Montroll, E.W.: On the theory of the Ising model of fer­romagnetism, Reviews of Modern Physics, 25, 353-389, 1953. Nelson, E.: Dynamical theories of Brownian motion, Princeton Univer­sity Press, 1967. Onsager L.: Crystal statistics. 1. A two-dimensional Ising model with an order-disorder transition, Physical Review, 65, 117-149, 1944. Orbach, R.: Linear antiferromagnetic chain with anisotropic coupling, Physical Review, 112, 309-316, 1958. Ornstein, D.: Ergodic theory, randomness and dynamical systems, Yale University Press, 1974. Pirogov, S.A., Sinai, J.G.: Phase diagrams of classical lattice systems, Theoretical and Mathematical Physics, 25, 1185-1192, 1975; and 26, 39-49, 1976. Both reprinted in [Si91]. Pfister, C., Yelenik, Y.: Mathematical theory of the wetting phenomenon in the 2D-Ising model, Helvetica Physica Acta, 69, 949-973, 1996; Large deviations and continuum limit in the 2D Ising model, Probability The­ory and Related Fields, 109, 435-506, 1997. Pfister, C., Yelenik, Y.: Interface, surface tension, and reentrant pinning in the 2D Ising model, in print in Communications in Mathematical Physics, 1999. Pais, A.: Subtle is the Lord: the science and the life of Albert Einstein, Oxford University Press, 1982. Peierls, R.: On Ising's model of ferromagnetism, Proceedings of the Cambridge Philosophical Society, 32, 477-481, 1936. Penrose, 0.: Convergence of the fugacity expansions for fluids and lat­tice gases, Journal of Mathematical Physics, 4, 1312-1320, 1963. See also The remainder in Mayer's fugacity series, Journal of Mathematical Physics, 4, 1488-1494, 1963. Perrin, J.: Les atomes, Gallimard, Paris, 1970 (reprint). Poincare, H.: Relations entre la physique experimentale et la physique mathimatique, in La Science et I 'Hypothese, p. 157-190, Champs­Flammarion, Paris, 1995 (Original 1900). And: The crises of math­ematical physics, in La Valeur de la Science, p. 129-150, Champs­Flammarion, Paris, 1994 (Original 1904). Russo, L., Monroy, G.: A family of codes between some Markov and Bernoulli schemes, Communications in Mathematical Physics, 43, 153-160, 1975. Ruelle, D., Takens, F.: On the nature of turbulence, 20, 167-192, 1971. And Note concerning our paper "On the nature of turbulence", Com­munications in Mathematical Physics, 23, 343-344, 1971: this note is important as it draws attention to several references that were not well known at the time. Rom-Kedar, Y., Thraev, D.: Big islands in dispersing billiard-like po­tentials, preprint, Weizmann Institute, April 2, 1998. Ruelle, D.: Cluster property of the correlation functions of classical gases, Reviews of Modern Physics, 36, 580-584, 1964. Ruelle, D.: Statistical mechanics of one-dimensional lattice gas, Com­munications in Mathematical Physics, 9, 267-278, 1968. Ruelle, D.: Statistical mechanics, Benjamin, New York, 1969. Ruelle, D.: Superstable interactions in classical statistical mechanics, Communications in Mathematical Physics, 18, 127-159, 1970. See also: Probability estimates for continuous spin systems, Communications in Mathematical Physics, 50, 189-194, 1976.

[Ru71 a]

[Ru71b]

[Ru73b]

[Ru76]

[Ru78a]

[Ru78b] [Ru78c]

[Ru79a]

[Ru79b]

[Ru80]

[Ru81]

[Ru89]

[Ru96a]

[Ru97a]

[Ru97b]

[Ru97c]

[SJ93]

[SML64]

[Sc86]

[Si68]

[Si70]

[Si72]

Bibliography 327

Ruelle, D.: Extension of the Lee~ Yang circle theorem, Physical Review Letters, 26, 303304, 1971. Ruelle, D.: Existence of a phase transition in a continuous system, Phys­ical Review Letters, 27, 1040~1041, 1971. Ruelle, D.: On the use of small external fields in problems of symme­try breakdown in statistical mechanics, Annals of Physics, 69, 364~374, 1972. Ruelle, D.: A measure associated with Axiom A attractors, American Journal of Mathematics, 98, 619~654, 1976. Ruelle, D.: Sensitive dependence on initial conditions and turbulent be­havior of dynamical systems, Annals of the New York Academy of Sci­ences, 356, 408-416, 1978. Ruelle, D.: Thermodynamic formalism, Addison Wesley, Reading, 1978. Ruelle, D.: What are the measures describing turbulence?, Progress of Theoretical Physics, (Supplement) 64, 339~345, 1978. Ruelle, D.: Ergodic theory of differentiable dynamical systems, Publica­tions Mathematiques de l'LH.E.S, 50, pp. 275~320, 1979. Russo, L.: The infinite cluster method in the two-dimensional Ising model, Communications in Mathematical Physics, 67, 251~266, 1979. Ruelle, D.: Measures describing a turbulent flow, Annals of the New York Academy of Sciences, volume Nonlinear dynamics, ed. R.H.G. Heileman, 357, 1--9, 1980. Russo, L.: On the critical percolation probabilities, Zeitschrift fur Wahr­scheinlichkeit und verwandter Gebiete, 56, 229~237, 1981. Ruelle, D.: Elements of differentiable dynamics and bifurcation theory, Academic Press, London, 1989. Ruelle, D.: Positivity of entropy production in nonequilibrium statistical mechanics, Journal of Statistical Physics, 85, 1-25, 1996. Ruelle, D.: Entropy production in non equilibrium statistical mechanics, Communications in Mathematical Physics, 189, 365~371, 1997. Ruelle, D.: Differ'entiation of SRB states, Communications in Math­ematical Physics, 187, 227-241, 1997. And Nonequilibrium statistical mechanics near equilibrium: computing higher order terms, preprint, IRES, July 1997. Ruelle, D.: New theon:tical ideas in non-equilibrium statistical mechan­ics, Lecture notes at Rutgers University, fall 1997. She, Z.S., Jackson, E.: Constrained Euler system for Navier~Stokes tur­bulence, Physical Review Letters, 70, 1255~ 1258, 1993. Schultz T.D., Mattis, D.C., Lieb, E.: The two-dimensional Ising model as a soluble problem of many fermions, Reviews of Modern Physics, 36, 856~871, 1964. Schwartz, J.: The pernicious influence of mathematics on science, in Discrete thoughts: essays in mathematics, science, and philosophy, M. Kac, G. Rota, and J. Schwartz, eds., pp. 19-25, Birkhiiuser, Boston, 1986. Sinai, y'G.: M ar'kov partitions and C -diffeomorphisms, Functional Anal­ysis and Applications, 2, 64-89, 1968, n.1; and: Construction of Markov partitions, Functional Analysis and Applications, 2, 70~80, 1968, n.2. Sinai, y'G.: Dynamical systems with elastic reflections. Ergodic proper­ties of dispersing billiards, Russian Mathematical Surveys, 25, 137~ 189, 1970. Sinai, y.G.: Gibbs measures in ergodic theory, Russian Mathematical Surveys, 27, 21~69, 1972.

328

[Si77]

[Si79]

[Si91]

[Sp71]

[Sp91]

[Su67]

[Su70] [TF61]

[Ta66]

[Th74]

[UF63]

[U030]

[Ul68],

[VB75]

[VB 77]

[VH50]

[VK64]

[VN55]

[VN79]

[VN87]

[VP92]

[VP94]

[VS06]

Bibliography

Sinai, Y.G.: Lectures in ergodic theory, Lecture Notes in Mathematics, Princeton University Press, 1977. Sinai, y'G.: Development of Krylov's ideas, in Krylov, N.S.: Works on the foundations in statistical physics, Princeton University Press, 1979, pp. 239-28l. Sinai y'G.: Mathematical problems of statistical mechanics, World Sci­entific, Singapore, 1991. Spitzer, F.: Markov random fields and Gibbs ensembles, American Mathematical Monthly, 78, 142-154, 1971. Spohn, H.: Large scale dynamics of interacting particles, Springer­Verlag, Berlin, 1991. Sutherland, W.: Exact solution of a two-dimensional model for hydro­gen-bonded crystals, Physical Review Letters, 19, 103-104, 1967. Sutherland, W.: Journal of Mathematical Physics, 11, 3183-3186, 1970. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechan­ics. An exact result, Philosophical Magazine, 6, 1061-1063, 1961. Taylor, W.: The kinetic theory of the dissipation of energy, reprinted (in English) in: S. Brush, Kinetic theory, 2, 176, Pergamon Press, Oxford, 1966. Thomson, W.: The kinetic theory of dissipation of energy, Proceedings of the Royal Society of Edinburgh, 8, 325-328, 1874, reprinted in [Br66]. Uhlenbeck, G.E., Ford, G.W.: Lectures in statistical mechanics, Amer­ican Mathematical Society, Providence, R.I., 1963, pp. 5, 16, 30. Uhlenbeck, G.E., Ornstein, 1.S.: On the theory of Brownian motion, Physical Review, 36, 823-841, 1930, reprinted in [Wa54]. Uhlenbeck, G.E.: An outline of statistical mechanics, in Fundamental problems in statistical mechanics, vol. II, ed. E.G.D. Cohen, North Hol­land, Amsterdam, 1968. van Beijeren, H.: Interface sharpness in the Ising system, Communica­tions in Mathematical Physics, 40, 1-6, 1975. van Beijeren, H.: Exactly solvable model for the roughening transition of a crystal surface, Physical Review Letters, 38, 993-996, 1977. van Hove, L.: Sur l'integrale de configuration pour les systemes de par­ticules Ii une dimension, Physica, 16, 137-143, 1950. van Kampen, N.G.: Condensation of a classical gas with long range attraction, Physical Review, 135A, 362-369, 1964. von Neumann, J.: Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, New Jersey, 1955. von Nageli, K.W.: Uber die Bewegungen kleinster Karper, Bayerische Akademie der Wissenschaften, Sitzungsberichte der Mathematisch-Phy­sikalischen Classe, 9, 389-453, 1879. van Beijeren, H., Nolden, I.M.: The roughening transition, in Topics in current physics, ed. W. Schommers, P. von Blankenhagen, Springer, Princeton, New Jersey, 1987. von Plato, J.: Boltzmann's ergodic hypothesis, Archive for the History of Exact Sciences, 44, 71-89, 1992. von Plato, J.: Creating modern probability, Cambridge University Press, New York, 1994. von Smoluchowski, M.: Zur kinetischen Theorie der Brownschen Mole­kularbewegung und der Suspensionen, Annalen der Physik, 21, 756-780, 1906.

[VW41]

[VW88]

[WF72]

[WL69]

[WL98]

Bibliography 329

van der Waerden, B.L.: Die lange Reichweite der regelmassigen Atom­anordnung in Mischcristallen, Zeitschrift fiir Physik, 118, 473-488, 1941. van der Waals, J.D.: On the continuity 'of the gaseous and liquid states, thesis, 1873, edited by J.S. Rowlinson, in Studies in statistical mechan­ics, vol. XIV, North Holland, Amsterdam, 1988. Wilson, K.G., Fisher, M.E.: Critical exponents in 3.99 dimensions, Physical Review Letters, 28, 240-243, 1972. Weijland, A., Van Leeuwen, J.: Non analytic behaviour of the diffusion coefficient of a Lorentz gas, Physica, 36, 457-490, 1967; and 38, 35-47, 1968; Hauge, E., Cohen, E.: Normal and abnormal diffusion in Ehren­fests's wind tree model, Journal of Mathematical Physics, ~8, 397-414, 1969; Hogey, W.: Convergent generalizations of the Boltzmann equation for a hard sphere gas, Physical Review, 185, 210-218, 1969. Woitkowsky, M.P., Liverani, C.: Conformally symplectic dynamics and symmetry of the Lyapunov spectrum, Communications in Mathematical Physics, 194, 47-60, 1998.

[WMTB76] Wu, T.T., McCoy, B.M., Tracy, C., Baruch, E.: Spin-spin correlation

[Wa54] [Wa59]

[We71]

[Wi83]

[Wi89]

[YY66]

[Ya52]

[Yo94]

[Za89]

functions for the two-dimensional Ising model: exact theory in the scal­ing region, Physical Review 13 B, 316-374, 1976. Wax, N.: Selected papers on noise and stochastic processes, Dover, 1954. Walker, L.R.: Antiferromagnetic linear chain, Physical Review, 116, 1089-1090, 1959. Wegner, F.J.: Duality in generalized Ising models and phase transitions without local order parameters, Journal of Mathematical Physics, 12, 2259-2272, 1971. Wilson, K.G.: The renormalization group and critical phenomena, Re­view of Modern Physics, 55, 583-600, 1983. Wittaker, E.T.: A treatise on the analytic dynamics of particles and rigid bodies, Cambridge University Press, 1989 (reprint). Yang, C.N., Yang, C.P.: Ground state energy of a Heisenberg-Ising lat­tice, Physical Review, 147,303-306, 1966; Physical Review, 150, 321-327, 327-339, 1966; Physical Review, 151, 258-264, 1966. Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model, Physical Review, 85, 808-816, 1952. Young, L.S.: Ergodic theory of differentiable dynamical systems, in Real and complex dynamical systems, ed. B. Branner, P. Hjorth, Nato ASI series, Kluwer, Dordrecht, 1995. And Ergodic theory of chaotic dynami­cal systems, Proceedings of the International Congress of Mathematical Physics, Brisbane, 1997, ed. A. Bracken. Zanetti, G.: Hydrodynamics of lattice gas automata, Physical Review, 40 A, 1539-1548, 1989.

Name Index

Anosov 261,265,266 Avogadro 1

Bachelier 234, 241 Baxter 217,231 Bethe 228 Boltzmann 1, 16, 18,20,22,27,38,41,

59,65,69,107,141,151,256,257,281, 295

Borel 40 Bowen 262 Brown 233 Brush 36,175 B6hm 248

Cantoni 233, 236 Cesaro 195 Clausius 129 Cohen 261

Dobrushin 154, 176, 189

Ehrenfest 40,41 Einstein 233, 237, 242 Euler 15,68

Fisher 108,176

Galgani 108 Gallavotti 261 Gibbs 18,21,22,26,37,39,41,58,69,

262,281 Grad 50 Green 176 Griffiths 176, 189

Helmholtz 37,38,253,256 Hilbert 234 Hoover 263 HultMn 231 Hurst 176

Jeans 108

Kac 175 Kadanoff 108 Kasteleyn 175 Kaufman 175 Kintchin 248, 249 Knudsen 55 Kramers 175 Krylov 44, 258 Kuhn 44

Lagrange 241 Lanford 35, 107, 154, 176, 179 Langevin 243 Lawler 176 Lebowitz 55 Lee 193 Leibnitz 38 Lieb 175,217,218,220 Liouville 29 Lucretius 141

Mattis 175 Maxwell 37,38 Minlos 198

Nelson Niigeli

Onsager Ornstein Oseledec

248 233,239

175,212 176,243,244 269

Pauling 217 Peierls 175, 189 Perrin 233, 236 Pirogov 207 Planck 43, 102 Plato 38 Poincare 233, 236

332 N arne Index

Poisson 48

Raoult 235 Ruelle 44, 154, 176, 179, 193, 258,

261-263,277

Schultz 175 Sinai 44,107,198,207,258,262,

270 Smale 273 Smoluchowski 233,239,240,242 Spohn 35 Sutherland 218 Sykes 176

Thomson 257

Uhlenbeck 243, 244

Van der Waals 9, 142 Van der Waerden 175 Van t'Hoff 235

Wannier 175 Ward 175 Wiener 242, 246 Wilson 108

Yang CN 175, 193, 212, 231 Yang CP 231

Zeno 05 Zermelo 41

Subject Index

absolute continuity 269 abstract nonsense 145 activity 125, 162 age of the Universe 15 allowed sequence 275 almost all points 278 analytic error 61,86,95 angular momentum 12 annoying error 104 Anosov map or system 261 Anosov system 262, 264, 266 Anosov system: def 265 ansatz 228 anthropocentrism 114 approach to equilibrium 16 approximate symmetry 207 Arnold's constant 203 artifact 139 atomic hypothesis 3,6,14,21,26,

234 atomic size 66 atomism 141,298 attracting set 262, 264 attracting set: def 264 attractive potential 155 attractor 264 attractor: def 264 average kinetic energy 20 average over time 10 average value 13, 17 Avogadro's number 1,10,15,69,91,

133,237 axiom A attractor 273 axiom A systems 262 axiom of choice 41,151

Bachelier 246 Bethe ansatz 227-229 big nonchaotic island 266 Birkhoff's ergodic theorem 153 black body 93, 102

Bogoliubov-Valatin transformation 225

Boltzmann's constant 7, 15,21,68,72, 75,144,235

Boltzmann's constant: universality 76 Boltzmann's equation 27, 30, 34, 48,

49,55,140,237,239 Boltzmann's formula 67 Boltzmann's H-theorem 31 Boltzmann's sea 24 Borel set 151 Bose condensation 97 boson 105 boundary condition 4,30,81,82,144,

177,202,276 boundary condition + 203 boundary condition - 203 boundary condition: periodic 4, 12 Boyle-Mariotte law 133 broken symmetry 146

canonical absolute temperature 70 canonical distribution 60,63 canonical ensemble 20, 25, 59, 60, 63,

68,69,89,104,106,111,116,117,143, 256

canonical entropy 70 canonical free energy 70 canonical internal energy 70 canonical pressure 70, 79 canonical specific volume 70 Carnot cycle 134 Carnot's principle 236 cell in hase space 271 cell in phase space 1,57,58,60, 103,

256,258,280 cell of Markov partition 289 cell size 2,61,67,70,94,95, 101 cell: coarse 258 cells permutation 280 Cesaro's limit 195

334 Subject Index

chain of oscillators 12 chaos 258 chaotic hypothesis 258, 261 chaotic system 47 charge 114 charged particles 116 chemical potential 144 classical thermodynamics 19,21 Clausius' virial theorem 130 close packing density 114, 123 cluster property: def 166 coalescence catastrophe 111 coarse grained cell 258, 270 coding & symbolic dynamics 275 coexistence line 201 coexisting phases 75, 213 collapse 112 collision 27 collision cross-section 28 collision volume 29 colloid 233, 234 combinatorial error 61,86,95 combinatorial factor 67 compatibility matrix 171 compatible sequence 275 compatible spins 171 compatible symbols 171 compressibility 125 condenser 114 conditional probability 154 conditional reversibility theorem 287 cone property 270 conservation law 31 constant of Boltzmann 7 constant of gas 91 constant of motion 11 constant of Planck 2, 9, 15 continuum 2 contraction rate of phase space 275 convexity 134 correlation function 16, 156, 179,213 Couette's flow 260 Coulomb interaction 65, 106, 116 covanance 243 critical exponent 220 critical parameter 134 critical point 108, 155, 156 crystal-liquid transition 108 crystalline solid 91 Curie-Weiss theory 206 cylindrical boundary condition 180

daemon 33, 288

Debye's screening 109 deflection angle 29 degeneracy 101 degeneration 99 degree of freedom: frozen 91 degree of freedom: oscillatory 89 density matrix 104 determinism 3 deterministic principle 3 diatomic gas 91 diffusion coefficient 235 Dirac's delta 244 discrete space 141 discrete time 141 distribution of Liouville 47 distribution SRB 262, 278, 287 DLR equation 153-155 doom 10 Doppler regime 243 Doppler's resistance 239,240 duality of Kramers-Wannier 187 Dulong-Petit law 92 dynamical base 268 dynamical base towards future 268 dynamical system 147 dynamics cycle 24, 25

Einstein's formula 240 Einstein-Smoluchowski relation 235 elastic collision 4 energy indetermination 8 enlarged ensembles 81 ensemble 17-19,39,58,67 ensemble equivalence 81,105 entropy 19,22,32,65,68,149,281,282 entropy creation 282 entropy decrease: forget 32 entropy generation rate 281 entropy increase 32 equation of Boltzmann 30, 140 equation of state 23,59,69, 161 equation of van der Waals 9 equilibrium property 25 equilibrium state 25, 179 equilibrium: approach to 107 equipartition 90,94, 103 equivalence of ensembles 68, 69, 75 equivalence of statistical ensembles

59,60,67-71,74,75,78,81,83,105, 111,117,142,143,147,158,296-298

ergo de 21,39 ergodic 37 ergodic decomposition 12, 18

ergodic distribution 17, 18 ergodic hypothesis 10,11,14,17-20,

23-25,33,35,38,40,256,257,264 ergodic hypothesis: critique 24 ergodic hypothesis: difficulties 24 ergodic problem 12, 13,26, 107 ergodic set 11 ergodic theorem 153,249 ergodic theory 59 ergodicity: def 11,17, 152 errors negligibility 61 essence of equivalence 70 etymology 39 Euler's equation 260 evaporation catastrophe 114 evolution map 57 exact solution, Ising 2D 211 exact solution: six-vertex 227 expansion rate of phase space 275 extremal distribution 151 extreme degeneracy 100

F model 217 fallacious argument 234 Fermi's condensation 98 Fermi's sphere 99 fermion 105,226 fermionic system 226 ferro electricity 218 Feynman-Kac formula 247 finite volume orthodicity 60 fixed particle boundary condition 77,

144,146,147,276 fixed spin boundary condition 171,

276 flight time 35 fluctuation theorem 284 fluctuation dissipation theorem 245 formalism 143 free boundary conditions 203 free energy 63,68, 177 free gas 68 free micro canonical energy 70 frozen degree 92 function: measurability 152

gamma function 15,68 gap 194 gas constant 91 Gauss' principle 283,285,297,311 Gaussian distribution 242 Gaussian process 243, 244 Gaussian thermostat 296, 297

Subject Index 335

ghastly generality 156 Gibbs distribution 154, 155 Gibbs distribution: def 146 Gibbs' potential 79,83 global iterated logarithm law 249 Grad's conjecture 30,50,55 Grad's limit 28,30,34,47,107 Grad-Boltzmann limit 55 grand canonical ensemble 20,77,117,

143 gravitation constant 113 gravitational force 114 Green-Kubo formula 261,293,294 ground state 60, 119,229

H-theorem 31 Hamilton equation 142 Hamilton's equation 4 hand waving 108 hard core 55, 113, 275 hard core interaction 136,141,143,

155 hard core lattice system 171 hard core potential 125 hard core: def 143 hard sphere 27, ~{O harmonic oscillator 94 Hausdorff' dimension 250 heat equation 247 heat theorem 22,47,253,257,261,264 Heisenberg's model 228 hidden variables 248 Hilbert's space 103 holode 22,39 homogeneous phase 147 hydrogen bond 217 hypothesis: ergodic 10 Holder's continuity 248

ice model 217 ice rule 217 imponderable 151 indetermination 8 indetermination principle 2, 105 indistinguishability 61, 144 infinite energy 94 infinite system 144 infinite volume 179 infrared bounds 196 infrared catastrophe 114 instability and boundary conditions

108 integrating factor 25

336 Subject Index

intensive quantity 82 interaction potential 134 interaction range 7 interaction strength 7 interesting case 10 intrinsic pulsation 12 invariant distribution 17 irreversibility 31,35, 107 Ising model: 2D solution 221 Ising's model 175,209,211,216,274 Ising's model: many spin interaction

170 iterated logarithm 248, 249

joint probability 245

Kac's potential 139 KDP model 217 KH2 P04 217 kinetic energy 3,57 kinetic theory 235 Kirkwood-Salsburg equations 163,

179 Knudsen's model 55

Landau's argument 136,158 Lanford's theorem 35,107,140 Langevin equation 243 large deviation 280 large matrix 104 large system 257 latent heat 219 lattice gas 160 lattice model 167 lattice spin system 160 law of conservation of difficulties 143 law of large numbers 66 law of motion 3 Lawler's theorem 250 least constraint principle 283, 285,

297,311 Lee-Yang theorem 193 Lennard-Jones potential 6,7,66,115,

129 Levy's regularity law 248 limit of Grad-Boltzmann 47 linear operator 104 Liouville's distribution 47,256 Liouville's theorem 4, 29 Liouville's volume 289 liquid gas transition 108 local distribution: def 145 local observable 145,280

local observable: def 145 locally finite configurations: def 144 long range order 160, 196 Lorentz' gas 47,49,55 Lorentz' model 31 Lyapunov's exponent 268

macroscopic equilibrium 19,58 macroscopic equilibrium state 18, 58 macroscopic observable 14, 16,24 macroscopic states 19 macroscopically negligible 21 Markov partition 272, 273, 289, 290 Markov pavement 272, 273 Markov's partition 270 Markov's partition: def 272 Maxwell's daemon 236 Maxwell's equations 93 Maxwell's rule 134, 139 Maxwell-Boltzmann distribution 32 Maxwell-Boltzmann law 87 Maxwellian distribution 154, 238 Maxwellian distribution: def 148 maxwellian momentum distribution

147 mean field 108, 137, 141 mean field theory 137,211,213 mean free path 28, 35, 311 measurable function: def 152 measurable set: def 151, 152 measure: def 151 Mermin-Wagner theorem 160, 161 metal 114 metaphysics 33 metastability 109, 134,206 microcanonical absolute temperature

70 microcanonical distribution 60 microcanonical ensemble 14,20,21,

24,25,59,60,63,68,69,117,143 micro canonical entropy 70,117,120 microcanonical internal energy 70 microcanonical partition function 120 micro canonical pressure 70 micro canonical specific volume 70 microscopic configuration 57,214 microscopic dynamics 19 microscopic state 1-3,57,61 minimal constraint principle 283, 285,

297,311 minus boundary condition 180 mixing: topological 274 mixture of phases 151

model of thermodynamics 17,20,21, 23,25,38,58-60,67,72,177

mole 1 molecular chaos 49,55,239 monoatomic gas 91 mono cyclic 37,44,46 mono cyclic system 37,253,256 mono cyclicity 255 mono de 17-19,37,58 Moon 26 motion reversibility 4 multifractal distribution 285 multiple collision 30

Navier-Stokes' equation 235,260 Navier-Stokes' fluid 260 nearest neighbor 157 neon 91 Newton's equation 32 no regret 162 noble picture 273 nOIse 243 non equivalence of ensembles 72, 75,

157, 159 nonergodicity 24 normal conditions 1 normal mode 12,91 nuclear size 66 number of cells 15 numerical density 119 numerical experiment 5 numerical simulation 136

observable 13, 18, 58 occupation number 86 Onsager's reciprocity 261 open boundary condition 179, 203 optimistic viewpoint 2 order of transition 219 order parameter: def 160 Ornstein-Uhlenbeck's process 245 orthode 19,39,59 orthodic 27,43 orthodic ensemble 75 orthodicity 19,21,22,26,39,59,60,

62-64,66,67,76,77,111,142 orthodicity in finite volume 60 oscillators chain 12 oscillatory degree of freedom 89,91,

94 Oseledec's theorem 269

pair correlation 16

Subject Index 337

pair interaction 1 paradox 95 paradox of Zeno 5 partition function 59,63,64,67,77,

82, 104, 106, 144 partition of phase space 270 partition sum 61 Pauli-Jordan transformation 223 Peierls' argument 189 perfect gas 89 perfect wall boundary condition 203 periodic boundary 30, 32 periodic boundary condition 4, 179,

214 periodic orbit 47,291 perpetual motion 237 Perron-Frobenius theorem 172 pfaffian 175 phase coexistence 196 phase separation 211 phase separation line 201 phase space 1,3,57 phase space contraction 283 phase transition 72,74,75,108,134,

188,205,211,219,280 phase transition: def 74,83,155,157,

158, 161, 183 phase: def 142 physical grounds 21 Pirogov-Sinai theory 207 Planck's constant 2, 10, 15, 101, 142 Planck's distribution 102 plus boundary condition 180 point process 155 Poisson's distribution 48, 49 positive definite potential 115 positive type potential 115 Postscript 273 pot 66 potential energy 3,57 potential of Kac 139 pressure 16 pressure ensemble 77,83 principle of D' Alembert 312 principle of equipartition 89 principle of Gauss 312 principle of indetermination 2 principle of least action 312 principle of least constraint 312 principle of Ruelle 277 principle of sufficient reason 26 probability distribution: def 151 process 243

338 Subject Index

propagator 243 proper frequencies 89 pure phase 151,190 pure phase: def 152

quadrature 246 quantum effects 107 quantum grand canonical ensemble

105 quantum mechanics 2,9,62,93,101 quantum micro canonical ensemble

105 quantum statistical mechanics 103 quasi-ergodic hypothesis 40

radiating cavity 94, 100 radiation 93 radiation distribution 102 range of interaction 6, 118 Raoult-van t'Hoff's law 234,235,238 rarefied gas 238 Rayleigh-Jeans' formula 94 reciprocity of Onsager 261 rectangular cell 270 recurrence 14 recurrence time 15,24,25,27,35,107 renormalization group 108 repulsive potential 115 reversibility 32, 33, 35, 277 reversibility paradox 237 reversible representation 280 reversible system 284 reversible thermostat 295, 296 rigid sphere 27,30,33 roughening transition 207 round-off error 5 Ruelle's theorem 193 ruler and compass 273

scale invariance 155 scale of time 5 Schr6dinger's operator 104,106 second law 19,32,256 second principle 22 sense of Fisher 123 set t-t-measurable: def 152 simplex 151 sinister axiom 151 six-vertex model 217 six-vertex model: solution 227 small system 257 Smoluchowski's theory 239 soluble model 108

specific heat 69,89,91,98 spectral theorem 273 spin configuration 184 spin system 108 spontaneous magnetization 212 spontaneous polarization 219 spontaneous symmetry breakdown

146 SRB distribution 278,287,289,308 stability 77,111,143 stability of a potential 57,63 stability of a potential: def 65 stability of matter 109 stable manifold 308 stable manifold, global def 265 star 66,114 stationary distribution 17,57 statistical ensemble 19,58 statistical ensembles equivalence 20 statistics 264 statistics of Bose-Einstein 43 statistics of motion 262 statistics of motion: def 262 statistics: quantum symmetry 106 Stirling's formula 15, 120 stochastic process 155 stochastic process: def 156, 243 stochastic thermostat 295 stock market 234, 242 Stokes law 234,235,238-240 Stokes regime 243 stosszahlansatz 30,32,34 straight line 201 strange idea 66 subshift of finite type 275 Sun 101 superstability 125, 145 superstability: def 143 surface tension 199,213 symbolic dynamics 270, 275 -symmetry 12 symmetry and phase transitions 160 symmetry of time reversal 277 symmetry: def 160

tautology 142 temperature 20, 22, 25 temperedness 77,111,143 temperedness of a potential 63 temperedness of a potential: def 65 theorem H 31 theorem H of Ruelle 283 theorem of Anosov 265, 275

theorem of heat 22 theorem of Lanford 35,140 theorem of Liouville 4, 29 theorem of Mermin-Wagner 160 theorem of van Hove 173 theory of speculation 241 thermodynamic formalism 142,280 thermodynamic limit 19, 20, 24, 25,

60,63,69,77,104,117,142,150 thermodynamics model 68 thermostat 296 third law 92 time average 10, 25 time of flight 28 time reversal 277,283,284,289,291 time scale 23, 33, 107 timing event 258 topological mixing 274 total energy 3 trace of matrix 104 tranformation of Bogoliubov-Valatin

225 transfer matrix 209 transfer matrix: def 172 transition matrix 274 transition matrix: mixing 274 transition matrix: transitive 274 transitive Anosov system 274 translation invariance 146 translation of an observable 160 translational degree of freedom 89 triviality theorem 196 typical cell 7

Uhlenbeck-Ornstein theory 242

Subject Index 339

ultraviolet catastrophe 112 universal constant 68 universal law 284 universe 114 unstable manifold 308 unstable manifold, global def 265 upsetting contradiction 93

van der Waals' equation 133,141 van Hove's theorem 136, 158, 173 van t'Hoff's law 234 variational principle 147, 148, 155 vertex model 6V 214 vertex model 8V 214 vertex models 214 virial series 133, 135 virial theorem 136 viscosity 234, 235 volume preserving map 4

wall potential 311 water 285 waterfall 288 wave functions 105 weight 60 white noise 243,244 Wien's distribution 102 Wiener process 246,248 Wiener theorem 246 Wiener's theory 246 wind tree model 48 wrong prediction 95 Wulff's construction 204

Zeno's paradox 5