Skip to main content
Log in

Abstracts of the workshop on statistical mechanics, dynamical systems, and turbulence

September 7–17, 1982

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. D. D. Joseph,Stability of Fluid Motion, Vols. I & II (Springer, Berlin, 1976).

    Google Scholar 

  2. H. Swinney and J. Gollub (eds.),Hydrodynamic Instabilities and the Transition to Turbulence. Springer Topics in Applied Physics (Springer, Berlin, 1981).

    Google Scholar 

  3. G. I. Barenblatt, G. Iooss, and D. Joseph (eds.),Nonlinear Dynamics and Turbulence (Pitman, London, 1983).

    Google Scholar 

References

  1. M. Van Dyke,An Album of Fluid Motions (Parabolic Press, Stanford, California, 1982).

    Google Scholar 

  2. D. R. Chapman, Computational Aerodynamic Development and Outlook,AIA J. 17:1293 (1979).

    Google Scholar 

  3. W. C. Reynolds, Computation of Turbulent Flows, inAnnu. Rev. Fluid Mech. 6 (1974).

  4. T. S. Lundgren, Strained Spiral Vortex Model for Turbulent Fine Structure,Phys. Fluids,12:2193 (1982).

    Google Scholar 

Reference

  1. J.-P. Eckmann,Rev. Mod. Phys. 53:643–654 (1981).

    Google Scholar 

Reference

  1. J.-P. Eckmann and P. Collet,Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980).

    Google Scholar 

References

  1. V. I. Arnol'd, Loss of Stability of Self-Oscillations Close to Resonance and Versal Deformations of Equivarient Vector Fields,Funct. Anal. Appl. 11:1–10 (1970).

    Google Scholar 

  2. Dan Henry,Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, Vol. 840 (Springer, Berlin, 1981).

    Google Scholar 

  3. G. Iooss and D. D. Joseph,Elementary Stability and Bifurcation Theory (Springer, Berlin, 1980).

    Google Scholar 

  4. O. E. Lanford III, Bifurcation of Periodic Solutions into Invariant Tori: The Work of Ruelle and Takens, inNonlinear Problems in the Physical Sciences and Biology, Springer Lecture Notes in Mathematics, Vol. 322 (Springer, Berlin, 1973).

    Google Scholar 

  5. J. Marsden and M. McCracken,The Hopf Bifurcation and its Applications (Springer, Berlin, 1976).

    Google Scholar 

  6. D. Ruelle and F. Takens, On the Nature of Turbulence,Comm. Math. Phys. 20:167–192 (1971);23:343–344 (1971).

    Google Scholar 

  7. D. Sattinger,Topics in Stability and Bifurcation Theory, Springer Lecture Notes in Mathematics, Vol. 309 (Springer, Berlin, 1972).

    Google Scholar 

  8. H. F. Weinberger, The Stability of Solutions Bifurcating from Steady or Periodic Solutions, inDynamical Systems, A. R. Bednarek and L. Cesari, eds. (Academic Press, New York, 1977), pp. 349–366.

    Google Scholar 

General References

  1. N. Kolmogorov,Dokl. Akad. Nauk. 98:527 (1954).

    Google Scholar 

  2. J. Moser,Nach. Akad. Wiss. Göttingen IIa:1 (1962).

    Google Scholar 

  3. V. Arnold,Russ. Math. Surv. 18(5):9 (1963);18(6):85 (1963).

    Google Scholar 

  4. J. Poschel, Ueber differenzierbare Faserung invarianter Tori, 1981, preprint, ETH-Zurich., and Integrability of Hamiltonian systems on Cantor Sets,Comm. Pure Appl. Math. 35:653–696 (1982).

  5. L. Chierchia and G. Gallavotti, Smooth Prime Integrals for Quasi integrable Hamiltonian Systems,Nuovo Cimento B67:277 (1982).

    Google Scholar 

  6. H. Poincare,Les Methodes Nouvelles de la Mecanique Celeste (Gouthier-Villars, Paris, 1892), Vol. I, Chap. V, p. 233.

    Google Scholar 

  7. J. Moser,Stable and Random Motions in Dynamical Systems, Ann. Math. Studies, (Princeton Univ. Press, Princeton, New Jersey, 1973).

    Google Scholar 

  8. D. Escande and F. Doveil,J. Stat. Phys. 26:257 (1981).

    Google Scholar 

  9. D. Escande, Renormalization Approach to Nonintegrable Hamiltonians, Austin Work-shop, March 1981, preprint Ecole Politecnique, Lab. Phys. Milieux Ionises, Palasieu, 1982.

  10. H. Whitney,Trans. Am. Math. Soc. 36:63 (1934).

    Google Scholar 

  11. H. Russman,Celestial Mech. 14:33 (1976);Comm. Pure Appl. Math. 29:755 (1976); andLecture Notes in Physics, Vol. 38, 1975, ed. J. Moser.

    Google Scholar 

  12. G. Gallavotti, Perturbation Theory for Classical Hamiltonian Systems, to appear inProgress in Physics, J. Frolich, ed. (Birkhauser, Boston).

References for the integrability (by quadratures) criteria

  1. H. Poincare,Methods Nouvelles de la Mecanique Celeste (Gauthier-Villars, Paris, 1897), Vol. 1, Chap. V.

    Google Scholar 

  2. G. Gallavotti,Meccanica Elementare (Boringhieri, Torino, 1980), Chap. V, Sec. 5.10.

    Google Scholar 

  3. J. Moser, Lectures on Hamiltonian Systems,Memoirs Am. Math. Soc,81 (1973).

  4. S. Shenker and L. Kadanoff,J. Stat. Phys. 27:631 (1982); S. Shenker, Scaling Behavior in a Map of a Circle onto Itself. Empirical Results, to appear inPhysica D; M. Feigenbaum, L. Kadanoff, and S. Shenker, Quasiperiodicity in Dissipative Systems. A Renormalization Group Analysis, preprint, Los Alamos, 1982, to appear inPhysica D; D. Rand, S. Ostlund, and E. Sethna, ITP (Santa Barbara) preprint.

    Google Scholar 

  5. N. Kolmogorov,Dokl. Akad. Nauk 98:27 (1954); V. Arnold,Russ. Math. Surveys 18(5) (1963); and J. Moser,Nach. Akad. Wiss. Göttingen IIa:1 (1962).

    Google Scholar 

  6. J. Poschel, Über differenzierbare Faserung invarianter Tori, ETA, Zurich, preprint (1981), and Integrability of Hamiltonian Systems on Cantor Sets,Comm. Pure Appl. Math. 35:653–696 (1982).

    Google Scholar 

  7. L. Chierchia and G. Gallavotti,Nuovo Cimento 67B:277 (1982).

    Google Scholar 

  8. G. Gallavotti, Perturbation Theory of Classical Hamiltonian Systems, to appear inProgress in Physics, J. Fröhlich, ed. (Birkhauser, Boston).

  9. P. Collet, H. Epstein, and G. Gallavotti, Perturbations of Geodesic Flows on Surfaces of Constant Negative Curvature, in preprint, Princeton, June 1982.

  10. H. McKean and E. Trubowitz,Comm. Math. Phys. 82:4 (1982).

    Google Scholar 

  11. G. Gallavotti, A Criterion of Integrability for Perturbed Nonresonant Harmonic Oscillators. “Wick Ordering” of the Perturbations in Classical Mechanics and Invariance of the Frequency Spectrum,Comm. Math. Phys. 87:365 (1982). The main result of this work has been previously obtained in reference 12.

    Google Scholar 

  12. H. Rüssmann, Math. Annalen169:55 (1967).

    Google Scholar 

References for the theory of the Hamilton-Jacobi equation associated with the geodesic flow on a surface of constant negative curvature

  1. V. Arnold and A. Avez,Ergodic Problems of Classical Mechanics (Benjamin, New York, 1967).

    Google Scholar 

  2. V. Guillemin and I. Kazhdan,Topology 19:301 (1980);19:291 (1980).

    Google Scholar 

  3. A. Livscic,Mat. Zemetki 10:555 (1971).

    Google Scholar 

  4. I. Gelfand and S. Fomin,AMS Translations, Ser. 2 Vol.1, p. 49 (1955).

    Google Scholar 

  5. I. Gelfand, M. Grev and N. Vilenkin,Generalized Functions, Vol. 5 (Academic Press, New York, 1966).

    Google Scholar 

  6. I. Gelfand, M. Graev, and I. Piateckii-Shhapiro,Representation Theory and Automorphic Functions (Saunders, Philadelphia, 1969).

    Google Scholar 

  7. H. Poincaré,Acta Math. 1:1 (1882).

    Google Scholar 

  8. V. Arnold,Russ. Math. Surveys 18(5):9 (1963).

    Google Scholar 

  9. J. Poshel, Über differenzierbare Faserung invarianter Tori, preprint, ETH-Zurich, 1981; L. Chierchia and G. Gallavotti,Nuovo Cimento 67B:277 (1982).

  10. H. McKean and E. Trubowitz,Comm. Math. Phys. 82:4 (1982).

    Google Scholar 

  11. G. Gallavotti, Perturbation Theory for Classical Hamiltonian Systems, preprint; in print in the series “Progress in Physics, PPhI”, in the collection of papers on rigorous results on the renormalization group methods edited by J. Fröhlich (Birkhauser, Boston).

  12. R. Bowen,Lecture Notes in Math., Vol. 470 (Springer Verlag, Berlin 1975).

    Google Scholar 

  13. P. Collet, H. Epstein, and G. Gallavotti, Perturbations of Geodesic Flows on Surfaces of Constant Negative Curvature and their Mixing Properties, preprint, Princeton University, July 1982.

References

  1. R. Bowen,On Axiom A Diffeomorphisms, CBMS Regional Conference Series in Mathematics, No. 35 (American Mathematical Society, Providence, Rhode Island, 1978).

    Google Scholar 

  2. V. Mel'nikov, On the Stability of the Center for Time-Periodic Perturbations,Trans. Moscow Math. Soc. 12:1–57 (1963).

    Google Scholar 

References

  1. E. Lorenz, Deterministic Non-Periodic Flow,J. Atmos. Sci. 20:130–141 (1963).

    Google Scholar 

  2. M. Hénon, A Two-Dimensional Mapping with a Strange Attractor,Commun. Math. Phys. 50:69–77 (1976).

    Google Scholar 

References

  1. E. Fermi,Thermodynamics (Dover, New York, 1936).

    Google Scholar 

  2. D. Ruelle,Statistical Mechanics: Rigorous Results (W. A. Benjamin, New York, 1969).

    Google Scholar 

  3. O. E. Lanford, Entropy and Equilibrium States in Classical Statistical Mechanics, inStatistical Physics and Mathematical Problems (Proceedings of the 1971 Battelle Rencontres), Springer Lecture Notes in Physics 20 (Springer, Berlin, 1972).

    Google Scholar 

References

  1. L. Gross,Thermodynamics, Statistical Mechanics and Random Fields, Lecture Notes in Mathematics, No. 929, Hennequin, Ed. (Springer-Verlag, Berlin, 1982).

    Google Scholar 

  2. R. B. Israel,Convexity in the Theory of Lattice Gases (Princeton Univ. Press, Princeton, New Jersey, 1979).

    Google Scholar 

  3. D. Ruelle,Statistical Mechanics (W. A. Benjamin, Inc., New York, 1969).

    Google Scholar 

  4. D. Ruelle,Thermodynamic Formalism (Addison-Wesley Pub. Co., Reading, Massachusetts, 1978).

    Google Scholar 

Reference GKS inequalities

  1. R. B. Griffiths, Correlations in Ising Ferromagnets,J. Math. Phys. 8:478–483 (1967).

    Google Scholar 

  2. D. Kelly and S. Sherman, General Griffiths Inequalities on Correlations in Ising Ferromagnets,J. Math. Phys. 9:466–484 (1968).

    Google Scholar 

  3. J. Ginibre, General Formulation of Griffiths Inequalities,Commun. Math. Phys. 16:310–328 (1970).

    Google Scholar 

  4. G. S. Sylvester, The Ginibre Inequality,Commun. Math. Phys. 73:105–114 (1980).

    Google Scholar 

References GHS Inequalities

  1. R. B. Griffiths, C. A. Hurst, and S. Sherman, Concavity of Magnetization of an Ising Ferromagnet in a Positive External Field,J. Math. Phys. 11:790–795 (1970).

    Google Scholar 

  2. R. S. Ellis, J. L. Monroe, and C. M. Newman, The GHS and Other Correlation Inequalities for a Class of Even Ferromagnets,Commun. Math. Phys. 46:167–182 (1976).

    Google Scholar 

  3. G. S. Sylvester, Inequalities for Continuous-Spin Ising Ferromagnets,J. Stat. Phys. 15:327–341 (1976).

    Google Scholar 

References FKG Inequalities

  1. C. M. Fortuin, P. W. Kastelyn, and J. Ginibre, Correlation Inequalities in Some Partially Ordered Sets,Commun. Math. Phys. 22:89–103 (1971).

    Google Scholar 

  2. S. Karlin and Y. Rimott, Classes of Orderings of Measures and Related Correlation Inequalities. I. Multivariate Totally Positive Distributions,J. Mult. Analysis 10:476–498 (1980).

    Google Scholar 

  3. L. Pitt, A Gaussian Correlation Inequality for Symmetric Convex Sets,Ann. Probability 5:470–474 (1977).

    Google Scholar 

  4. J. Lebowitz, Bounds on Correlations and Analyticity Properties of Ferromagnetic Ising Spin Systems,Commun. Math. Phys. 28:313–321 (1972).

    Google Scholar 

  5. C. M. Newman, Normal Fluctuations and the FKG Inequalities,Commun. Math. Phys. 74, 119–128 (1980).

    Google Scholar 

References Lee-Yang Theorem

  1. T. D. Lee and C. N. Yang, Statistical Theory of Equations of State and Phase Transitions, II, Lattice Gas and Ising Model,Phys. Rev. 87:408–419 (1952).

    Google Scholar 

  2. C. M. Newman, Zeros of the Partition Function for Generalized Ising Systems,Commun. Pure Appl. Math. 27:143–159 (1974).

    Google Scholar 

  3. E. H. Lieb and A. D. Sokal, A General Lee-Yang Theorem for One-Component and Multicomponent Ferromagnets,Commun. Math. Phys. 44:223–235 (1975).

    Google Scholar 

  4. C. M. Newman, Gaussian Correlation Inequalities for Ferromagnets,Z. Wahr. 33:75–93 (1975).

    Google Scholar 

References

  1. F. Spitzer, Introduction aux processes de Markov a parametres dans Zv, inÉcole d'Été de Probabilités de Saint-Flour, Springer Lecture Notes in Mathematics #390 (Springer, New York, 1974).

    Google Scholar 

  2. J. Fröhlich, B. Simon, and T. Spencer, Infrared Bounds, Phase Transitions, and Continuous Symmetry Breaking,Commun. Math. Phys. 50:79–85 (1976).

    Google Scholar 

  3. J. Fröhlich, R. Israel, E. Lieb, and B. Simon, Phase Transitions and Reflection Positivity I. General Theory and Long Range Lattice Models,Commun. Math. Phys. 62:1–34 (1978).

    Google Scholar 

  4. A. Klein, Gaussian OS-Positive Processes,Z. Wahrsch. Verw. Gebiete 40:115–124 (1977).

    Google Scholar 

References

  1. D. B. Abraham, On Correlation Functions of the Ising andX-Y Models,Studies Appl. Math. 51:179–209 (1972).

    Google Scholar 

  2. D. B. Abraham, Pair Function for the Rectangular Ising Ferromagnet,Comm. Math. Phys. 60:181–191 (1978).

    Google Scholar 

  3. D. B. Abraham, Odd Operators and Spinor Algebras in Lattice Statistics: n-Point Functions for the Rectangular Ising Model,Comm. Math. Phys. 59:17–34 (1978).

    Google Scholar 

  4. D. B. Abraham,n-Point Functions for the Rectangular Ising Ferromagnet,Phys. Lett. A 61:271–274 (1977).

    Google Scholar 

  5. D. B. Abraham,n-Point Functions for the Rectangular Ising Ferromagnet,Comm. Math. Phys. 60:205–213 (1978).

    Google Scholar 

  6. D. B. Abraham and A. Martin-Löf, The Transfer Matrix for a Pure Phase in the Two-Dimensional Ising Model,Comm. Math. Phys. 32:245–268 (1973).

    Google Scholar 

  7. M. Aizenman, Instability of Phase Coexistence and Translation Invariance in Two Dimensions,Phys. Rev. Lett. 43:407–409 (1973).

    Google Scholar 

  8. M. Aizenman, Translation Invariance and Instability of Phase Coexistence in the Two-Dimensional Ising System,Comm. Math. Phys. 73:83–94 (1980).

    Google Scholar 

  9. R. Z. Bariev, On the Rotational Symmetry of the Spin-Spin Correlation Function of the Two-Dimensional Ising Model,Phys. Lett. A 55:456–458 (1976).

    Google Scholar 

  10. R. Z. Bariev, Many-Point Correlation Functions of the Two-Dimensional Ising Model,Phys. Lett. A 64:169–171 (1977).

    Google Scholar 

  11. E. Barouch, B. M. McCoy, and T. T. Wu, Zero-Field Susceptibility of the Two-Dimensional Ising Model NearT c,Phys. Rev. Lett. 31:1409–1411 (1973).

    Google Scholar 

  12. R. Brauer and H. Weyl, Spinors inn-Dimensions,Amer. J. Math. 57:425–449 (1935).

    Google Scholar 

  13. H. Cheng, and I. I. Wu, Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model III,Phys. Rev. 164:719–735 (1967).

    Google Scholar 

  14. R. L. Dobrushin, The Description of a Random Field by Means of Conditional Probabilities and Conditions of its Regularity,Teor. Verojatnost. i Printenen. 13:201–229 (1968). [English translation.Theory Probab. Appl. 13:197–224 (1968)].

    Google Scholar 

  15. R. L. Dobrushin, The Gibbsian Random Fields for Lattice Systems with Pairwise Interactions,Funktional Anal. i Prilozen. 2:31–43 (1968). [English translation,Functional Anal. Appl. 2:292–301 (1968).]

    Google Scholar 

  16. R. L. Dobrushin, Auto Model Generalized Random Fields and their Renorm-Group, inMulticomponent Random Systems, Advances in Probability and Related Topics, Vol. 6 (R. L. Dobrushin and Ya G. Sinai, Eds.) (Dekker, New York, 1980).

    Google Scholar 

  17. A. Erdelyi,Higher Transcendental Functions, Vol. II (McGraw-Hill, New York, 1953).

    Google Scholar 

  18. M. E. Fisher, Correlation Functions and the Critical Region of Simple Fluids,J. Math. Phys. 5:944–962.

  19. R. Garnier, Sur des equations differentielles du troisieme order dont l'integrale generale est uniforme et sur une classe d'equations nouvelles d'ordre superieur dont l'integrale generale a ses points critiques fixes,Inn. Sci. Ecole Norm. Sup. 29:1–126 (1912).

    Google Scholar 

  20. I. M. Gelfand and N. Ya. Vilenkin,Generalized Functions, Vol. 4 (Academic Press, New York, 1964).

    Google Scholar 

  21. J. Glimm and A. Jaffe, Critical Exponents and Renormalization in the ϕ4 Scaling Limit, presented at the Conference on Quantum Dynamics: Models and Mathematics, Bielefeld, 1975.

  22. J. Glimm and A. Jaffe, Particles and Scaling for Lattice Fields and Ising Models.Comm. Math. Phys. 51:1–13 (1976).

    Google Scholar 

  23. C. Gruber, A. Hinifrmann, and D. Merlini,Group Analysis of Classical Lattice Systems (Springer-Verlag, Berlin, 1977).

    Google Scholar 

  24. T. Hida,Brownian Motion (Springer-Verlag, New York, 1980).

    Google Scholar 

  25. E. L. Ince,Ordinary Differential Equations (Dover, New York, 1947).

    Google Scholar 

  26. M. Jimbo, T. Miwa, and M. Sato, Holonomic Quantum Fields—The Unanticipated Link between Deformation Theory of Differential Equations and Quantum Fields, inMathematical Problems in Theoretical Physics (K. Osterwalder, Ed.), Lecture Notes in Physics No. 116 (Springer-Verlag, Berlin, 1980).

    Google Scholar 

  27. J. D. Johnson, S. Krinsky, and B. M. McCoy, Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of thex-y-z Hamiltonian,Phys. Rev. A 8:2526–2547 (1973).

    Google Scholar 

  28. M. Kac and J. C. Ward, A Combinatorial Solution of the Two-Dimensional Ising Model,Phys. Rev. 88:1332–1337 (1952).

    Google Scholar 

  29. L. P. Kadanoff, Spin-Spin Correlations in the Two-Dimensional Ising Model,Nuovo Cimento B 44:276–304 (1966).

    Google Scholar 

  30. L. P. Kadanoff, Scaling Laws for the Ising Models nearT c,Physics 2, No. 6 263–272 (1966).

    Google Scholar 

  31. L. P. Kadanoff and H. Ceva, Determination of an Operator Algebra for the Two-Dimensional Ising Model.Phys. Rev. B 3:3918–3938 (1971).

    Google Scholar 

  32. P. W. Kasteleyn, Dimer Statistics and Phase Transitions,J. Math. Phys. 4:287–293 (1963).

    Google Scholar 

  33. T. Kato,Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1976).

    Google Scholar 

  34. B. Kaufman, Crystal Statistics II, Partition Function Evaluated by Spinor Analysis,Phys. Rev. 76:1232–1243 (1949).

    Google Scholar 

  35. B. Kaufman and L. Onsager, Crystal Statistics III, Short-Range Order in a Binary Ising Lattice,Phys. Rev. 76:1244–1252 (1949).

    Google Scholar 

  36. D. G. Kelly and S. Sherman, General Griffiths' Inequalities on Correlations in Ising Ferromagnets,J. Math. Phys. 9:466–484 (1968).

    Google Scholar 

  37. M. G. Krein, Integral Equations on a Half-Line with Kernel Depending upon the Difference of the Arguments,Trans. Amer. Math. Soc. 22:163–288 (1962).

    Google Scholar 

  38. O. E. Lanford and D. Ruelle, Observables at Infinity and States with Short Range Correlations in Statistical Mechanics,Comm. Math. Phys. 13:194–215 (1969).

    Google Scholar 

  39. J. L. Lebowitz and A. Martin-Löf, On the Uniqueness of the Equilibrium State for Ising Spin Systems,Comm. Math. Phys. 25:276–282 (1972).

    Google Scholar 

  40. B. M. McCoy and T. T. Wu,The Two-Dimensional Ising Model (Harvard Univ. Press, Cambridge, Massachusetts, 1973).

    Google Scholar 

  41. B. M. McCoy, C. A. Tracy, and T. T. Wu, Spin-Spin Correlation Functions for the Two-Dimensional Ising Model, inStatistical Mechanics and Statistical Methods in Theory and Application (U. Landman, Ed.), pp. 83–97 (Plenum, New York, 1977).

    Google Scholar 

  42. B. M. McCoy, C. A. Tracy and T. T. Wu, Painlevé Functions of the Third Kind,J. Math. Phys. 18:1058–1092 (1977).

    Google Scholar 

  43. B. M. McCoy, C. A. Tracy, and T. T. Wu, Two-Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory. Explicit Formulas forn-Point Functions,Phys. Rev. Lett. 38:793–796 (1977).

    Google Scholar 

  44. B. M. McCoy and T. T. Wu, Two-Dimensional Ising Field Theory forT<T c: String Structure of the Three-Point Function.Phys. Rev. D 18:1243–1252 (1978).

    Google Scholar 

  45. B. M. McCoy and T. T. Wu, Two-Dimensional Ising Field Theory forT<T c: Green's-Functions Strings inn-Point Functions,Phys. Rev. D 18:1253–1258 (1978).

    Google Scholar 

  46. B. M. McCoy and T. T. Wu, Two-Dimensional Ising Field Theory in a Magnetic Field: Breakup of the Cut in the Two-Point Function,Phys. Rev. D 18:1259–1267 (1978).

    Google Scholar 

  47. B. M. McCoy and T. T. Wu, Two-Dimensional Ising Model nearT c: Approximation for Small Magnetic Field,Phys. Rev. D 18:4886–4901 (1978).

    Google Scholar 

  48. B. M. McCoy and T. T. Wu, Non-Perturbative Quantum Field Theory,Sci. Sinica 22:1021–1032 (1979)

    Google Scholar 

  49. E. W. Montroll, R. B. Potts, and J. C. Ward, Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model,J. Math. Phys. 4:308–322 (1963).

    Google Scholar 

  50. C. R. Nappi, On the Scaling Limit of the Ising Model,Nuovo Cimento A 44:392–400 (1978).

    Google Scholar 

  51. C. M. Newman, Gaussian Correlation Inequalities for Ferromagnets,Z. Wahrsch. Verw. Gebiete. 33:75–93 (1975).

    Google Scholar 

  52. L. Onsager, Crystal Statistics I. A Two-Dimensional Model with an Order-Disorder Transition,Phys. Rev. 65:117–149 (1944).

    Google Scholar 

  53. L. Onsager, Discussion,Nuovo Cimento 6 Suppl.:261 (1949).

    Google Scholar 

  54. K. Osterwalder and R. Schrader, Axioms for Euclidean Green's Functions,Comm. Math. Phys. 31:83–112 (1973);42:281–305 (1975).

    Google Scholar 

  55. J. Palmer, Complex Structures and External Fields,Arch. Rat. Mech. Anal. 76 (1980).

  56. J. Palmer, Products in Spin Representations,Adv. Appl. Math. 2:290–328 (1981).

    Google Scholar 

  57. M. Sato, T. Miwa, and M. Jimbo, Studies on Holonomic Quantum Fields, I.Proc. Jpn. Acad. A 53:6–10 (1977).

    Google Scholar 

  58. M. Sato, T. Miwa, and M. Jimbo, Studies on Holonomic Quantum Fields, II-XV,Proc. Jpn. Acad. A 53:147–152 (1977);53:153–158 (1977);53:183–185 (1977);53:219–224 (1977);54:1–5 (1978),54:36–41 (1978);54:221–225 (1978);54:263–268 (1978);54:309–313 (1978);55:6–9 (1979);55:73–77 (1979);55:115–120 (1979);55:157–162 (1979);55:267–272 (1979).

    Google Scholar 

  59. M. Sato, T. Miwa, and M. Jimbo, Holonomic Quantum Fields, I.Publ. RIMS, Kyoto Univ. 14:223–267 (1978).

    Google Scholar 

  60. M. Sato, T. Miwa, and M. Jimbo, Holonomic Quantum Fields, II,Publ. RIMS, Kyoto Univ. 15:201–278 (1979).

    Google Scholar 

  61. M. Sato, T. Miwa, and M. Jimbo, Holonomic Quantum Fields, III,Publ. RIMS, Kyoto Univ. 15:577–629 (1979).

    Google Scholar 

  62. M. Sato, T. Miwa, and M. Jimbo, Holonomic Quantum Fields, IV,Publ. RIMS, Kyoto Univ. 15:871–972 (1979).

    Google Scholar 

  63. M. Sato, T. Miwa, and M. Jimbo, Holonomic Quantum Fields, V,Publ. RIMS Kyoto Univ. 16:531–584 (1980).

    Google Scholar 

  64. L. Schlesinger, Uber eine klasse von Differentialsystemen beliebiger Ordnung mit festen knitschen Punkten,J. Reine Angew. Math. 141:96–145 (1912).

    Google Scholar 

  65. T. D. Schultz, D. C. Mattis, and E. H. Lieb, Two-Dimensional Ising Model as a Soluble Problem of Many Fermions,Rev. Mod. Phys. 36:856–871 (1964).

    Google Scholar 

  66. D. Shale and F. Stinespring, Spinor Representations of Infinite Orthogonal Groups,J. Math. Mech. 14:315–322 (1965).

    Google Scholar 

  67. B. Simon, Notes on Infinite Determinants of Hilbert Space Operators,Adv. Math. 24:244–273 (1977).

    Google Scholar 

  68. C. A. Tracy and B. M. McCoy, Neutron Scattering and the Correlation Functions of the Ising Model nearT c,Phys. Rev. Lett. 31:1500–1504 (1973).

    Google Scholar 

  69. C. A. Tracy and B. M. McCoy, Examination of the Phenomenological Scaling Functions for Critical Scattering,Phys. Rev. B 12:368–387 (1975).

    Google Scholar 

  70. C. A. Tracy, Painlevé Transcendents and Scaling Functions of the Two-Dimensional Ising Model, in “Nonlinear Equations in Physics and Mathematics” (A. O. Barut, ed.), pp. 221–237 (Reidel, Dordrecht, Holland, 1978).

    Google Scholar 

  71. H. Widom, Asymptotic Behavior of Block Toeplitz Matrices and Determinants, II,Adv. Math. 21:1–29 (1976).

    Google Scholar 

  72. T. T. Wu, Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model, I,Phys. Rev. 149:380–401 (1966).

    Google Scholar 

  73. T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Spin-Spin Correlation Functions for the Two-Dimensional Ising Model: Exact Theory in the Scaling Region,Phys. Rev. B 13:316–374 (1976).

    Google Scholar 

References

  1. C. T. Thompson,Contemp. Phys. 19:203–274 (1978). A very brief clear description of basic ideas.

    Google Scholar 

  2. Shang-keng Ma,Modern Theory of Critical Phenomena (W. A. Benjamin, New York, 1976).

    Google Scholar 

  3. C. Domb and M. S. Green,Phase Transitions and Critical Phenomena (in five volumes) (Academic Press, New York). Series started in 1972 and contains references to all the older physics literature.

    Google Scholar 

  4. P. Collet and Jean-Pierre Eckmann,A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics, Springer-Verlag Lecture Notes in Physics #74 (Springer, Berlin, 1978). The only completely solved, nontrivial example.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lanford, O.E., Lundgren, T., Eckmann, J.P. et al. Abstracts of the workshop on statistical mechanics, dynamical systems, and turbulence. J Stat Phys 32, 169–202 (1983). https://doi.org/10.1007/BF01009428

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01009428

Keywords

Navigation