============================================================================= M393C (52395), Statistical Mechanics, Spring 1998 ============================================================================= SYLLABUS (approx) (N/5 indicates fraction adapted from references) Thermodynamic Formalism thermodynamic (TD) variables and potentials 1st and 2nd law of TD convexity, Legendre transform, theorem of Fenchel references: (3/5) introduction by A. Wightman in [Israel], generic books on statistical mechanics Statistical Mechanics Formalism prepared ensembles, probabilities, observables information and entropy classical ensembles, partition functions, etc equivalence of ensembles quantum ensembles references: (2/5) [Ruelle 1], generic books on statistical mechanics Some Simple Models classical ideal gas lattice gases and spin systems Ising model in d=1 dimensions Curie-Weiss mean field model quantum ideal gases Bose gas - grand canonical ensemble the Gaussian model references: (2/5) [Thompson], [Huang], generic books on statistical mechanics The Thermodynamic Limit van Hove potentials TD limit of free energy and pressure equivalence of 2 ensembles general lattice spin systems TD limit of pressure list of models covered references: (4/5) [Thompson], [Ruelle 1], [Israel] Correlations correlations and connected correlations example: the Gaussian model ferromagnetic interactions GKS correlation inequalities TD limit of correlations references: (3/5) [Glimm, Jaffe], generic books on statistical mechanics High Temperature Expansion HT expansion for for continuous models van Hove potentials HT expansion for lattice spin systems finite-range models references: (4/5) D. Brydges in [OS1] Low Temperatures: The Peierls Argument ferromagnetic Ising model antiferromagnetic Ising model references: (5/5) J. Ginibre in [MD], many other books Invariant States Gibbs states, equilibrium states DLR equation tangent to the pressure variational equations pure states, clustering references: (5/5) [Sinai], [Israel], [Ruelle 1,2] Phase Transitions phenomenology symmetry breaking, long range order domain walls, spin waves references: (2/5) [Sinai], [Huang], generic books on statistical mechanics Discrete Spins in d=1 Dimensions absence of phase transitions references: (5/5) [Ruelle 1] Continuous Symmetries absence of long range order (LRO) in d<3 X-Y model infrared bounds, LRO in d>2 references: (4/5) [Mermin], [Grosse], (references in) [FFS] The Lee-Yang Theorem one-component ferromagnets references: (3/5) [LS], [Ruelle 1] Critical Phenomena phenomenology fluctuation-dissipation relations critical indices, universality, etc scaling hypothesis and its consequences references: (3/5) [FFS], [Huang], generic books Renormalization general formalism RG transformations, fixed points, etc the ultraviolet (UV) fixed point, spectrum the IR fixed point, expected results references: (0/5) [FFS], generic books on renormalization The Hierarchical Model reduction of the RG transformation the UV fixed point, spectrum the IR fixed point, spectrum references: (1/5) [KW1], [KW2] Disordered Systems, Neural Networks ? references: J. Froehlich in [OS2], [Petritis], ... Models Realated to Tilings (?) Models Related to Number Theory (?) Non-Equilibrum Statistical Mechanics (?) ============================================================================= SOME REFERENCES [Ruelle 1] D. Ruelle, "Statistical Mechanics. Rigorous Results", Mathematical Physics Monograph Series, W.A. Benjamin, 1969. 530.132 R837S [Thompson] C.J. Thompson, "Mathematical Statistical Mechanics", MacMillan, 1972. Princeton University Press, 1972. QC20 T47 [Grosse] H. Grosse, "Models in Statistical Physics and Quantum Field Theory", Trieste Notes in Physics, Springer Verlag, 1988. QC174.8 G76 [Israel] R.B. Israel, "Convexity in the Theory of Lattice Gases", Princeton University Press, 1979. QC174.85 L38 I87 [Simon] B. Simon, "The Statistical mechanics of Lattice Gases, Vol 1", Princeton University Press, 1993. QC174.85 L38 S6 [Sinai] Ya.G. Sinai, "Theory of Phase Transitions: Rigorous Results", International Series in Natural Philosophy, Vol. 108, Pergamon Press, 1982. QC175.16 P5 S615 [FFS] R. Fernandez, J. Froehlich, A.D. Sokal, "Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory", Springer Verlag, 1992. QC174.45 F46 [Glimm, Jaffe] J. Glimm, A. Jaffe, "Quantum Physics: A Functional Integral Point of View", Springer Verlag, 1981 (2nd edition 1987). QC 174.45 G49 [Ruelle 2] D. Ruelle, "Thermodynamic Formalism", Encyclopedia of Mathematics and its Applications, Vol. 5, Addison-Wesley, 1987. QC174.86 C6 R83 [LS] E.H. Lieb, A.D. Sokal, "A General Lee-Yang Theorem for One-Component and Multicomponent Ferromagnets", Commun. Math. Phys. 80, 153 (1981) [Mermin] N.D. Mermin, "Absence of Ordering in Certain Classical Systems", Journal of Mathematical Physics 8, 1061 (1967) [KW1] H. Koch, P. Wittwer, "On the Renormalization Group Transformation for Scalar Hierarchical Models", mp_arc 91-1, Commun Math Phys 138, 537 (1991) [KW2] H. Koch, P. Wittwer, "A Nontrivial Renormalization Fixed Point for the Dyson-Baker Hierarchical Model", mp_arc 93-230, Commun. Math. Phys. 164, 627 (1994) ------------------------------------------------------------------------------ [Huang] K. Huang, "Statistical Mechanics", John Wiley & Sons, 1963. QC174.8 H83 [Chandler] D. Chandler, "Introduction to Modern Statistical Mechanics", Oxford University Press, 1987. QC174.8 C47 [Stanley] H.E. Stanley, "Introduction to Phase Transitions and Critical Phenomena", Oxford University Press, 1971. 536.401 ST25I [Ma] S.-K. Ma, "Modern Theory of Critical Phenomena" Frontiers in Physics, Lecture Notes Series, Benjamin/Cummings, 1976. QC173.4 C74 M3 [Balian] R. Balian, "From Microphysics to Macrophysics, Methods and Applications of Statistical Physics", Vol 1-2, Springer Verlag, 1991. QC174.8 B35 [Pathria] R.K. Pathria, "Statistical Mechanics", International Series in natural Philosophy, Volume 45, Pergamon Press, 1972. QC175 P35 [Cardy] J.L. Cardy, "Scaling and renormalization in statistical physics", Cambridge University Press (1996). [CFP] R.J. Creswick, H.A. Farach, C.P. Poole, "Introduction to Renormalization Group Methods", John Wiley & Sons, 1992. QC20.7 R43 C74 [de Gennes] P.G. de Gennes, "Scaling Concepts in Polymer Physics", Cornell University Press, 1979. QD381 G45 ------------------------------------------------------------------------------ [F] J. Froehlich (ed), "Scaling and Self-Similarity in Physics, Renormalization in Statistical Mechanics and Dynamics", Birkhaeuser, 1983. QC20.7 R43 S33 [DG] C. Domb, M.S. Green (eds), "Phase Transitions and Critical Phenomena" Vol. 1-8, Academic Press, 1972-1984. QC175 D59 [DL] C. Domb, J.L. Lebowitz (eds) "Phase Transitions and Critical Phenomena" Vol. 9- , Academic Press, 1972-1984. QC175 D59 [L] A. Lenard (ed), "Statistical Mecanics and Mathematical Problems. Battelle Seattle 1971 Rencontres", Lecture Notes in Physics, Springer Verlag, 1973. QC174.82 B37 [OS1] K. Osterwalder, R. Stora (eds), "Critical Phenomena, Random Systems, Gauge Theories (Les Houches 1984)" Part 1, North-Holland, 1986. QC173.4 C47 P47 PT.1 [OS2] K. Osterwalder, R. Stora (eds), "Critical Phenomena, Random Systems, Gauge Theories (Les Houches 1984)" Patr 2, North-Holland, 1986. QC173.4 C47 P47 PT.2 [ZS] J.B. Zuber, R. Stora (eds), "Recent Advances in Field Theory and Statistical Mechanics (Les Houches 1982), North-Holland, 1986. QC174.45 A1 D48 [LM] M. Levy, P. Mitter (eds) "New developments in Quantum Field Theory and Statistical mechanics (Cargese 1976)", Plenum, 1977. QC174.45 A1 N47 [MD] L. Michel, D. Ruelle (eds) "Systemes a un Nombre Infini de Degrees de Liberte", ... (?) ------------------------------------------------------------------------------ [Petritis 1] D. Petritis, "Equilibrium Statistical Mechanics of Frustrated Spin Glasses: A Survey of Mathmatical Results", mp_arc 94-363 [Petritis 2] D. Petritis, "Thermodynamic Formalism of Neural Computing", mp_arc 95-221 [Baxter] R.B. Baxter, "Exactly Solved Models in Satistical Mechanics", Academic Press, 1982. QC 174.8 B39 [Baker] G.A. Baker, "Quantitative Theory of Critical Phenomena", Academic Press, 1990. QC173.4 C74 B35 [Katz] Amnon Katz, "Principles of Statistical mechanics; the Information Theory Approach", 1967. QC175 K33 [Penrose] O. Penrose, "Foundations of Statistical Mechanics", Pergamon Press, 1970. QC175 P45 (pma library) [Hobson] A. Hobson, "Concepts in Statistical Mechanics", Gordon and Breach, 1971. QC174.8 H6 ==============================================================================