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M393C (52395), Statistical Mechanics, Spring 1998
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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 (?)

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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)

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[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

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[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", ... (?)

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[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

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