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ChE/Ch 164 Introduction to Statistical Thermodynamics

The principal texts are D.A. McQuarrie, Statistical Mechanics, Harper & Row, 1976, and D. Chandler, Introduction to Modern Statistical Mechanics, Oxford, New York, 1987. Portions of the material to be discussed are treated in these texts, in the pages cited below after "Ref. I" and "Ref. II", respectively. McQuarrie's text covers many topics, apart from those related to the Ising model and some very modern results. Chandler's excellent text provides much of this material, and so is a very useful supplement. Several other books, referred to and used particularly in latter half of the course, are given at the end of this syllabus. Nonequilibrium statistical mechanics is treated in Ch 165. Much of the relevant part for equilibrium statistical mechanics is contained in McQuarrie’s Statistical Thermodynamics, which is the first part of his Statistical Mechanics.

1. Introduction. Aims of statistical mechanics, various ensembles and thermodynamics, relevant mathematics (probability distributions, Stirling's approximation, binomial distribution, Lagrange multipliers, relation between partial derivatives), brief thermodynamics review. Ref. I, 13-25; Ref. II, Chaps. 1 and 3; F. Chap. 1.

2. Canonical ensemble. Distribution function (derivation), evaluation of Lagrange multipliers using second law, expressions for thermodynamic functions in terms of canonical partition function, third law and entropy, remark on microcanonical ensemble, physical meaning of partition function. Ref. I, 35-47; Ref. II, Chap. 3; F, Chap. 1.

3. Grand canonical ensemble. Distribution function (derivation), evaluation of Lagrange multipliers, expressions for thermodynamic functions in terms of grand canonical partition function, isothermal-isobaric ensemble. Ref. I, 51-57; F, Chap. 1.

4. Various statistics. Indistinguishability of particles and Fermi-Dirac, Einstein-Bose and Boltzmann distribution functions for noninteracting particles obtained via grand canonical partition function, thermodynamic properties. Ref. I, 68-76; R, Chap. 9D.

5. Noninteracting molecules. Relation of canonical partition function to molecular partition function and thermodynamic properties, monatomic, diatomic and polyatomic gases, symmetry of wave functions, ortho-para hydrogen, chemical equilibria, phase equilibria. Ref. I, 81-89, 91-109, 129-140, 142-157; Ref. II, Chap. 4; F, Chap. 2.

6. Independent modes or particles. Fermi-Dirac system (electron gas), Einstein crystal, Debye crystal, specific heat, free electron model of metal, photon gas (black body radiation) (only briefly mentioned). Ref. I, 177-182, 194-206; Ref. II, Chap. 4; F. Chap. 2.

7. Classical statistical mechanics. Phase space, classical partition function, Ref. I, 113-123, 185-189; F, Chap. 3.

8. Fluctuations in energy (in canonical ensemble), in magnetization (canonical ensemble), in number of particles (grand canonical ensemble and, more generally, in other properties ("thermodynamically conjugate variables"), smallness of fluctuations, and equivalence of different ensembles. Ref. I, 57-64; Ref. II, Chap. 3; F, Chap. 1.

9. Interacting particles. Lattice statistics, Ising model for spin systems, partition function for one-dimensional system, broken symmetry, mean field theory, Curie-Weiss law, discontinuities at the critical point Bragg-Williams approach to "mean field" theory, adiabatic demagnetization, free energy. Ref. II, Chap. 5; Re, part of Chap. 9F; Hi, part of Chap. 7; Z, part of Chap. 5; Hu, Chap. 16; P, part of Chap. 11; PB, part of Chap. 3.

10. Applications to lattice-like or mean field problems. Regular solutions, order-disorder in binary alloys, lattice gas, van der Waals equation, polymer-solvent solutions (Flory-Huggins model), liquid crystals. See refs. to Sect. 11. Hi II, part of Chap. 21.

11. More on Ising model. Correlation length, transfer matrix method, renormalization group theory and application to Ising model. TKS, Chaps. 4.4, 4.5; Ma, part of Chap. 17; P, part of Chap. 12 and 13. H. J. Maris and L. J. Kadanoff, Am. J. Phys. 46, 652 (1978); TKS, 171-176. (Application of renormalization group theory to polymers is given in G, Chap. 11).

12. Interacting particles (imperfect gases). Thermodynamic perturbation theory, cumulant expansion, Gibbs-Bogoliubov bound. Ref. I, 302-306; Ref. II, Chap. 5.5; R, pp. 387-389; AT, pp. 37-41; Z, 220-223; F, Chap. 9.

Other texts referred to above are H.L. Friedman, A Course in Statistical Mechanics (Prentice-Hall, 1985) (F); M. Toda, R. Kubo and N. Saito, Statistical Physics.I. (Springer-Verlag, 1980) (TKS); L. E. Reichl, A Modern Course in Statistical Physics (U. Texas Press, 1980) (R); R. Abe and Y. Takahashi, Statistical Mechanics (U. Tokyo Press, 1975) (AT); T. L. Hill, Statistical Mechanics (McGraw-Hill,1956) (Hi); J. M. Ziman, Models of Disorder (Cambridge U.P., 1979) (Z); K. Huang, Statistical Mechanics (Wiley, 1965) (Hu); M. Plischke and B. Bergersen, Equilibrium Statistical Mechanics (Prentice-Hall, 1989) (PB); R. K. Pathria, Statistical Mechanics, 2nd ed., (Butterworth, 1996) (P); T. L. Hill, Introduction to Statistical Thermodynamics (Addison-Wesley, 1960) (Hi II). Other useful texts include D.A. McQuarrie, Introduction to Statistical Thermodynamics (Harper, 1973); N. Davidson, Statistical Mechanics (McGraw-Hill, 1962); G. S. Rushbrooke, Introduction to Statistical Mechanics (Oxford U.P., 1951); H. Eyring, D. Henderson, B. J. Stoner, E.M. Eyring, Statistical Mechanics and Dynamics (Wiley, 1982), 2nd. ed.; P. de Gennes, Scaling Concepts in Polymer Physics (Cornell U.P., 1979).

A topic not covered in the above syllabus is cluster expansions, used, for example, in treatments of non-ideal gases, electrolyte solutions, etc. Treatments are given Ref. I, Chap. 12, and in Friedman, Chaps. 6, 7, and in other books. Monatomic liquids are treated in Ref. I, Chap. 13.

Texts on Reserve in Millikan, First Floor
McQuarrie, Statistical Mechanics, overnight.
McQuarrie, Statistical Thermodynamics, overnight.
Davidson, Statistical Mechanics, overnight.
Friedman, A Course in Statistical Mechanics, overnight.
Eyring, Statistical Mechanics & Dynamics, 2nd ed., overnight.
Chandler, Introduction to Modern Statistical Mechanics, overnight.

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