Advanced Statistical Mechanics

Statistical Mechanics is the study of systems where the number of interacting particles becomes infinite. In the last fifty years tremendous advances have been made which have required the invention of entirely new fields of mathematics such as quantum groups and affine Lie algebras. They have engendered remarkable discoveries concerning non-linear differential equations and algebraic geometry, and have produced profound insights in both condensed matter physics and quantum field theory. Unfortunately, none of these advances are taught in graduate courses in statistical mechanics. This book is an attempt to correct this problem. It begins with theorems on the existence (and lack) of order for crystals and magnets and with the theory of critical phenomena, and continues by presenting the methods and results of fifty years of analytic and computer computations of phase transitions. It concludes with an extensive presentation of four of the most important of exactly solved problems: the Ising, 8 vertex, hard hexagon and chiral Potts models.

1. Basic Principles. 2. Reductionism, Phenomena and Models. 3. Stability, Existence and Uniqueness. 4. Theorems on Order. 5. Critical Phenomena and Scaling Theory. 6. Mayer Virial Expansions and Groenevelt's Theorems. 7. Ree-Hoover Virial Expansion and Hard Spheres. 8. High Density Expansions. 9. High Temperature Expansions for Magnets at H=0. 10. The Ising Model in Two Dimensions, Summary of Results. 11. The Pfaffian Solution of the Ising Model. 12. Ising Model Spontaneous Magnetization, Form Factors and Susceptibility. 13. The Star-Triangle (Yang-Baxter) Equation. 14. The Eight Vertex and XYZ models. 15. The RSOS and the Chiral Potts models. 16. Conclusion.

Barry M. McCoy is currently Distinguished Professor of Physics at C.N.Yang Institute for Theoretical Physics, at the State University of New York. He took his PhD at Harvard University in 1967, and has since had various positions, including the Editorial Board of the Ramanujan Journal, the Editorial Board of Journal of Physics A, Resident at the Bellagio Conference and Study Center of the Rockefeller Foundation, and Miller Professor at University of California at Berkeley. In 1999 he won the Dannie Heineman Prize for Mathematical Physics.