# Introduction to Mathematical Statistical Physics

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*R. A. Minlos*

This book presents a mathematically rigorous approach to the
main ideas and phenomena of statistical physics. The introduction
addresses the physical motivation, focussing on the basic concept of
modern statistical physics, that is the notion of Gibbsian random
fields.

Properties of Gibbsian fields are analyzed in two ranges of
physical parameters: “regular” (corresponding to
high-temperature and low-density regimes) where no phase transition is
exhibited, and “singular” (low temperature regimes) where
such transitions occur.

Next, a detailed approach to the analysis of the phenomena of phase
transitions of the first kind, the Pirogov-Sinai theory, is
presented. The author discusses this theory in a general way and
illustrates it with the example of a lattice gas with three types of
particles. The conclusion gives a brief review of recent developments
arising from this theory.

The volume is written for the beginner, yet advanced students will
benefit from it as well. The book will serve nicely as a supplementary
textbook for course study. The prerequisites are an elementary
knowledge of mechanics, probability theory and functional analysis.

#### Reviews & Endorsements

This book presents a mathematically rigorous approach to the main ideas and phenomena of statistical physics … The book will serve nicely as a supplementary textbook for course study.

-- Zentralblatt MATH

The author presents a concise introduction to the subject … This new textbook will surely find its place among existing monographs.

-- European Mathematical Society Newsletter

This work can serve as a clear and concise introduction for researchers interested in the mathematical theory of classical statisical mechanics.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to Mathematical Statistical Physics

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preliminaries vii8 free
- Part 1. The Subject and the Main Notions of Equilibrium Statistical Physics 110 free
- Lecture 1. Typical Systems of Statistical Physics (Phase Space, Dynamics, Microcanonical Measure) 312
- Lecture 2. Statistical Ensembles (Microcanonical and Canonical Ensembles, Equivalence of Ensembles) 918
- Lecture 3. Statistical Ensembles—Continuation (the System of Indistinguish-able Particles and the Grand Canonical Ensemble) 1524
- Lecture 4. The Thermodynamic Limit and the Limit Gibbs Distribution 2130

- Part 2. The Existence and some Ergodic Properties of Limiting Gibbs Distributions for Nonsingular Values of Parameters 2534
- Lecture 5. The Correlation Functions and the Correlation Equations 2736
- Lecture 6. Existence of the Limit Correlation Function (for Large Positive / μ or Small β) 3544
- Lecture 7. Decrease of Correlations for the Limit Gibbs Distribution and Some Corollaries (Representativity of Mean Values, Distribution of Fluctuations, Ergodicity) 4150
- Lecture 8. Thermodynamic Functions 4756

- Part 3. Phase Transitions 5362
- Lecture 9. Gibbs Distributions with Boundary Configurations 5564
- Lecture 10. An Example of Nonuniqueness of Gibbs Distributions 5968
- Lecture 11. Phase Transitions in More Complicated Models 6776
- Lecture 12. The Ensemble of Contours (Pirogov-Sinai Theory) 7382
- Lecture 13. Deviation: the Ensemble of Geometric Configurations of Contours 7988
- Lecture 14. The Pirogov-Sinai Equations (Completion of the Proof of the Main Theorem) 8796
- Lecture 15. Epilogue. What is Next? 91100

- Bibliography 99108
- Index 101110
- Back Cover Back Cover1114