eBook ISBN:  9781470421663 
Product Code:  ULECT/19.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
eBook ISBN:  9781470421663 
Product Code:  ULECT/19.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 

Book DetailsUniversity Lecture SeriesVolume: 19; 2000; 103 ppMSC: Primary 82
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 hightemperature and lowdensity 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 PirogovSinai 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.
ReadershipGraduate students and research mathematicians interested in statistical mechanics and the structure of matter; physicists, chemists, and computer scientists interested in networks.

Table of Contents

Part 1. The subject and the main notions of equilibrium statistical physics

Lecture 1. Typical systems of statistical physics (phase space, dynamics, microcanonical measure)

Lecture 2. Statistical ensembles (microcanonical and canonical ensembles, equivalence of ensembles)

Lecture 3. Statistical ensembles—continuation (the system of indistinguishable particles and the grand canonical ensemble)

Lecture 4. The thermodynamic limit and the limit Gibbs distribution

Part 2. The existence and some ergodic properties of limiting Gibbs distributions for nonsingular values of parameters

Lecture 5. The correlation functions and the correlation equations

Lecture 6. Existence of the limit correlation function (for large positive $\mu $ or small $\beta $)

Lecture 7. Decrease of correlations for the limit Gibbs distribution and some corollaries (representativity of mean values, distribution of fluctuations, ergodicity)

Lecture 8. Thermodynamic functions

Part 3. Phase transitions

Lecture 9. Gibbs distributions with boundary configurations

Lecture 10. An example of nonuniqueness of Gibbs distributions

Lecture 11. Phase transitions in more complicated models

Lecture 12. The ensemble of contours (PirogovSinai theory)

Lecture 13. Deviation: The ensemble of geometric configurations of contours

Lecture 14. The PirogovSinai equations (completion of the proof of the main theorem)

Lecture 15. Epilogue. What is next?


Additional Material

Reviews

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


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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 hightemperature and lowdensity 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 PirogovSinai 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.
Graduate students and research mathematicians interested in statistical mechanics and the structure of matter; physicists, chemists, and computer scientists interested in networks.

Part 1. The subject and the main notions of equilibrium statistical physics

Lecture 1. Typical systems of statistical physics (phase space, dynamics, microcanonical measure)

Lecture 2. Statistical ensembles (microcanonical and canonical ensembles, equivalence of ensembles)

Lecture 3. Statistical ensembles—continuation (the system of indistinguishable particles and the grand canonical ensemble)

Lecture 4. The thermodynamic limit and the limit Gibbs distribution

Part 2. The existence and some ergodic properties of limiting Gibbs distributions for nonsingular values of parameters

Lecture 5. The correlation functions and the correlation equations

Lecture 6. Existence of the limit correlation function (for large positive $\mu $ or small $\beta $)

Lecture 7. Decrease of correlations for the limit Gibbs distribution and some corollaries (representativity of mean values, distribution of fluctuations, ergodicity)

Lecture 8. Thermodynamic functions

Part 3. Phase transitions

Lecture 9. Gibbs distributions with boundary configurations

Lecture 10. An example of nonuniqueness of Gibbs distributions

Lecture 11. Phase transitions in more complicated models

Lecture 12. The ensemble of contours (PirogovSinai theory)

Lecture 13. Deviation: The ensemble of geometric configurations of contours

Lecture 14. The PirogovSinai equations (completion of the proof of the main theorem)

Lecture 15. Epilogue. What is next?

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