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Introduction to Mathematical Statistical Physics
R. A. Minlos, Institute for Problems of Information Transmission, Moscow, Russia
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University Lecture Series
2000; 103 pp; softcover
Volume: 19
ISBN-10: 0-8218-1337-4
ISBN-13: 978-0-8218-1337-9
List Price: US$27
Member Price: US$21.60
Order Code: ULECT/19
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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 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.

Readership

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

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

Table of Contents

The subject and the main notions of equilibrium statistical physics
  • Typical systems of statistical physics (Phase space, dynamics, microcanonical measure)
  • Statistical ensembles (Microcanonical and canonical ensembles, equivalence of ensembles)
  • Statistical ensembles-Continuation (the system of indistinguishable particles and the grand canonical ensemble)
  • The thermodynamic limit and the limit Gibbs distribution
The existence and some ergodic properties of limiting Gibbs distributions for nonsingular values of parameters
  • The correlation functions and the correlation equations
  • Existence of the limit correlation function (for large positive \(\mu\) or small \(\beta\))
  • Decrease of correlations for the limit Gibbs distribution and some corollaries (Representativity of mean values, distribution of fluctuations, ergodicity)
  • Thermodynamic functions
Phase transitions
  • Gibbs distributions with boundary configurations
  • An example of nonuniqueness of Gibbs distributions
  • Phase transitions in more complicated models
  • The ensemble of contours (Pirogov-Sinai theory)
  • Deviation: The ensemble of geometric configurations of contours
  • The Pirogov-Sinai equations (Completion of the proof of the main theorem)
  • Epilogue. What is next?
  • Bibliography
  • Index
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