Mathematical Foundations of Quantum Statistical Mechanics: Continuous Systems

by D.Y. Petrina

Overview

This monograph is devoted to quantum statistical mechanics. It can be regarded as a continuation of the book "Mathematical Foundations of Classical Statistical Mechanics. Continuous Systems" (Gordon & Breach SP, 1989) written together with my colleagues V. I. Gerasimenko and P. V. Malyshev. Taken together, these books give a complete presentation of the statistical mechanics of continuous systems, both quantum and classical, from the common point of view. Both books have similar contents. They deal with the investigation of states of in­ finite systems, which are described by infinite sequences of statistical operators (reduced density matrices) or Green's functions in the quantum case and by infinite sequences of distribution functions in the classical case. The equations of state and their solutions are the main object of investigation in these books. For infinite systems, the solutions of the equations of state are constructed by using the thermodynamic limit procedure, accord­ ing to which we first find a solution for a system of finitely many particles and then let the number of particles and the volume of a region tend to infinity keeping the density of particles constant. However, the style of presentation in these books is quite different.

Product Details

ISBN-13:	9789401040839
Publisher:	Springer Netherlands
Publication date:	10/14/2012
Series:	Mathematical Physics Studies , #17
Edition description:	Softcover reprint of the original 1st ed. 1995
Pages:	445
Product dimensions:	6.30(w) x 9.45(h) x 0.04(d)

Table of Contents

1. Evolution of States of Quantum Systems of Finitely Many Particles.- 1. Principal Concepts of Quantum Mechanics.- 2. Evolution of States of Quantum Systems with Arbitrarily Many Particles.- 3. Evolution of States in the Heisenberg Representation and in the Interaction Representation.- Mathematical Supplement I.- References.- 2. Evolution of States of Infinite Quantum Systems.- 4. Bogolyubov Equations for Statistical Operators.- 5. Solution of the Bogolyubov Equations.- 6. Gibbs Distributions.- Mathematical Supplement II.- Mathematical Supplement III.- 3. Thermodynamic Limit.- 7. Thermodynamic Limit for Statistical Operators.- 8. Statistical Operators in the Case of Quantum Statistics.- 9. Bogolyubov’s Principle of Weakening of Correlations.- Mathematical Supplement IV.- References.- 4. Mathematical Problems in the Theory of Superconductivity.- 10. Fröhlich Model.- 11. Bogolyubov’s Compensation Principle for “Dangerous” Diagrams. Compensation Equations.- 12. Bardeen-Cooper-Schrieffer (BCS) Hamiltonian.- 13. Microscopic Theory of Superfluidity.- Mathematical Supplement V.- 5. Green’s Functions.- 14. Green’s Functions. Equations for Green’s Functions.- 15. Investigation of the Equations for Green’s Functions in the Theory of Superconductivity and Superfluidity.- 16. Green’s Functions in the Thermodynamic Limit.- 6. Exactly Solvable Models.- 17. Description of the Hamiltonians of Model Systems.- 18. Functional Spaces of Translation-Invariant Functions.- 19. Model Hamiltonians in the Spaces of Translation Invariant Functions.- 20. Model BCS Hamiltonian in the Space hT. Equivalence of General and Model Hamiltonians in the Space of Pairs.- 21. Equations for Green’s Functions and Their Solutions.- Mathematical Supplement VI.- 7. Quasiaverages. Theorem onSingularities of Green’s Functions of 1/q2 -Type.- 22. Quasiaverages.- 23. Green’s Functions and Their Spectral Representations.- 24. Theorem on Singularities of Green’s Functions of 1/q2 -Type.

Editorial Reviews

A monograph devoted to the study of equilibrium and nonequilibrium states of infinite continuous systems in quantum statistical mechanics. The states of these systems are described by infinite sequences of statistical operators (reduced density matrices) or Green's functions which satisfy the infinite hierarchy of integro-differential equations. The author focuses on the investigation of these equations and the construction of their solutions, and studies in detail model systems in the theories of superconductivity and superfluidity and other exactly solvable models. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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