Mathematical Problems of Statistical Mechanics and Dyanamics: A Collection of Surveys (Paperback, Softcover Repri)

'Phase Diagrams for Continuous-Spin Models: An Extension of the Pirogov-Sinai Theory.- 1. Formulation of the Main Result.- 1.1. Configurations.- 1.2. States.- 1.3. Hamiltonians.- 1.4. Gibbs States.- 1.5. Assumptions of the Main Theorem.- 1.6. The Main Theorem.- 1.7. Strategy of the Proof.- 2. Preliminaries.- 2.1. Abstract Polymer Models and Cluster Expansions.- 2.2. Gaussian Gibbsian Fields: The Correlation Decay.- 2.3. Estimates of Semi-Invariants: Cluster Expansions of Perturbed Gaussian Fields.- 3. The Main Lemma.- 3.1. The Geometry of Configurations.- 3.2. Reduction to the Main Lemma.- 4. Proof of the Main Lemma.- 4.1. Contours.- 4.2. Reduction to a Contour Model.- 4.3. Estimates of the Main Term G ? (?). Decomposition of the Contour Energy.- 4.4. Boundary Terms of Partition Functions of Contour Models.- 4.5. Conclusion of the Proof of the Main Lemma.- References.- Space-Time Entropy of Infinite Classical Systems.- 1. Introduction.- 2. Statistical Estimates of the Gibbs Distribution.- 3. Reduction to Partial Flows.- 4. Estimate of Space-Time Entropy.- References.- Spectrum Analysis and Scattering Theory for a Three-Particle Cluster Operator.- 1. Introduction. A General Definition of the Cluster Operator.- 2. Three-Particle Cluster Operators.- 3. Equations for the Resolvent of a Self-Adjoint Three-Particle Cluster Operator.- 4. Study of Equations (3.4)-(3.6).- 5. The Main Result.- 6. Proof of Theorem 5.11 (Scattering Theory).- References.- Stochastic Attractors and their Small Perturbations.- 1. Introduction.- 2. Dynamical Systems with Stochastic Attractors.- 3. Stochastic Perturbations (Regular Case).- 4. The Law of Exponential Decay and Small Stochastic Perturbations.- 5. Stochastic Perturbations (Singular Case).- 6. Small Quasi-Stochastic Perturbations.- 7. Ergodic Properties of Dynamical System Discretizations.- References.- Statistical Properties of Smooth Smale Horseshoes.- 1. General Background.- 1.1. Structures in the Product ? = $$ \mathbb{Z} = \mathbb{X}\,x\mathbb{Y} $$.- 1.2. Uniformly Hyperbolic Transformations of ? = $$ \mathbb{Z} = \mathbb{X}\,x\mathbb{Y} $$.- 1.3. A Sufficient Condition for Uniform Hyperbolicity in ? = $$ \mathbb{Z} = \mathbb{X}\,x\mathbb{Y} $$.- 1.4. Leaves and Rectangles.- 1.5. The Smale Horseshoe.- 2. Expanding and Contracting Fibrations of a Smale Horseshoe.- 2.1. The Smoothness of Expanding and Contracting Fibres.- 2.2. Expanding and Contracting Fibrations are Holderian.- 2.3. The Local Smoothness of Expanding and Contracting Fibrations.- 2.4. The Holder Property of the Canonical Isomorphism Defined by a Fibration.- 3. Smooth Invariant Conditional Probability Distributions on Fibrations.- 3.1. The Evolution of Densities of Conditional Probability Distributions on Fibres Induced by ?.- 3.2. The Existence of a T-Invariant Smooth Family of Probability Distributions on Fibres at D(?).- 3.3. Comparison of Densities of Conditional Probability Distributions on Different Fibres.- 3.4. The Dependence of T-Invariant Conditional Densities on the Number of the Fibre.- 4. Smooth Non-Singular Probability Distributions on a Smale Horseshoe.- 4.1. Defining Measures on Measurable Rectangles in Terms of Conditional Probability Distributions on Fibres.- 4.2. An Average Description of the Evolution of Measures from the Class G.- 4.3. The Construction of an Eigenmeasure for a Smale Horseshoe.- 5. A Natural Invariant Probability Distribution on the Hyperbolic Set of a Smale Horseshoe.- 5.1. The Sequence of Probability Distributions $$ {\hat \mu _{\left( m ight)}} $$.- 5.2. The Computation of the Asymptotics of $$ {\hat \mu _{\left( m ight)}} $$ via the Matrix Technique.- 5.3. The Weak Limit μ0{ - } of the Sequence of Measures μ(m){-}.- 6. Some Properties of the Constructed Limit Probability Distributions on a Smale Horseshoe.- 6.1. The T-Invariant Conditional Probability Distributions P{ - ?I} on Expanding F