Mathematical Problems of Statistical Mechanics and Dyanamics: A Collection of Surveys

ISBN-13: 9789401085403 / Angielski / Miękka / 2011 / 280 str.

R. L. Dobrushin

Opis

Approach your problems from the It isn't that they can't see the solution. right end and begin with the answers. It is that they can't see the problem. Then one day, perhaps you will find the final question. G. K. Chesterton. The Scandal of Father Brown 'The point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Spis treści

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 ? = $$ \mathbb = \mathbb\,x\mathbb $$.- 1.2. Uniformly Hyperbolic Transformations of ? = $$ \mathbb = \mathbb\,x\mathbb $$.- 1.3. A Sufficient Condition for Uniform Hyperbolicity in ? = $$ \mathbb = \mathbb\,x\mathbb $$.- 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 Hölderian.- 2.3. The Local Smoothness of Expanding and Contracting Fibrations.- 2.4. The Hölder 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 $$ {\hat \mu _{\left( m \right)}} $$.- 5.2. The Computation of the Asymptotics of $$ {\hat \mu _{\left( m \right)}} $$ 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 Fibres of the Hyperbolic Set ?.- 6.2. Weak Bernoulli Partition for the T-Invariant Measure µ0.- 6.3. The Eigenfunction e(I) of a Smale Horseshoe.- 7. Evolution of Probability Distributions on a Smale Horseshoe.- 7.1. Asymptotic Inequalities for Measures and Integrals.- 7.2. The Asymptotics of Integrals.- 7.3. Mappings Which Possess a Smale Horseshoe.- Appendix:Ergodic Properties of Positive Matrices with Bounded Ratio of Rows.- References.- Author Index.