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

Front Cover
R.L. Dobrushin
Springer Science & Business Media, Oct 31, 1986 - Science - 280 pages
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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.
 

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Contents

Phase Diagrams for ContinuousSpin Models An Extension of the PirogovSinai Theory
1
Formulation of the Main Result
4
13 HAMILTONIANS
5
15 ASSUMPTIONS OF THE MAIN THEOREM
6
16 THE MAIN THEOREM
10
17 STRATEGY OF THE PROOF
11
THE CORRELATION DECAY
17
CLUSTER EXPANSIONS OF PERTURBED GAUSSIAN FIELDS
30
Small QuasiStochastic Perturbations
180
Ergodic Properties of Dynamical System Discretizations
184
References
196
Statistical Properties of Smooth Smale Horseshoes
199
General Background
203
12 UNIFORMLY HYPERBOLIC TRANSFORMATIONS OF Z X x Y
204
13 A SUFFICIENT CONDITION FOR UNIFORM HYPERBOLICITY IN Z X x Y
206
14 LEAVES AND RECTANGLES
207

The Main Lemma
62
32 REDUCTION TO THE MAIN LEMMA
64
Proof of The Main Lemma
76
42 REDUCTION TO A CONTOUR MODEL
79
43 ESTIMATES OF THE MAIN TERM G₂𝛤 DECOMPOSITION OF THE CONTOUR ENERGY
88
44 BOUNDARY TERMS OF PARTITION FUNCTIONS OF CONTOUR MODELS
95
45 CONCLUSION OF THE PROOF OF THE MAIN LEMMA
106
References
122
SpaceTime Entropy of Infinite Classical Systems
125
Statistical Estimates of the Gibbs Distribution
126
Reduction to Partial Flows
129
Estimate of SpaceTime Entropy
132
References
136
Spectrum Analysis and Scattering Theory for a ThreeParticle Cluster Operator
139
ThreeParticle Cluster Operators
140
Equations for the Resolvent of a SelfAdjoint ThreeParticle Cluster Operator
143
Study of Equations 3436
146
The Main Result
155
Proof of Theorem 511 Scattering Theory
157
References
160
Stochastic Attractors and their Small Perturbations
161
Dynamical Systems with Stochastic Attractors
164
Stochastic Perturbations Regular Case
170
The Law of Exponential Decay and Small Stochastic Perturbations
176
Stochastic Perturbations Singular Case
179
15 THE SMALE HORSESHOE
209
Expanding and Contracting Fibrations of a Smale Horseshoe
211
22 EXPANDING AND CONTRACTING FIBRATIONS ARE HÖLDERIAN
213
24 THE HÖLDER PROPERTY OF THE CANONICAL ISOMORPHISM DEFINED BY A FIBRATION
215
Smooth Invariant Conditional Probability Distributions on Fibrations
216
32 THE EXISTENCE OF A TINVARIANT SMOOTH FAMILY OF PROBABILITY DISTRIBUTIONS ON FIBRES AT
217
33 COMPARISON OF DENSITIES OF CONDITIONAL PROBABILITY DISTRIBITIONS ON DIFFERENT FIBRES
218
34 THE DEPENDENCE OF TINVARIANT CONDITIONAL DENSITIES ON THE NUMBER OF THE FIBRE
219
Smooth NonSingular Probability Distributions on a Smale Horseshoe
220
42 AN AVERAGE DESCRIPTION OF THE EVOLUTION OF MEASURES FROM THE CLASS
222
43 THE CONSTRUCTION OF AN EIGENMEASURE FOR A SMALE HORSESHOE
224
A Natural Invariant Probability Distribution on the Hyperbolic Set of a Smale Horseshoe
226
52 THE COMPUTATION OF THE ASYMPTOTICS OF VIA THE MATRIX TECHNIQUE
228
53 THE WEAK LIMIT 𝜇₀ OF THE SEQUENCE OF MEASURES 𝜇ₘ
233
Some Properties of the Constructed Limit Probability Distributions on a Smale Horseshoe
235
62 WEAK BERNOULLI PARTITION FOR THE TINVARIANT MEASURE 𝜇₀
236
63 THE EIGENFUNCTION el OF A SMALE HORSESHOE
239
Evolution of Probability Distributions on a Smale Horseshoe
240
72 THE ASYMPTOTICS OF INTEGRALS
244
73 MAPPINGS WHICH POSSESS A SMALE HORSESHOE
250
Ergodic Properties of Positive Matrices with Bounded Ratio of Rows
253
References
256
Author Index
257
Subject Index
259
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Page 256 - On infinite processes leading to differentiability in the complement of a point, Differential and combinatorial topology (A symposium in honor of Marston Morse), Princeton University Press, Princeton, NJ, 1965, pp.

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