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Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics.
This important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniq...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore :
World Scientific Publishing Company,
2002.
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| Colección: | Series on partial differential equations and applications.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Ch. 1. Introduction. 1.1. Laplace integrals. 1.2. The problems in statistical mechanics. 1.3. Semi-classical analysis and transfer operators. 1.4. About the contents
- ch. 2. Witten Laplacians approach. 2.1. De Rham Complex. 2.2. Witten Complex. 2.3. Witten Laplacians. 2.4. Semi-classical considerations. 2.5. An alternative point of view: Dirichlet forms. 2.6. A nice formula for the covariance. 2.7. Notes
- ch. 3. Problems in statistical mechanics with discrete spins. 3.1. The Curie-Weiss model. 3.2. The 1-d Ising model. 3.3. The 2-d Ising model. 3.4. Notes
- ch. 4. Laplace integrals and transfer operators. 4.1. Introduction. 4.2. Classical Laplace method. 4.3. The method of transfer operators. 4.4. Elementary properties of operators with integral kernels. 4.5. Elementary properties of the transfer operator. 4.6. Operators with strictly positive kernel and application. 4.7. Thermodynamic limit. 4.8. Mean value. 4.9. Pair correlation. 4.10. 2-dimensional lattices. 4.11. Notes
- ch. 5. Semi-classical analysis for the transfer operators. 5.1. Introduction. 5.2. Explicit computations for the harmonic Kac operator. 5.3. Harmonic approximation for the transfer operator. 5.4. WKB constructions for the transfer operator. 5.5. The case of the Schrödinger operator in dimension 1. 5.6. Harmonic approximation for the transfer operator: upper bounds. 5.7. First conclusions about the splitting. 5.8. Some elements about the decay. 5.9. Splitting revisited. 5.10. Notes
- ch. 6. Basic facts in spectral theory and on the Schrödinger operator. 6.1. Introduction. 6.2. Selfadjoint operators, spectrum and spectral decomposition. 6.3. Discrete spectrum, essential spectrum. 6.4. Essentially selfadjoint operators. 6.5. Examples. 6.6. More on selfadjointness. 6.7. The max-min principle. 6.8. Compactness. 6.9. Notes
- ch. 7. Log-Sobolev inequalities. 7.1. Introduction. 7.2. Log-Sobolev inequalities in the strictly convex case. 7.3. Around Herbst's argument : necessary conditions for log-Sobolev inequalities. 7.4. Extension of the Bakry-Emery argument : convexity at infinity. 7.5. The case of the circle. 7.6. The case of the line. 7.7. General remarks. 7.8. Notes
- ch. 8. Uniform decay of correlations. 8.1. Introduction. 8.2. Lower bound for the spectrum of the Witten Laplacian. 8.3. Uniform estimates for a family of 1-dimensional Witten Laplacians. 8.4. A proof of the decay of correlations. 8.5. Generalized Brascamp-Lieb inequality. 8.6. Notes
- ch. 9. Uniform log-Sobolev inequalities. 9.1. Introduction and preliminaries. 9.2. Some log-Sobolev inequality for effective single spin phase. 9.3. The role of the decay estimates for log-Sobolev inequality. 9.4. Second part of the proof of the log-Sobolev inequality. 9.5. Conclusion. 9.6. Notes.