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  • Book
  • Aug 2013

Stochastic Processes on a Lattice and Gibbs Measures

Authors:

  1. Bernard Prum

    1. Université Paris V, Paris, France

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    You can also search for this author in PubMed   Google Scholar

  2. Jean Claude Fort

    1. Université Paris V, Paris, France

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Mathematical Physics Studies (MPST, volume 11)

  • 1488 Accesses

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-ix
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  2. Aspects of the Ising Model

    • Bernard Prum, Jean Claude Fort
    Pages 1-18

  3. Gibbs Measures

    • Bernard Prum, Jean Claude Fort
    Pages 19-39

  4. The Existenee of Gibbs Measures

    • Bernard Prum, Jean Claude Fort
    Pages 40-57

  5. Phase Transitions - 1: Methods of Convex Analysis

    • Bernard Prum, Jean Claude Fort
    Pages 58-81

  6. Other Inequalities

    • Bernard Prum, Jean Claude Fort
    Pages 82-98

  7. Phase Transition 2: Phase Diagrams and Perturbed Hamiltonians

    • Bernard Prum, Jean Claude Fort
    Pages 99-119

  8. Phase Transition 3: Reflexive Positivity

    • Bernard Prum, Jean Claude Fort
    Pages 120-135

  9. Continuous Symmetry and Other Methods

    • Bernard Prum, Jean Claude Fort
    Pages 136-160

  10. The Dynamics of Ising Models

    • Bernard Prum, Jean Claude Fort
    Pages 161-173

  11. Statistics and Applications

    • Bernard Prum, Jean Claude Fort
    Pages 174-204

  12. Back Matter

    Pages 205-220
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About this book

In many domains one encounters "systems" of interacting elements, elements that interact more forcefully the closer they may be. The historical example upon which the theory offered in this book is based is that of magnetization as it is described by the Ising model. At the vertices of a regular lattice of sites, atoms "choos e" an orientation under the influence of the orientations of the neighboring atoms. But other examples are known, in physics (the theories of gasses, fluids, .. J, in biology (cells are increasingly likely to become malignant when their neighboring cells are malignant), or in medecine (the spread of contagious deseases, geogenetics, .. .), even in the social sciences (spread of behavioral traits within a population). Beyond the spacial aspect that is related to the idea of "neighboring" sites, the models for all these phenomena exhibit three common features: - The unavoidable ignorance about the totality of the phenomenon that is being studied and the presence ofa great number of often unsuspected factors that are always unquantified lead inevitably to stochastic models. The concept of accident is very often inherent to the very nature of the phenomena considered, so, to justify this procedure, one has recourse to the physicist's principle of indeterminacy, or, for example, to the factor of chance in the Mendelian genetics of phenotypes. 
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Keywords

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Authors and Affiliations

  • Université Paris V, Paris, France

    Bernard Prum, Jean Claude Fort

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