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2881 results in Probability theory and stochastic processes

Page 35 of 145

  • 7 - Pirogov–Sinai Theory

  • Sacha Friedli, École Polytechnique Fédérale de Lausanne, Yvan Velenik, Université de Genève
  • Book:
    Statistical Mechanics of Lattice Systems
    Published online:
    17 November 2017
    Print publication:
    23 November 2017, pp 346-408

  • Summary

    As we have already discussed several times in previous chapters, a central task of equilibrium statistical physics is to characterize all possible macroscopic behaviors of the system under consideration, given the values of the relevant thermodynamic parameters. This includes, in particular, the determination of the phase diagram of themodel. This can be tackled in at least two ways, as already seen in Chapter 3. In the first approach, one determines the set of all infinite-volume Gibbs measures as a function of the parameters of the model. In the second approach, one considers instead the associated pressure and studies its analytic properties as a function of its parameters; of particular interest is the determination of the set of values of the parameters at which the pressure fails to be differentiable.

    Our goal in the present chapter is to introduce the reader to the Pirogov–Sinai theory, in which these two approaches can be implemented, at sufficiently low temperatures (or in other perturbative regimes), for a rather general class of models. This theory is one of the few frameworks in which first-order phase transitions can be established and phase diagrams constructed, under general assumptions.

    To make the most out of this chapter, readers should preferably be familiar with the results derived for the Ising model in Chapter 3, as those provide useful intuition for the more complex problems addressed here. They should also be familiar with the cluster expansion technique described in Chapter 5, the latter being the basic tool we will use in our analysis. However, although it might help, a thorough understanding of the theory of Gibbs measures, as given in Chapter 6, is not required.

    Conventions.We know from Corollary 6.41 that one-dimensional models with finite-range interactions do not exhibit phase transitions and thus possess a trivial phase diagram at all temperatures. We will therefore always assume, throughout the chapter, that d ≥ 2.

    It will once more be convenient to adopt the physicists’ convention and let the inverse temperature β appear as a multiplicative constant in the Boltzmann weights and in the pressures. To lighten the notations, we will usually omit to mention β and the external fields, especially for partition functions.

    Introduction

    Most of Chapter 3 was devoted to the study of the phase diagram of the Ising model as a function of the inverse temperature β and magnetic field h.

  • 3 - The Ising Model

  • Sacha Friedli, École Polytechnique Fédérale de Lausanne, Yvan Velenik, Université de Genève
  • Book:
    Statistical Mechanics of Lattice Systems
    Published online:
    17 November 2017
    Print publication:
    23 November 2017, pp 82-176

  • Summary

    In this chapter, we study the Ising model on Zd, which was introduced informally in Section 1.4.2.We provide both precise definitions of the concepts involved and a detailed analysis of the conditions ensuring the existence or absence of a phase transition in this model, therefore providing full rigorous justification to the discussion in Section 1.4.3. Namely,

  • • In Section 3.1, the Ising model on Zd is defined, together with various types of boundary conditions.

  • • In Section 3.2, several concepts of fundamental importance are introduced, including: the thermodynamic limit, the pressure and the magnetization. The latter two quantities are then computed explicitly in the case of the onedimensionalmodel (Section 3.3).

  • • The notion of infinite-volume Gibbs state is given a precise meaning in Section 3.4. In Section 3.6, we discuss correlation inequalities, which play a central role in the analysis of ferromagnetic systems like the Isingmodel.

  • • In Section 3.7, the phase diagram of the model is analyzed in detail. In particular, several criteria for the presence of first-order phase transitions, based on the magnetization and the pressure of the model, are introduced in Section 3.7.1. These are used to prove the existence of a phase transition when h = 0 (Sections 3.7.2 and 3.7.3) and the absence of a phase transition when h ≠ 0 (Section 3.7.4). A summary with a link to the discussion in the Introduction is given in Section 3.7.5.

  • • Finally, in Section 3.10, the reader can find a series of complements to this chapter, in which a number of interesting topics, related to the core of the chapter but usually more advanced or specific, are discussed in a somewhat less precise manner.

    • Weemphasize that some of the ideas and concepts introduced in this chapter are not only useful for the Ising model, but are also of central importance for statistical mechanics in general. They are thus fundamental for the understanding of other parts of the book.

    Finite-Volume Gibbs Distributions

    In this section, the Ising model on Zd is defined precisely and some of its basic properties are established. As a careful reader might notice, some of the definitions in this chapter differ slightly from those of Chapter 1. This is done for later convenience.

  • Appendix A - Notes

  • Sacha Friedli, École Polytechnique Fédérale de Lausanne, Yvan Velenik, Université de Genève
  • Book:
    Statistical Mechanics of Lattice Systems
    Published online:
    17 November 2017
    Print publication:
    23 November 2017, pp 506-512

  • Summary

    Chapter 1

    [1](p. 4) The property described in (1.1) is usually referred to as additivity rather than extensivity. Extensivity of the energy is often valid and equivalent to additivity in the thermodynamic limit, at least for systems with finite-range interactions, as considered usually in this book. For systems with long-range interactions, extensivity does not always hold.

    [2](p. 20) This terminology was introduced by Gibbs [137], but the statistical ensembles were first introduced by Boltzmann under a different name (ergode for themicrocanonical ensemble and holode for the canonical).

    [3](p. 21)We adopt here the following point of view explained by Jaynes in [181]:

    This problem of specification of probabilities in cases where little or no information is available, is as old as the theory of probability. Laplace's “Principle of Insufficient Reason” was an attempt to supply a criterion of choice, in which one said that two events are to be assigned equal probabilities if there is no reason to think otherwise.

    Of course, some readers might not consider such a point of view to be fully satisfactory. In particular, one might dislike the interpretation of a probability distribution as a description of a state of knowledge, rather than as a quantity intrinsic to the system. After all, there is amore fundamental theory and it would be satisfactory to derive this probability distribution from the latter. Many attempts have been done, but no fully satisfactory derivation has been obtained. We will not discuss such issues further here, but refer the interested reader to the extensive literature on this topic; see for example [130].

    [4](p. 22) Historically, the entropy of a probability density had already been introduced by Gibbs in [137].

    [5](p. 39) In our brief description of a ferromagnet and its basic properties, we are neglecting many physically important aspects of the corresponding phenomena. Our goal is not to provide a faithful account, but rather to provide the uninitiated readerwith an idea of what ferromagnetic and paramagnetic behaviors correspond to. We refer readers who would prefer a more thorough description to any of the many books on condensed matter physics, such as [1, 14, 65, 356].

Statistical Mechanics of Lattice Systems
  • A Concrete Mathematical Introduction
  • Sacha Friedli, Yvan Velenik
  • Published online:
    17 November 2017
    Print publication:
    23 November 2017
  • This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kać interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.

Page 35 of 145