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2193 results in Statistical Physics

Page 36 of 110

  • 1 - Introduction

  • from Part I - Atomic and Mesoscopic Viewpoint
  • Yoshitsugu Oono, University of Illinois, Urbana-Champaign
  • Book:
    Perspectives on Statistical Thermodynamics
    Published online:
    24 November 2017
    Print publication:
    14 December 2017, pp 3-11

  • Summary

    Summary

    * Science is an empirical endeavor.

    * The world in which we live seems to permit microscopic, mesoscopic, and macroscopic descriptions.

    * The macroscopic description is in terms of observables and concepts meaningful at the scale of our daily lives.

    * The microscopic description is in terms of particles and fields that underlie our world.

    * We see why atoms andmolecules are so small compared to us. Thus, the law of large numbers is crucial to relate the macro and micro worlds.

    * The mesoscopic description describes fluctuations deviating from the law of large numbers.

    * The study of large deviation is the key to understanding fluctuations.

    Key words

    three levels of description: microscopic, mesoscopic, macroscopic

    Everybody knows that the materials we see around us are made of atoms and molecules. We could even see them, for example, with the aid of atomic force microscopes. However, only 50 years ago no one could see atoms. About 100 years ago the existence of atoms was still disputed.

    Atomisms, Ancient and Modern

    The idea that the world is made of indivisible and unchanging minute particles (atomism) is, however, not a very unique idea. After all, it seems that there are only two choices: (i) the world is infinitely divisible and continuous or (ii) the world is made of indivisible units separated by void (we ignore various easy combinations of these ideas here).

    Leucippus (fifth century BCE) is usually credited with inventing atomism in Greece. His disciple Democritus (c. 460–370 BCE) systematized his teacher's theory. The early atomists tried to account for the formation of the natural world by means of atoms and void alone. The void space is described simply as nothing, or the negation of body. Atoms are, by their nature, intrinsically unchanging, but can differ in size, shape, spatial position (and orientation), etc. They move in the void and can temporarily make clusters according to their shapes and surface structures. The changes in the world of macroscopic objects were understood to be caused by rearrangements of the atomic clusters. Thus, atomism explains changes in the macroscopic world without creating new substance. All the macroscopic phenomena are naturally ephemeral.

Perspectives on Statistical Thermodynamics
  • Yoshitsugu Oono
  • Published online:
    24 November 2017
    Print publication:
    14 December 2017
  • This original text develops a deep, conceptual understanding of thermal physics, highlighting the important links between thermodynamics and statistical physics, and examining how thermal physics fits within physics as a whole, from an empirical perspective. The first part of the book is devoted to elementary, mesoscopic topics such as Brownian motion, which leads to intuitive uses of large deviation theory, one of the pillars of modern probability theory. The book then introduces the key concepts behind statistical thermodynamics, and the final part describes more advanced and applied topics from thermal physics such as phase transitions and critical phenomena. This important subject is presented from a fresh perspective and in a highly pedagogical manner, with numerous worked examples and relevant cultural side notes throughout, making it ideal as either a textbook for advanced thermal physics courses or for self-study by undergraduate and graduate students in physics and engineering.

  • 5 - Cluster Expansion

  • 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 232-261

  • Summary

    Introduction

    The cluster expansion is a powerful tool in the rigorous study of statistical mechanics. It was introduced by Mayer during the early stages of the study of the phenomenon of condensation and remains widely used nowadays. In particular, it remains at the core of the implementation of many renormalization arguments in mathematical physics, yielding rigorous results that no other methods have yet been able to provide.

    Simply stated, the cluster expansion provides a method for studying the logarithm of a partition function. We will use it in various situations, for instance to obtain new analyticity results for the Isingmodel in the thermodynamic limit.

    In a first application, we will obtain new results on the pressure, completing those of Chapter 3. There, we saw that the pressure is analytic in the half-space, but the techniqueswe used did not provide further quantitative information. Here wewill use the cluster expansion to compute the coefficients of the expansion of, in terms of the variable:

    In our second application, wewill fix h = 0, and study the analyticity of. We will obtain, when is sufficiently small, an expansion in terms of the variable,

    One might hope that this series converges for all. Unfortunately, the method developed in this chapter will guarantee analyticity only when, where is some number, strictly smaller than the critical value. We will call a regime such as a regime of very high temperature to distinguish it from the high-temperature regime used in earlier chapters.

    Similarly, we will also obtain, when is sufficiently large, an expansion in terms of the variable,

    Once again, this series will be guaranteed to converge at very low temperature, that is, for all, where.

    Of course, the cluster expansion is not limited to the study of the pressure in these different regimes and we will show how it can be used to extract additional information on other quantities of interest. In particular, at very low temperatures, we will derive a series expansion for the spontaneous magnetization and prove exponential decay of the truncated 2-point function.

    We hope that this sample of applications will convince the reader that the cluster expansion is a versatile tool that, even though applicable only in restricted regions of the space of parameters, provides precious information often unavailable when using other techniques.

  • Preface

  • 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 xv-xvii

  • Summary

    Equilibrium statistical mechanics is a field that has existed for more than a century. Its origins lie in the search for a microscopic justification of equilibrium thermodynamics, and it developed into a well-established branch of mathematics in the second half of the twentieth century. The ideas and methods that it introduced to treat systems with many components have now permeated many areas of science and engineering, and have had an important impact on several branches of mathematics.

    There exist many good introductions to this theory designed for physics undergraduates. It might, however, come as a surprise that textbooks addressing it from a mathematically rigorous standpoint have remained rather scarce. A reader looking for an introduction to its more advanced mathematical aspects must often either consult highly specialized monographs or search through numerous research articles available in peer-reviewed journals. It might even appear as if the mastery of certain techniques has survived from one generation of researchers to the next only by means of oral communication, through the use of chalk and blackboard…

    It seems a general opinion that pedagogical introductory mathematically rigorous textbooks simply do not exist. This book aims at starting to bridge this gap. Both authors graduated in physics before turning to mathematical physics. As such, we have witnessed this lack from the student's point of view, before experiencing it, a few years later, from the teacher's point of view. Above all, this text aims to provide the material we would have liked to have had at our disposal when entering this field.

    Although we hope that it will also be of interest to students in theoretical physics, this is in fact a book on mathematical physics. There is no general consensus on what this term actually refers to. In rough terms, what it means for us is: the analysis of problems originating in physics, at the level of rigor associated with mathematics. This includes the introduction of concepts and the development of tools enabling such an analysis. It is unfortunate thatmathematical physics is often held in rather low esteem by physicists, many of whom see it as useless nitpicking and as dealing mainly with problems that they consider to be already fully understood. There are, however, very good reasons for these investigations.

  • 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.

Page 36 of 110