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This chapter provides an overview of mechanisms and theoretical models of pattern formation in dense gravity-driven granular flows. A wide range of phenomena, from one- and two-dimensional avalanches flowing upon inclined planes, instabilities and fingering of avalanche fronts, flows in rotating drums to self-organized criticality and statistics of granular avalanches, are considered. In the majority of gravity-driven granular flows, the motion is confined at the surface of the granular system. Various approaches are used to describe surface flows, including depth-averaged hydrodynamic equations (the Saint-Venant model), two-phase models rolling and static fractions of erodible granular flows (Bouchaud, Cates, Ravi Prakash, and Edwards equations), and order parameter models for partially fluidized granular flows.

This chapter contains a general discussion of the phenomenon of BEC, under conditions much broader than those realized in the equilibrium noninteracting system introduced in Chapter 1. A definition of BEC in a general (nonequilibrium, noninteracting) Bose system is given in terms of the eigenvalues of the single-particle density matrix; alternative definitions are also discussed. The important concepts of order parameter and superfluid velocity are introduced. The question is raised: why should BEC occur, and when does it (not)? For Fermi systems, Cooper pairing is defined in terms of the eigenvalues of the two-particle density matrix; again, alternative definitions are briefly discussed, and possible reasons for the occurrence of the phenomenon reviewed. The chapter concludes with an overview of the consequences of BEC/Cooper pairing, and with a discussion of some unusual cases in which the BEC is “fragmented”. Two appendices review the second-quantization formalism and the properties of number and phase operators.

This chapter introduces the basic concepts and language that will be needed later on: order, symmetry, invariance, broken symmetry, Hamiltonian, condensed matter, order parameter, ground state, and several thermodynamic terms. It also presents the necessary concepts from thermodynamics and statistical mechanics that will be needed later. It boils down the latter to its most elemental and essential ingredient: that of temperature as controlling the relative probabilities of configurations of different energies. For much of statistical mechanics, all else is commentary. This is sufficient to present an intuitive understanding of why and how matter organizes itself into different phases as temperature varies, and leads to the all-important concept of a phase transition.

Statistical-mechanical systems often involve discrete elementary degrees of freedom such as spins in the Ising model. Field theories, on the other hand, have continuous fields, defined over the whole space-time or part of it, as fundamental degrees of freedom. These two seemingly different descriptions of physical phenomena can be related close to the critical point. The present chapter summarizes how the description by continuous fields emerges from discrete degrees of freedom in a more systematic manner than in previous chapters. The phenomenological Landau-Ginzburg approach, based on the concept of order parameter, is expanded to generate effective field theories. The important roles of symmetry and topology are also elucidated in some detail. Also shown are some of the important consequences of having a broken-symmetry phase, such as long-range order, the emergence of Nambu-Goldstone modes when the symmetry involved is continuous, and topological defects.

This chapter introduces a quantum field theory for interacting boson systems. It develops a mean-field theory to study the superfluid phase. A path integral formulation is then developed to re-derive the superfuid phase, which results in a low energy effective non-linear sigma model. A renormalization group approach is introduced to study the zero temperature quantum phase transition between superfluid and Mott insulator phase, and finite temperature phase transition between superfluid and normal phase. The physics and the importance of symmetry breaking in phase transitions and in protecting gapless excitations are discussed. The phenomenon of superfluidity and superconductivity is also discussed, where the coupling to U(1) gauge field is introduced.

As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.

The physics of dipolar glasses is briefly discussed. It is shown that the local dipolar structure can be characterized by a local polarization distribution function and not by the homogeneous polarization as in classical ferroelectrics. The glassy structure is determined by the Edwards–Anderson order parameter. The dielectric properties and NMR characteristics of dipolar glasses and inhomogeneous ferroelectrics are presented. The difference between the field‐cooled and zero‐field‐cooled dielectric constant is stressed. It is also shown that the local polarization distribution function and the Edwards–Anderson order parameter can be determined by NMR. The random‐bond–random‐field Ising model theory of dipolar glasses is presented.

The present chapter explains the mean-field approximation, the Landau theory, the infinite-range model, and the Bethe approximation, and shows that all these (mean-field) theories are essentially equivalent to each other. The Landau theory is a phenomenological approach that uses the concept of symmetry and the order parameter, a measure of the breaking of that symmetry, as fundamental collective degrees of freedom. Also described are the Landau theory of tricritical behaviour, correlation functions, the limit of applicability of the mean-field theory, known as the Ginzburg criterion, and dynamic critical phenomena. Mean-field theories yield the exact critical exponents for dimensions larger than the upper critical dimension, and their solutions provide a reasonable starting point for more advanced methods including the renormalization group.

This chapter applies the Green function formalism to the BCS theory of superconductivity — the Gor'kov equations are derived which make the basis for the further analysis. The Green functions are used to derive the expressions for such physical quantities as the superconducting order parameter, the electric current, the electron density, and the thermodynamic potential. The Bogoliubov–de Gennes equations are derived from the Gor'kov equations. The Gor'kov theoy is used to derive the Green functions in a homogeneous state, the gap function, the critical temperature, the supercurrent, etc.

This chapter introduces order parameters -- the reduction of a complex system of interacting particles into a few fields that describe the local equilibrium behavior at each point in the system. It introduces an organized approach to studying a new material system -- identify the broken symmetries, define the order parameter, examine the elementary excitations, and classify the topological defects. It uses order parameters to describe crystals and liquid crystals, superfluids and magnets. It touches upon broken gauge symmetries and the Anderson/Higgs mechanism and an analogue to braiding of non-abelian quantum particles. Exercises explore sound, second sound, and Goldstone’s theorem; fingerprints and soccer balls; Landau theory and other methods for generating emergent theories from symmetries and commutation relations; topological defects in magnets, liquid crystals, and superfluids, and defect entanglement.

Liquid crystal is a state of matter which has an intermediate order between liquids and crystals. While fluid in nature, the materials in liquid crystals posses an order in molecular orientation. As a result, the molecular orientation of liquid crystal is easily controlled by weak forces, a property that is extensively used in the application of liquid crystals to display devices. Liquid crystal is an example that the collective nature of soft matter is created by phase transition. This chapter discusses how the interaction between individual molecules creates spontaneous macroscopic ordering, and how it affects the material response to external forces. The phase transition in liquid crystals is an example of order–disorder transition, the general aspects of which can be seen in liquid crystals.

This chapter describes random magnetic systems, ‘spin glasses’, by special random ensembles of factor graphs. It also studies the glass phase, characterized by a freezing of the spins, in the framework of equilibrium statistical physics. It describes the two types of spin glass phase transitions that have been encountered in mean field models, and introduces the relevant order parameters to describe them based on the notion of overlap. Special attention is given to the concept of frustration — a basic ingredient of spin glasses — which is discussed in conjunction with gauge transformations.

This chapter reviews a few properties, from the point of view of phase transitions, of simple ferromagnetic lattice models. In systems with finite range interactions a transfer matrix can be defined. It first examines its properties in a finite volume. In the infinite volume limit, low and high temperature considerations provide convincing evidence of the existence of phase transitions in Ising-like systems. The notion of order parameter is related to cluster properties in the low temperature broken phase. It is shown in a simple example that phase transitions indeed correspond to breaking of ergodicity. The chapter extends the analysis to ferromagnetic systems with continuous symmetries. The appendix contains a brief discussion of quenched disorder.

QENS on diffusing atoms with a sizable coherent neutron scattering cross section yields information on interference effects and correlated motion contained in the coherent scattering function S(Q, ω). As an introduction to this complex situation, this chapter begins by describing first coherent elastic scattering due to short-range order. The structure factor S_SROM(Q) for short-range order of a lattice fluid can be expressed in terms of Cowley's short-range order parameters. For a diffusing lattice fluid, coherent QENS consists of a single Lorentzian: its intensity is given by S_SRO and its width Λ and thus the collective diffusion coefficient D_c exhibit the so-called de Gennes narrowing: at the Q value where S_SRO has a maximum, Λ and D_c exhibit a minimum. Furthermore, elastic distortion scattering (Huang scattering) and quasielastic scattering on distortions combined with diffusion is dealt with.

Liquid crystals (LCs) may be divided into two subgroups: (1) lyotropic LCs, formed by mixing rigid rodlike molecules with a solvent, and (2) thermotropic LCs, formed by heating. One finds in the literature such terms as mesomorphs, mesoforms, mesomorphic states, and anisotropic liquids. The molecules in LCs have an orderly arrangement, and different orders of structures (nematic, smectic, or cholesteric structure) have been observed, as schematically shown in Figure 9.1. The kinds of molecules that form LCs generally possess certain common molecular features. The structural characteristics that determine the type of mesomorphism exhibited by various molecules have been reviewed. At present, our understanding of polymeric liquid crystals, often referred to as liquid-crystalline polymers (LCPs), is largely derived from studies of monomeric liquid crystals. However, LCPs may exhibit intrinsic differences from their monomeric counterparts because of the concatenation of monomers to form the chainlike macromolecules. The linkage of monomers inevitably means a loss of their translational and orientational independence, which in turn profoundly affects the dynamics of polymers in the liquid state. These intramolecular structural constraints are expressed in the flexibility of the polymer chain. Generally speaking, the chemical constitution of the monomer determines the flexibility and equilibrium dimensions of the polymer chain (Gray 1962). Figure 9.2 illustrates the variability of chain conformation (flexible chain, semiflexible chain, and rigid rodlike chain) forming macromolecules. Across this spectrum of chain flexibility, the persistence in the orientation of successive monomer units varies from the extreme of random orientation (flexible chains) to perfect order (the rigid rod). Hence, efforts have been made to synthesize LCPs that consist of rigid segments contributing to the formation of a mesophase and flexible segments contributing to the mobility of the entire macromolecule in the liquid state (Ober et al. 1984). From the point of view of molecular architecture, as schematically shown in Figure 9.3, two types of LCP have been developed: (1) main-chain LCPs (MCLCPs), having the monomeric liquid crystals (i.e., mesogenic group) in the main chain of flexible links, and (2) side-chain LCPs (SCLCPs), having the monomeric liquid crystals attached, as a pendent side chain, to the main chain.

The core of the exposition of the theory of conformal symmetry in statistical mechanics are the concepts of correlation functions of order parameter fields, whose behaviour under conformal transformations are the defining characteristic of conformal field theories. Chapter 7 discusses the transformation properties of the energy-momentum tensor, the conformal Ward identities, and the operator product expansion lead to the loop or Witt algebra with central extension, the Virasoro algebra, allowing the characterization of the possible universality classes, in particular through the conformal anomaly or central charge. It discusses how the finite-size corrections to thermodynamic quantities, obtained from conformal transformations to finite geometries, can be used to determine critical parameters, especially the central charge.

At a phase transition two or more different phases may coexist, such as vapour and liquid. Phase transitions can be classified according to their order. A phase transition is of first order if going from one phase to the other involves a discontinuous change in entropy, and, thus, a finite amount of latent heat; higher-order phase transitions do not involve latent heat but exhibit other types of discontinuities. This chapter investigates the necessary conditions for the coexistence of phases, and how phases are represented in a phase diagram. The order of a phase transition is defined with the help of the Ehrenfest classification. The chapter discusses the Clausius–Clapeyron relation which, for a first-order phase transition, relates the discontinuous changes in entropy and volume. Finally, this chapter considers the Ising ferromagnet as a simple model which exhibits a second-order phase transition. It also introduces the notion of an order parameter.

Chapter 1 introduces various essential ideas on second-order phase transitions and the theoretical challenges that accompany them. Furthermore, it focuses on important issues, for example, correlation length, correlation functions, scaling laws and behaviour, energy minimization, entropy maximization, dimensionality of space and order parameters and critical exponents, etc. It introduces and also devotes a short discussion to the Ising model and its most significant developments during the years of its study, as well as a short background about Ising himself. The chapter also contains two appendices that summarize all relevant results of ensembles of classical statistical mechanics and quantum statistical mechanics.