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This chapter starts by listing the common sources of noise in the EM and how they can be addressed by averaging techniques. Conditions for digital sampling, or for the representation of an effectively continuous image by an array of discrete density measurements, are discussed. The concept of image alignment is defined, and alignment is introduced as a precondition for averaging as well as for making any meaningful comparison of experimental images. The cross-correlation function is then introduced as one of the most important tools to achieve alignment. Averages of aligned images are characterized by statistical measures such as variance and signal-to-noise ratio. Measures of resolution are introduced based on a comparison, in Fourier space, of two independent averages from halfsets of the data. Among these are the differential phase residual and the Fourier ring correlation. The chapter ends with a discussion of the resolution-limiting factors, and with an outline of rank sum analysis, a method of statistical validation.

This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.

The description of the thermodynamics of the strongly ionized plasma starts by discussing the most popular and well-studied model of the one-component plasma (OCP), which represents a system of point ions placed in a homogeneous medium of charges of opposite sign. The results of calculations by the Monte Carlo method of the binary correlation function, static structure factor, dielectric permeability, isothermal compressibility, and internal and free energies are presented. The region of existence of Wigner crystallization is determined. Pseudopotential models of multicomponent plasma are considered. The advantages and disadvantages of the quasiclassical approximation, density functional, and quantum Monte Carlo methods are discussed. A number of the proposed models in the region of increased nonideality lose thermodynamic stability, which is attributed to the possibility of a phase transition and the separation of the system into phases of different densities.

This chapter presents some additional complications that arise in addition to the simple Luttinger liquid theory for one-dimensional systems. Examples of pitfalls into which the bosonization beginners can fall are discussed. This chapter also explores one of the most important consequences of the existence of a lattice, namely the possibility for the electrons to give an insulator driven by the interactions known as the Mott insulator. The discussion focuses on what happens when one breaks the spin rotation invariance in a Luttinger liquid with spin, and considers the logarithmic corrections of correlation functions.

This chapter presents a universal definition of linear response, which at first is studied for the example of the damped oscillator. From the quantal time evolution operator lowest order transition rates are deduced, together with microscopic response functions. Their fundamental properties are studied, in particular, the meaning of their reactive and dissipative parts for energy transfer. Correlation functions are introduced and related to response functions by way of the fluctuation dissipation theorem. For a random matrix model, the latter is shown to also hold true for microcanonical ensembles. Connections to thermal Green functions are established and an extension of response theory to unstable modes is discussed. With the help of the Mori product, relations among the static response, the adiabatic and the isothermal susceptibilities are given. Time dependent irreversible processes are studied by introducing relaxation functions and by following the variation of the density operator and of entropy. Onsager relations are presented and Kubo's expressions for transport coefficients are given.

It has been shown for molecular fluids that the velocity-time correlation function in ddimensions exhibits a power-law dependence for t >> 1. This function reveals a long time tail compared with an exponentially decaying function. According to the fluctuation-dissipation theorem, the kinetic coefficients are expressed by time integrals of the corresponding correlation functions. For d > 2, these integrals converge; for two-dimensional systems the convergence of these integrals is problematic and hence, the existence of the kinetic coefficients is questionable. For force-free granular fluids, one can expect convergence of these integrals, since there exists an additional decay of correlation functions due to the decay of temperature. This problem has not been addressed for granular gases. This chapter presents the results for d-dimensionless gases.

Dynamic heterogeneity is now recognized as a central aspect of structural relaxation in disordered materials with slow dynamics, and was the focus of intense research in the last decade. Here, this chapter describes how initial, indirect observations of dynamic heterogeneity have recently evolved into well‐defined, quantitative, statistical characterizations, in particular through the use of high‐order correlation and response functions. The chapter highlights both recent progress and open questions about the characterization of dynamic heterogeneity in glassy materials. The chapter also discusses the limits of available tools and describe a few candidates for future research in order to gain a deeper understanding of the origin and nature of glassiness in disordered systems.

Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one or more dimensions. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra. These are related to nonlinear correlation functions of excitation and response. This chapter discusses convolution in linear and nonlinear systems, along with the convolution theorem, linear system analysis, nonlinear cross-correlation, correlation theorem, Laplace transformation, Hankel transformation, Abel transformation, z transformation, Hadamard transformation, and wavelet transformation.

Chapter 25 established the scaling behaviour of correlation functions at criticality, T = T_c . This chapter studies the critical domain which is defined by the property that the correlation length is large with respect to the microscopic scale, but finite. Using results proven in Chapter 10, it first demonstrates strong scaling above T_c : in the critical domain above T_c , all correlation functions, after rescaling, can be expressed in terms of universal correlation functions, in which the scale of distance is provided by the correlation length. The first part of the chapter is restricted to Ising-like systems. It then generalizes the results to N-component order parameters in Section 26.6. In Section 26.7, shows how to expand the universal two-point function when T approaches T_c , using the short distance expansion. Finally, the appendix discusses the energy correlation function when the specific heat exponent vanishes.

The goal of this chapter is to review recent analytical results about the growth of a (static) correlation length in glassy systems, and the connection that can be made between this length scale and the equilibrium correlation time of its dynamics. The definition of such a length scale is first given in a generic setting, including finite‐dimensional models, along with rigorous bounds linking it to the correlation time. We then present some particular cases (finite connectivity mean‐field models, and Kac limit of finite‐dimensional systems) where this length can be actually computed.

Matters within the realm of spectroscopy—the interaction of molecules with radiation—are considered. The treatment focuses on three areas: the absorption and emission of infrared light, the scattering of light, and the scattering of neutrons. The theory is developed around a consideration of the time correlation functions representing the evolution of molecular dipole moments, polarizabilities, and positions and orientations; the corresponding spectra are the spatial and temporal Fourier transforms of these quantities. Particular attention is paid to the estimation of correlation functions from their short-time behaviour which may be calculated using spectral moments: these can be computed using equilibrium statistical mechanics. It is shown that such studies of allowed and induced spectra can yield valuable information concerning molecular interactions in both gases and liquids.

The present chapter is an introductory account of the basic concepts and important consequences of conformal symmetry, i.e. the invariance under local scale transformations, in field theories characterizing critical behaviour. The goal is to catalogue universality classes as a list of possible values of critical exponents and to find restrictions on the functional forms of correlation functions, which satisfy conformal Ward identities. From a mathematics standpoint, conformal symmetry applies to continuum theories, and therefore its obvious application to critical phenomena is formulated in the language of field theory. The energy-momentum tensor plays a fundamental role in defining the conformal generators that satisfy the Virasoro algebra, and any conformal field theory is characterized by the central charge a number that is important to classify critical field theories. One of the most remarkable applications of conformal field theory is found in the analysis of finite-size effects.

This chapter explores the development of infants' attention to object function and how function is used by infants in categorizing objects. It proposes a developmental progression wherein infants attend first to the structural properties of objects, then to both structural and functional properties, and finally to the correlation between structural and functional properties. Data is presented showing that infants are capable of categorizing objects based on structural properties prior to categorizing based on functional properties, and that infants treat functional properties of objects as more central to category membership than structural properties. Finally, the chapter reviews findings that infants' attention to structure-function correlations is initially ‘atheoretical’ and only later conforms to the kinds correlations found in the real world. The ages at which any changes are observed will depend on how categorization is assessed and the kinds of objects that infants are categorizing.

This chapter introduces path integral and various correlation functions at zero and non-zero temperatures to study interacting quantum systems. Semi-classical approximation and instanton effects are used to evaluate path integrals. The relation between correlation functions and physical measurements is discussed. The path integral method is then applied to a few simple systems, including a quantum system with friction and a quantum electric circuit.

This chapter discusses quantum theory of many-fermion systems. Various correlation functions and their related physical measurements are calculated for free fermion systems at both zero and non-zero temperatures. It is pointed out that many free fermion systems resemble critical points which are on verge to change into other qualitatively different phases. The topological properties of a filled band and its relation to quantized Hall conductance are also discussed. The chapter also stresses that a many-fermion system in two and higher dimension is not really a local quantum system. In one dimension, a many-fermion system can be viewed as a local quantum system, and the Jordan–Wigner transformation is introduced to transform a 1D many-fermion system to a 1D local many-boson system. The chapter points out that the Fermi/Bose statistics are a dynamical property of the hopping Hamiltonian and come from the statistical hopping algebra.

This chapter provides a simple physical interpretation to the formal continuum limit that has led, from an integral over position variables corresponding to discrete times, to a path integral. It shows that the integral corresponding to discrete times can be considered as the partition function of a classical statistical system in one space dimension. The continuum limit, then, corresponds to a limit where the correlation length, which characterizes the decay of correlations at large distance, diverges. This limit has some universality properties in the sense that different discretized forms lead to the same path integral. In this statistical framework, the correlation functions that have been introduced earlier appear as continuum limits of the correlation functions of classical statistical models on a one-dimensional lattice. Thus, the path integral can be used to exhibit a mathematical relation between classical statistical physics on a line and quantum statistical physics of a point-like particle at thermal equilibrium.

This chapter discusses ways to describe theoretically neutron scattering as a probe for condensed matter, first the scattering on a rigidly bound and isolated nucleus by the neutron-nucleus interaction as well as coherent and incoherent neutron scattering. Then the corresponding matrix element, in terms of perturbation theory of quantum mechanics, is derived which yields the double differential scattering cross-section of the sample as a function of momentum and energy transfer of the scattered neutrons. With the concept of the van Hove correlation functions, this matrix formulation is finally transformed into classical correlation functions, which are convenient if the particle motion can be described by classical dynamical models. In this way, the coherent scattering function S(Q, ω) is the double Fourier transform of the correlation function G(r,t), whereas the incoherent scattering function Si(Q, ω) is the double Fourier transform of the self-correlation function Gs(r,t). The convolution approximation relates S(Q, ω) to Si(Q, ω). The scattering intensity decreases with increasing Q due to the Debye–Waller factor which is directly connected to lattice vibrations.

This chapter discusses the non-linear s-model, a field theory characterized by an orthogonal O(N) symmetry acting non-linearly on the fields. The study has several motivations. From the viewpoint of statistical physics, the model appears in the study of the large-distance properties, in the ordered phase at low temperature, of lattice spin models with O(N) symmetry and short-range interactions. Indeed, in the case of continuous symmetries, the whole low-temperature phase has a non-trivial large-distance physics due to the presence of Goldstone modes with vanishing mass or infinite correlation length. Moreover, the model possesses, in two dimensions, the property of asymptotic freedom (the Gaussian fixed point is marginally stable for the large-momentum or short-distance behaviour) and the spectrum is non-perturbative. These properties are shared, in dimension 4, by quantum chromodynamics (QCD), a non-Abelian gauge theory and a piece of the Standard Model of fundamental interactions describing physics at the microscopic scale.

This chapter begins with a review of the semi-classical model of the interaction of charged particles with light, and then proceeds to the full quantum theory by replacing the classical fields with the corresponding field operators. A presentation of the quantum Maxwell equations and their behavior under parity and time reversal transformations is followed by a discussion of stationary density operators. The notion of positive- and negative-frequency parts of field operators is extended to interacting fields and used to define multi-time correlation functions that describe experimental results. The perturbation expansion is formulated by means of the interaction picture and combined with the dipole approximation to calculate the Einstein A- and B-coefficients. The chapter ends with a discussion of spontaneous emission in a cavity and Raman scattering.

This chapter reviews the basic building blocks of the regularization of Quantum Field Theories (QFT) on a space-time lattice. It assumes some familiarity with QFT in the continuum. In an introductory section, the path integral formulation is reviewed, focusing on important aspects such as the transfer matrix, the relation of correlation functions and physical observables, the perturbative expansion, and the key issue of renormalization and the Wilsonian renormalization group. It then considers in detail the lattice formulation of scalar, fermion and gauge field theories, paying careful attention to their physical interpretation, and the continuum limit. The difficulty of discretizing chiral fermions is discussed in detail, and various fermion discretizations are described. The strong coupling expansion is introduced in the context of lattice Yang-Mills theory and the criteria for confinement and for the presence of a mass gap are presented. It concludes with a description of Wilson's formulation of lattice QCD and a brief overview of its applications.