There is a close relationship between random graphs and percolation. In fact, percolation and random graphs have been viewed as “the same phenomenon expressed in different languages” (Albert and Barabási, ). Early ideas on percolation (although not under that name) in molecular chemistry can be found in the articles by Flory () and Stockmayer ().

The magnetic properties in solids originate mainly from the magnetic moments associated with electrons. The nuclei in solids also carry a magnetic moment. That, however, varies from isotope to isotope of an element. The nuclear magnetic moment is zero for a nucleus with even numbers of protons and neutrons in its ground state. The nuclei can have a non-zero magnetic moment if there are odd numbers of either or both neutrons and protons. However, the magnetic moment of a nucleus is three orders of magnitude less than that of the electron.

The microscopic theory of magnetism is based on the quantum mechanics of electronic angular momentum, which has two distinct sources: orbital motion and the intrinsic property of electron spin [1]. The spin and orbital motion of electrons are coupled by the spin–orbit interaction. The magnetism observed in various materials can be fundamentally different depending on whether the electrons are free to move within the material (such as conduction electrons in metals) or are localized on the ion cores. In a magnetic field, bound electrons undergo Larmor precession, whereas free electrons follow cyclotron orbits. The free-electron model is usually a starting point for the discussion of magnetism in metals. This leads to temperature-independent Pauli paramagnetism and Landau diamagnetism. This is the case with noble metals and alkali metals. On the other hand, localized non-interacting electrons in 3d-transition metals, 4f-rare earth elements, 5f-actinide elements, and their alloys and intermetallic compounds with incompletely filled inner shells exhibit Curie paramagnetism. Many transition metal-based insulating oxide and sulfide compounds also show Curie paramagnetism. In the presence of magnetic interactions, many such systems eventually develop long-range magnetic order if the magnetic interaction can overcome thermal fluctuations in some temperature regimes.

Against the above backdrop, in the next three chapters, we will introduce the readers to the basic phenomenology of magnetism, concentrating mainly on solid materials with some electrons localized on the ion cores. There are some excellent textbooks available on the subject, including those by J. M. D. Coey [1], B. D. Cullity and C. D. Graham [2], D. Jiles [3], S. J. Blundell [4], and N. W. Ashcroft and N. D. Mermin [5].

Introduction

The main objective of this chapter is familiarization with a variety of numerical methods that are essential for solving advanced problems of applied physics and engineering. With the help of suitable examples, basic skills on appropriately using these methods for various applications in physics are provided.

The chapter focuses on the following special second order differential equations, which are known to have standard functional form and/or analytical solutions.

The solutions of these equations are referred to as ‘special functions’, which are significantly different from standard functions like sine/cosine, exponential and logarithmic functions. This chapter also describes the use of quadrature methods of integration for calculating improper integrals, which are either infinite in the interval of integration, or the interval of integration has an infinite bound. The quadrature methods discussed in this chapter are as follows:

The chapter has been written in a manner so as to develop the necessary skills of the reader to evaluate certain integrals that are generally not discussed in introductory physics classes because they involve advanced calculations.

Bessel Function of the First Kind

Bessel functions have several applications in physics. They arise while solving Laplace's and Helmholtz equations in spherical and cylindrical coordinates. The functions are also useful while solving problems based on electromagnetic wave propagation and Schrödinger's equation.

Introduction

Differential equations are key mathematical tools for modelling physics problems. They are frequently used in all branches of physics while expressing the variation in one quantity w.r.t. the other. There are various kinds of differential equations, such as:

This chapter introduces the necessary numerical tools for determining approximate solutions of ordinary differential equations. It focuses only on initial and boundary value problems involving first and second order ordinary linear differential equations. These equations contain functions of one independent variable, and derivatives in that variable.

There are several numerical techniques for determining the solutions of differential equations. In this chapter, some commonly used methods have been explained. Section 4.2 shows the use of Euler's method to determine the solution of a differential equation. This is followed by modified Euler's method in Section 4.3, Runge–Kutta second order method in Section 4.4 and Runge–Kutta fourth order method in Section 4.5. A graphical comparison of these four methods is presented in Section 4.6. In Section 4.7, a quick review of the finite difference method has been provided for second order boundary value problems. Some advanced application problems of physics involving the first and second order differential equations have been discussed in Section 4.8.

This chapter uses the plotting skills developed in the second chapter. The reader is encouraged to refine their understanding of the plotting techniques.

Euler's Method

This is the most basic method of numerical integration. It is a first order method for approximating solutions of differential equations. This method uses the initial value as the starting point and approximates the next point of the solution curve using a tangent line to that point. The accuracy crucially depends on the step size used to approximate the subsequent point on the solution curve.

The algorithm for writing a Scilab program based on Euler's method is explained in Section 4.2.1 (first order) and in Section 4.2.2 (second order) with the help of suitable examples.

Film-based holography employs the use of high-resolution films such as the use of photopolymers or photorefractive materials for recording. These materials, while having high resolution, have a couple of drawbacks. The film-based techniques are typically slow for real-time applications and difficult to allow direct access to the recorded hologram for manipulation and subsequent processing. With recent advances in high-resolution solid-state 2-D sensors and the availability of ever-increasing power of computers and digital data storage capabilities, holography coupled with electronic/digital devices has become an emerging technology with an increasing number of applications such as in metrology, nondestructive testing, and 3-D imaging. While electronic detection of holograms by a TV camera was first performed by Enloe et al. in 1966, hologram numerical reconstruction was initiated by Goodman and Lawrence. In digital holography, it has meant that holographic information of 3-D objects is captured by a CCD, and reconstruction of holograms is subsequently calculated numerically. Nowadays, digital holography means the following situations as well. Holographic recording is done by an electronic device, and the recorded hologram can be numerically reconstructed or sent to a display device (called a spatial light modulator) for optical reconstruction. Or, hologram construction is completely numerically simulated. The resulting hologram is sent subsequently to a display device for optical reconstruction. This aspect of digital holography is often known as computer-generated holography.

Percolation can be defined more generally than as a process on ${\mathbb{Z}}^d$, $d\ge 2$. In this chapter, we motivate the main ideas and theory of percolation on more general graphs by application to polymer gelation and amorphous computing.

In this chapter, we discuss various issues that arise when networks increase in size. What does it mean for a network to increase in size and how would we visualize that process? Can a sequence of networks, increasing in size, converge to a limit, and what would such a limit look like? We discuss the transformation of an adjacency matrix to a pixel picture and what it means for a sequence of pixel pictures to increase in size. If a limit exists, the resulting function is called a limit graphon, but it is not itself a network. Estimation of a graphon is also discussed and methods described include an approximation by SBM and a network histogram.

As we have seen, networks, such as the Internet and World Wide Web, social networks (e.g., Facebook and LinkedIn), biological networks (e.g., gene regulatory networks, PPI networks, networks of the brain), transportation networks, and ecological networks are becoming larger and larger in today’s interconnected world. Some of these networks are truly huge and difficult, if not impossible, to analyze completely and efficiently. In this chapter, we discuss some of the issues involving comparing networks for similarity or differences, including choice of similarity measures, exchangeable random structures of networks, and property testing in networks.