The theory of probability is so useful that it is required in almost every branch of science. In physics, it is of basic importance in quantum mechanics, kinetic theory, and thermal and statistical physics to name just a few topics. In this chapter the reader is introduced to some of the fundamental ideas that make probability theory so useful. We begin with a review of the definitions of probability, a brief discussion of the fundamental laws of probability, and methods of counting (some facts about permutations and combinations), probability distributions are then treated.

A notion that will be used very often in our discussion is ‘equally likely’. This cannot be denned in terms of anything simpler, but can be explained and illustrated with simple examples. For example, heads and tails are equally likely results in a spin of a fair coin; the ace of spades and the ace of hearts are equally likely to be drawn from a shuffled deck of 52 cards. Many more examples can be given to illustrate the concept of ‘equally likely’.

A definition of probability

Now a question that arises naturally is that of how shall we measure the probability that a particular case (or outcome) in an experiment (such as the throw of dice or the draw of cards) out of many equally likely cases that will occur. Let us flip a coin twice, and ask the question: what is the probability of it coming down heads at least once. There are four equally likely results in flipping a coin twice: HH, HT, TH, TT, where H stands for head and T for tail.

Linear vector space is to quantum mechanics what calculus is to classical mechanics. In this chapter the essential ideas of linear vector spaces will be discussed. The reader is already familiar with vector calculus in three-dimensional Euclidean space E3 (Chapter 1). We therefore present our discussion as a generalization of elementary vector calculus. The presentation will be, however, slightly abstract and more formal than the discussion of vectors in Chapter 1. Any reader who is not already familiar with this sort of discussion should be patient with the first few sections. You will then be amply repaid by finding the rest of this chapter relatively easy reading.

Euclideann-spaceEn

In the study of vector analysis in E3, an ordered triple of numbers (a1, a2, a3) has two different geometric interpretations. It represents a point in space, with a1, a2, a3 being its coordinates; it also represents a vector, with a1, a2, and a3 being its components along the three coordinate axes (Fig. 5.1). This idea of using triples of numbers to locate points in three-dimensional space was first introduced in the mid-seventeenth century. By the latter part of the nineteenth century physicists and mathematicians began to use the quadruples of numbers (a1, a2, a3, a4) as points in four-dimensional space, quintuples (a1, a2, a3, a4, a5) as points in fivedimensional space etc. We now extend this to n-dimensional space En, where n is a positive integer. Although our geometric visualization doesn't extend beyond three-dimensional space, we can extend many familiar ideas beyond three-dimensional space by working with analytic or numerical properties of points and vectors rather than their geometric properties.

Group theory did not find a use in physics until the advent of modern quantum mechanics in 1925. In recent years group theory has been applied to many branches of physics and physical chemistry, notably to problems of molecules, atoms and atomic nuclei. Mostly recently, group theory has been being applied in the search for a pattern of ‘family’ relationships between elementary particles. Mathematicians are generally more interested in the abstract theory of groups, but the representation theory of groups of direct use in a large variety of physical problems is more useful to physicists. In this chapter, we shall give an elementary introduction to the theory of groups, which will be needed for understanding the representation theory.

Definition of a group (group axioms)

A group is a set of distinct elements for which a law of ‘combination’ is well defined. Hence, before we give ‘group’ a formal definition, we must first define what kind of ‘elements’ do we mean. Any collection of objects, quantities or operators form a set, and each individual object, quantity or operator is called an element of the set.

A group is a set of elements A, B, C,…, finite or infinite in number, with a rule for combining any two of them to form a ‘product’, subject to the following four conditions:

1. (1) The product of any two group elements must be a group element; that is, if A and B are members of the group, then so is the product AB.

2. (2) The law of composition of the group elements is associative; that is, if A, B, and C are members of the group, then (AB)C = A(BC).

3. […]

This book evolved from a set of lecture notes for a course on ‘Introduction to Mathematical Physics’, that I have given at California State University, Stanislaus (CSUS) for many years. Physics majors at CSUS take introductory mathematical physics before the physics core courses, so that they may acquire the expected level of mathematical competency for the core course. It is assumed that the student has an adequate preparation in general physics and a good understanding of the mathematical manipulations of calculus. For the student who is in need of a review of calculus, however, Appendix 1 and Appendix 2 are included.

This book is not encyclopedic in character, nor does it give in a highly mathematical rigorous account. Our emphasis in the text is to provide an accessible working knowledge of some of the current important mathematical tools required in physics.

The student will find that a generous amount of detail has been given mathematical manipulations, and that ‘it-may-be-shown-thats’ have been kept to a minimum. However, to ensure that the student does not lose sight of the development underway, some of the more lengthy and tedious algebraic manipulations have been omitted when possible.

Each chapter contains a number of physics examples to illustrate the mathematical techniques just developed and to show their relevance to physics. They supplement or amplify the material in the text, and are arranged in the order in which the material is covered in the chapter. No effort has been made to trace the origins of the homework problems and examples in the book. A solution manual for instructors is available from the publishers upon adoption.

Abstract

The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions Dq and the f(α) singularity spectrum can be readily determined from the scaling behaviour of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of ‘generalized boxes’. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent applications of the wavelet transform modulus maxima method to fully developed turbulence data. Then we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic information about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered from the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period- doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos.

The reader might want to jump right into the book, but I decided to give a guided tour (which one may leave and rejoin at will of course) through the chapters, to whet the reader's taste.

Antoine opens in Chapter 1 with a brief survey of the basic properties of wavelet transforms, both continuous (CWT) and discrete (DWT). In the latter case one learns about the intuitively very appealing concept of multiresolution analysis. Section 1.4 looks ahead to the two- and more-dimensional versions, and summarily brings out connections with well known symmetry groups of physics, and the theory of coherent states.

In the second chapter, also by Antoine, the 2-D wavelet transform is treated. Here the characterization as mathematical microscope must be further qualified, because it misses the new and important property of orientability of the 2-D wavelets, which the 1-D case lacks. A real-world microscope is not more sensitive in one direction than in another one, it is ‘isotropic’. But the mathematical microscope as embodied in 2-D wavelets has an extra feature: these wavelets can be designed in such a way that they are directionally selective. Apart from dilation and translation, one can now also rotate the wavelet, which makes possible a sensitive detection of oriented features of a signal (a 2- D image). In many texts the 2-D case is still limited to the DWT, and the wavelets are usually formed by taking tensor products of 1-D wavelets in the x and y-direction, thereby giving preference to horizontal, vertical and diagonal features in the plane.

Abstract

In the field of atomic and solid state physics, wavelet analysis has been applied so far in three different directions: (i) time-frequency analysis of harmonic generation in laser-atom interactions; (ii) ab initio electronic structure calculations in atoms and molecules; and (iii) construction of localized bases for the lowest Landau level of a 2-D electron gas submitted to a strong magnetic field. We survey these three types of applications, with more emphasis on methods than on precise results.

Introduction

There are two ways in which wavelets could play a role in atomic physics and possibly in solid state physics.

First one may envisage them as physical objects, namely quantum states or wave functions. It is commonplace to remark that coherent states (CS) have a privileged role in atomic physics. Laser-atom interactions, revival phenomena, Rydberg wave packets and various semi-classical situations are all instances in which a coherent state description is clearly well-adapted. Of course, what is implied here are canonical CS, associated to the harmonic oscillator or the electromagnetic field [36]. But wavelets are also coherent states, namely those associated to the affine groups in various space dimensions, as we have seen in Chapter 2 (see [1] for a review on coherent states). Thus wavelets could well be thought of as convenient substitutes for canonical CS. However, this suggestion is still speculative at the present moment, very little has been achieved in this direction.

Abstract

Two fundamental properties of turbulence are intermittency and non-linearity. They imply that the standard Fourier spectral techniques are inadequate for its analysis. Spectral analysis based on wavelets provides a means to handle intermittency. New tools are required to handle non-linearity.

In this chapter, we redesign spectral analysis in terms of wavelet methods, paying particular attention to statistical stability, error estimates and nonlinearity. The application to both computer simulations and measurements carried out in fusion plasmas provide some illustrative examples.

Introduction

Although the phenomenon of turbulence is only partially understood, there seems to be consensus on several aspects. First, that intermittency is a basic property of turbulence. This means that the characteristics of the turbulence (spectral distribution, amplitude etc.) vary on a short time scale. Analysis techniques that rely on the accumulation of data over time scales larger than this characteristic time scale will then average out much of the dynamics and obliterate relevant information (as may occur with Fourier analyses). Wavelet analysis provides an interesting starting point for redesigning the standard analysis techniques in order to tackle this problem. In this chapter we shall redefine some basic Fourier analysis techniques in terms of wavelets, such as cross coherence. We shall emphasize the need for statistical stability and provide noise level estimates. Finally, we provide some examples of these techniques.

Second, it is generally accepted that turbulence only arises in non-linear systems. Therefore, to understand the nature of turbulence, it is essential to employ analysis tools that are capable of handling this non-linearity.

Abstract

We have used wavelets to analyse, model and compute turbulent flows. The theory and open questions encountered in turbulence are presented. The wavelet-based techniques that we have developed to study turbulence are explained and the main results are summarized.

Introduction

In this chapter we will summarize the ten years of research we have done to try to better understand, model and compute fully developed turbulent flows using wavelets and wavelet packets. Fully developed turbulence is a highly nonlinear regime (very large Reynolds number tending to infinity) and is distinct from the transition to turbulence (low Reynolds number). We have chosen to present a personal point of view concerning the current state of our understanding of fully developed turbulence. It may not always coincide with the point of view of other researchers in this field because many issues we are addressing in this chapter are still undecided and highly controversial. This paper is a substantially revised and extended version of: Wavelets and Turbulence by Farge, Kevlahan, Perrier and Goirand which appeared in Proceedings of the IEEE, vol. 84, no. 4, April 1996, pp. 639–669.

After more than a century of turbulence study [30], [173], no convincing theoretical explanation has produced a consensus among physicists (for a historical review of various theories of turbulence see [160], [158], [72], [91]). In fact, a large number of ad hoc ‘phenomenological’ models exist that are widely used by fluid mechanicians to interpret experiments and to compute many industrial applications (in aeronautics, combustion, meteorology …) where turbulence plays a role.