Scale Invariance E-Book

Dick Henriksen

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 534

Veröffentlichungsjahr: 2015

Beschreibung

Bringing the concepts of dimensional analysis, self-similarity, and fractal dimensions together in a logical and self-contained manner, this book reveals the close links between modern theoretical physics and applied mathematics. The author focuses on the classic applications of self-similar solutions within astrophysical systems, with some general theory of self-similar solutions, so as to provide a framework for researchers to apply the principles across all scientific disciplines. He discusses recent advances in theoretical techniques of scaling while presenting a uniform technique that encompasses these developments, as well as applications to almost any branch of quantitative science. The result is an invaluable reference for active scientists, featuring examples of dimensions and scaling in condensed matter physics, astrophysics, fluid mechanics, and general relativity, as well as in mathematics and engineering.

Leseprobe

Table of Contents

Cover

Related Titles

Title Page

Copyright

Dedication

Preface

Acknowledgments

Introduction

Chapter 1: Arbitrary Measures of the Physical World

1.1 Similarity

1.2 Dimensional Similarity

1.3 Physical Equations and the ‘Pi’ Theorem

1.4 Applications of the Pi Theorem

References

Chapter 2: Lie Groups and Scaling Symmetry

2.1 The Rescaling Group

2.2 Familiar Physical Examples

2.3 Less Familiar Examples

References

Chapter 3: Poincaré Group Plus Rescaling Group

3.1 Galilean Space-Time

3.2 Minkowski Space-Time

3.3 Kinematic General Relativity

References

Chapter 4: Instructive Classic Problems

4.1 Introduction

4.2 Ideal Fluid Flow Past a Wedge: Self-Similarity of the ‘Second Kind’

4.3 Boundary Layer on a Flat Plate: the Blasius Problem

4.4 Adiabatic Self-Similarity in the Diffusion Equation

4.5 Waves in a Uniformly Rotating Fluid

References

Chapter 5: Variations on Lie Self-Similarity

5.1 Variations on the Boltzmann–Poisson System

5.2 Hydrodynamic Examples

5.3 Axi-Symmetric Ideal Magnetohydrodynamics

References

Chapter 6: Explorations

6.1 Anisotropic Self-Similarity

6.2 Mathematical Variations

6.3 Periodicity and Similarity

References

Chapter 7: Renormalization Group and Noether Invariants

7.1 Hybrid Lie Self-Similarity/Renormalization Group

References

Chapter 8: Scaling in Hydrodynamical Turbulence

8.1 General Introduction

8.2 Homogeneous, Isotropic, Decaying Turbulence

8.3 Dimensional Phenomenology of Stationary Turbulence

8.4 Structure in 2D Turbulence

References

Epilogue

Appendix: Examples from the Literature

Index

End User License Agreement

Pages

xi

xiii

xv

xvi

xvii

xviii

xix

xx

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

34

28

29

30

31

32

33

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

267

268

269

270

271

272

273

274

275

276

277

278

Guide

Cover

Table of Contents

Preface

Introduction

Begin Reading

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 3.1

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 5.1

Figure 6.1

Figure 7.1

Figure 7.2

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

List of Tables

Table 4.1

Related Titles

Wolf, J.P.

The Scaled Boundary Finite Element Method

2003

Print ISBN: 978-0-471-48682-4 Adobe PDF ISBN: 978-0-470-86148-6

Krantz, W.B.

Scaling Analysis in Modeling Transport and Reaction Processes

A Systematic Approach to Model Building and the Art of Approximation

2007

Print ISBN: 978-0-471-77261-3, also available in digital formats

Logan, J.

Applied Mathematics, Fourth Edition

Fourth Edition 2013

Print ISBN: 978-1-118-47580-5, also available in digital formats

Abry, P., Goncalves, P., Vehel, J. (eds.)

Scaling, Fractals and Wavelets

2008

Print ISBN: 978-1-848-21072-1, also available in digital formats

Richard N. Henriksen

Scale Invariance

Self-Similarity of the PhysicalWorld

Author

Richard N. Henriksen

Queen's University

Department of Physics

Engineering Physics and Astronomy

Kingston, ON

Canada

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.

© 2015 Wiley-VCH Verlag GmbH & Co.

KGaA, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Print ISBN: 978-3-527-41335-5

ePDF ISBN: 978-3-527-68733-6

ePub ISBN: 978-3-527-68735-0

Mobi ISBN: 978-3-527-68736-7

oBook ISBN: 978-3-527-68734-3

To colleagues past, Judith, my parents and my children.

Preface

This book attempts to reconcile a physical understanding of Similarity, Self-Similarity and Scale Invariance with a more formal mathematical description. The latter treatment is implicit in the analysis of differential equations by Lie groups, but here that method is tailored to address more physical problems in an original manner.

The book begins with an extensive presentation of the physical formulation based on Unit invariance. Subsequently, it presents the more mathematical Lie algebra formulation and compares the two methods through physical examples. The Lie algebra method advocated in the book is not restricted to reproducing results found by conventional analyses. It emphasizes that the invariants associated with the rescaling symmetry are the physical content. Moreover, the Lie method is shown to lead to generalizations such as the complete symmetry in conjunction with the Rotation and Translation groups, a technique for coarse graining, anisotropic rescaling, and useful contact with the Renormalization group. The book also demonstrates a novel treatment of scaling symmetry in Special and General Relativity. Familiar complications such as multi-variable Self-Similarity and Self-Similarity of the ‘second kind’ are readily treated and clarified through examples.

The actual application of these symmetries in the literature has been of incomparable importance in building intuition about myriad physical situations. A legion of authors has contributed. This book is ultimately a credit to these authors. In the appendix, a mere sampling of the ingenuity of these works is listed.

Kingston, Ontario

October 2014

Richard N. Henriksen

Acknowledgments

Much of what I understand of Scale Invariance has been learned from astrophysical colleagues. The ingenuity exercised in myriad studies is inspiring. Students have been tolerant. My wife Judith never fails to contribute, encourage and love.

Introduction

The real voyage of discovery consists not in seeking new lands but seeing with new eyes - Marcel Proust

This book is a compendium of Dimensional analysis techniques elaborated both formally and through examples. The examples include elementary illustrations, and are also meant to reveal what this analysis can do for advanced and important current problems. It is difficult for one person to treat examples from all possible disciplines. The examples chosen are, when rigorous, mainly from hydrodynamics, astrophysics and relativity. One hopes that the techniques developed are more universal. In this f03, we try to list the objectives and the accomplishments of each chapter, in order that the reader may have more than an index from which to choose.

Chapter 1 is based on a classical theorem, usually attributed to Buckingham. It discusses the reason for the importance of Dimensional analysis and gives a simple intuitive proof of the theorem. Unlike some other treatments, great stress is placed on forming an accurate ‘catalogue’ of physical quantities that affect a system's behaviour. This disciplined step is essential to obtaining accurate conclusions, and it generally requires substantial physical insight.

Examples in this chapter are taken from engineering fluid problems such as pipe flow (even from the Canadian tar sands or a micrometeorite puncture in a space suit) and flow past rigid bodies. The latter flow allows the f03 of Similarity symmetry between separate systems. Diffusion of vorticity is used to introduce Self-Similarity in a single system. Examples include illustrations from fundamental physics plus speculation (in the spirit of Dimensional analysis) on the metabolic rate variation between mammals. We also use the combined ‘catalogue theorem’ analysis to discuss the drag reduction in the flow past flexible bodies and ponder Leonardo's observation about trees.

Chapter 2 presents the Lie theory approach introduced by Carter and Henriksen (see Ref. [1] in Chapter 2) in the context of Similarity and Self-Similarity. This is a more algorithmic, algebraic, approach than that of Chapter 1. It is important to realize that although most of the results can be obtained in other ways, they normally require more subtle argument than does this algorithmic approach. This technique allows all possible rescaling symmetries to be readily identified, and it can be used (see subsequent chapters) to generalize the usual Self-Similar Rescaling group to include the Poincaré/Galilean group. The Lie algebra affords a useful classification of Similarity and Self-Similarity in terms of a Self-Similarity classification. The ‘class’ is the ratio of temporal to spatial rescaling.

The first two sections of Chapter 2 give the mathematical foundations. They are a sub-set of the Lie group method for solving differential equations (e.g. Bluman and Cole, see Ref. [2] in Chapter 2). The novelty is to introduce Dimensional covectors into the rescaling algebra. All of the conventional Dimensional analysis is hidden in the calculation of these vectors for each physical quantity (including the base manifold of space-time in relativity).

Once these Dimensional covectors are obtained, the variation of any physical object along the ‘direction’ (in base Dimension space) imposed by the Lie Self-Similar algebra may be calculated. The corresponding physical invariants along the symmetry direction are then readily found. This practical simplicity is due to the Lie derivative taking a very simple form, when these invariants are used as the description of the set of physical objects. The set of invariant functions and the relation between them is the generalization of the typical Universal function of a Self-Similar variable (equal here to the Lie group path parameter), which contains the physical essence of a system.

We proceed first by illustrative elementary examples, which in some cases re-work problems of Chapter 1. However, gradually novel applications are introduced. These include a Self-Similar solution of the Schrödinger equation and an f03 to the Self-Similar Symmetry of Burgers equation. Multi-variable Self-Similarity is introduced in the context of the Boltzmann–Poisson (or Vlasov–Poisson) collisionless, self-gravitating, spherical system.

In these introductory two chapters, presenting the two techniques of Dimensional analysis is the principal goal. The results found in the various applications are less authoritative, being principally an examination of one or the other technique in action. Despite sometimes being speculative, they may however be inspirational to the receptive mind.

In Chapter 3, the most general symmetry group of Galilean space-time is discussed with the Lie method, which requires adding the Rescaling group action to that of the Poincaré/Galilean group. In parallel with the analysis of the Poincaré Rescaling group in Galilean space-time, we have extended the analysis to Minkowski space-time. We give a Lie group derivation of the Lorentz boost. We also take the opportunity to digress briefly from our main objective, in order to discuss the Lorentz boost as complex rotation in a complex Minkowski space. This includes the compactification of Minkowski space by stereographic projection.

The Lie technique allows us to readily examine the Minkowski space Self-Similar symmetry as a combination of rotation and Lorentz boost. In this manner, we derive coordinate transformations that express this combined Symmetry. These formulae include the time dilation formulae in various combined examples.

Clues from the discussion of Galilean space-time are used to discover a general type of covariant Self-Similarity in the Riemannian space-time of General Relativity. This Self-Similar symmetry was puzzingly absent for a rather long period, even after the work of Cahill and Taub (see Ref. [7] in Chapter 3). We refer to the abstract description of this Symmetry as kinematic Self-Similarity. It exists also in Galilean space-time.

The more general Self-Similarity in relativistically gravitating matter generalized the homothetic variety that was introduced by Cahill and Taub. In the special case of geometrical spherical symmetry, this results in one of the Lie invariants being a ratio of time to a power of radius. This more general General Relativistic Symmetry corresponds to the general Symmetry known to exist classically. Its previous apparent absence from General Relativity was the source of the puzzle referred to above. Here the algorithmic study has produced a result that generalizes what had been already found by Lynden-Bell and Lemos (see Ref. [10] in Chapter 3).

In order to obtain this result as well as some more elementary cases elsewhere, we were obliged to introduce the notion of ‘hidden constants’. By choosing Units appropriately, one can always remove constants that appear explicitly in the physical equations of a system. However, this will not, in general, remove the constraints that they impose. This is the case when they are essential to the Dimensional coherence of the governing equations. In the context of a Dimensionally consistent Lie group Symmetry, this is not always the case. In developing kinematic Self-Similarity, we used the ability to set c = G = 1, while simultaneously imposing the new Symmetry through Lie derivatives of the Dimensionally coherent metric.

The algorithmic Dimensional analysis allows us to introduce readily the notion of ‘running constants’. These are not usually fundamental constants, but are rather contingent constants, such as a fluid viscosity (including the parameter in Burgers equation), or resistivity in a conducting fluid. However, fundamental coupling constants in field theory are also ‘running’. There is an evident connection in this context between the Rescaling group and the Renormalization group that we explore subsequently.

In Chapter 4, we turn to a collection of more practical problems. These are chosen to illustrate the nuances that arise in the application of rescaling symmetry. Some notions introduced by Barenblatt (see Ref. [2] in Chapter 1) are shown to be treatable by the Lie method. These include describing Self-Similar solutions as ‘intermediate asymptotes’ and the notion of Self-Similarity of the ‘second kind’. We prefer to describe ‘incomplete Self-Similarity’ as multi-variable Self-Similarity, wherein conservation laws and physical intuition can suggest the form of special solutions. This is illustrated in the example of a point source of waves in a rotating medium due to Sobolev.

The unifying concept in all of these examples is the concept of ‘Self-Similarity class’. We give an example where this class may be considered as adiabatically varying in response to boundary conditions that are varied adiabatically. This is a new concept inspired by the Lie technique. This same example indicates numerically how an envelope solution may be a renormalized approximation to a series of perturbative local solutions.

Another novel use of Self-Similarity in this chapter is the f03 of anisotropic ‘classes’. This finds expression in the discussion of the boundary layer on a flat plate due to Blasius. Not all lengths need be scaled similarly. This leads to an expanded dimensional algebra vector wherein a given component is particular to a given spatial dimension. As a practical aside, we are able to suggest analytic approximations to the Blasius boundary-layer equation.

Chapter 5 is really an extension of the examples in Chapter 4 to more advanced problems. We begin with a detailed study of an isotropic, spherically symmetric, collisionless system of particles.

This first example has some scientific interest, but it is used mainly to show how various Self-Similar classes arise from the same set of equations. There is a real advantage gained by solving the system with an initially general class. In particular, the Self-Similar class is sometimes related to the more commonly used ‘polytropic index’, but it also proves to be more general. The polytropes are not in fact globally Self-Similar. Nevertheless, there is a sense in which they are ‘close’ to this symmetry, and a Self-Similar class appears asymptotically. The transformed variables that are the key to our method are themselves both general and easily made ‘close’ to Self-Similar variables. The use of these variables sometimes allows the use of Self-similar class when the system is only partially Self-Similar.

A particular detailed example is the ‘Schuster polytrope’. This demonstrates the novel concept alluded to above in that we are able to study this non-Self-Similar system by using the class concept to remain ‘close’ to Self-Similar symmetry in the group-based variables. The Schuster and the corresponding Self-Similar distribution functions share the same dependence on energy, but the potential differs essentially. Our method gives an unusual form for the Schuster polytrope, which has the virtue that it is easy to verify.

The preceding globally Self-Similar systems are all infinite spheres. The question also treated in this extended discussion is ‘can Self-Similar symmetry be present in finite spheres?’ We show this to be possible by taking the limit of vanishing spatial scaling, that is infinite Self-Similar class. This yields the polytrope with index unity, but with a non-Self-Similar potential. This argument introduces the concept of ‘running’ rescaling algebra components, which is much used subsequently.

The polytropes may be proved to be asymptotically Self-Similar by pursuing the other ‘running’ extreme of infinite spatial scale, that is zero Self-Similar class. This only applies for the polytropes that extend to infinity with infinite mass, but in these cases the asymptotic potential is correctly found. This essentially introduces the idea of a power series solution in powers of the reciprocal scale. This procedure proves useful in later examples and represents a novel kind of ‘coarse graining’. All of this is accomplished by remaining ‘close’ to Self-Similarity through the use of suitably Lie group-based coordinates and variables.

The final study of the finite collisionless sphere is used as an example of Pi theorem Renormalization. We use the Units freedom as a way of quantitatively renormalizing the equations. This leads to a power series approximation to the system. It represents in one case the power law solution of the isothermal sphere, as was found by Chandrasekhar (see Ref. [3] in Chapter 5) using other methods. It turns out to be an example of a development that is exterior to the Lie algebra approach to Self-Similarity.

We turn finally in this chapter to hydrodynamic examples. Couette flow is used to show the effect of a variable viscosity. This viscosity is an example of a running constant. The only true Self-Similar flow without time-dependent boundaries is in fact the steady state. The more general Self-Similar Symmetry in the Couette geometry is restricted to expanding or contracting cylindrical waves.

Perhaps the most important example studied in this chapter is that of the steady, laminar wake, at large distance behind an arbitrarily shaped body. We show that such a wake can be studied algorithmically as another example of anisotropic Self-Similarity. Once this is applied systematically, all standard results (given for example in the text by Landau and Lifshitz) are recovered.

We are able to apply our method to study the time dependence of such a wake while retaining Self-Similar Symmetry. The solution is given in a power series in powers of a reciprocal spatial scale. This is small at large scales. The expansion does not converge in time so that it is only locally valid. This situation is exactly that which may be treated by the envelope renormalization scheme. We apply this procedure to show that one can find a convergent global solution in this way. The wake is found to decay on a convection timescale.

Chapter 6 begins with the formal discussion of anisotropic spatial scaling. Ultimately this is described by generalizing the rescaling algebra component δ to a strain tensor δ. A brief discussion of the generation of distinct, but Similar, anisotropic structures is also given.

A following major section of this chapter compares the more formal mathematical technique of solving partial differential equations using Lie Symmetry ‘generators’ and the technique advocated in this book. This is done in some detail in the context of the Liouville equation. Some well-known simple solutions are re-derived.

The chapter continues with a discussion of discrete Self-Similarity as illustrated by periodic systems. We find that this notion corresponds to complexifying the rescaling algebra. The harmonic, Mathieu, diffusion and wave equations are used as examples.

Chapter 7 addresses the relation between Self-Similar Symmetry and the iterative Renormalization group. We show that the two techniques may be combined to suggest a hybrid Self-Similar/Renormalization group. The idea requires additional exploration and numerical demonstration. However, an abbreviated procedure following Sedov (see Ref. [4] in Chapter 1) is introduced in Chapter 8. The ideas are used to derive the fractal path of a free quantum particle using the Schrödinger equation.

The second part of this chapter compares Self-Similarity with Noether symmetry in the context of Lagrangian mechanics. The Noether conserved quantities are demonstrated to be more general than Self-Similar scale invariance and allow more general solution types. In particular, the Noether integrals exist in conservative systems where a Dimensional constant exists independently of the energy. Ab initio Dimensional Self-similarity is ambiguous in such cases unless appropriate limits are taken.

The ultimate Chapter 8 attempts an overview of the use of scale invariance in turbulence theory. Due respect is paid to the appearance of broken Self-Similarity in higher order structure functions, although no serious solution is offered. Much of the discussion is confined to the second- and third-order structure or correlation functions, where the Rescaling Symmetry is at least approximate. The Lie group analysis is based on the von Karman–Howarth equation. Attempts are made to re-derive both the Kolmogorov second-order velocity scaling and the third-order ‘four fifths’ law. In some derivations of standard phenomenology, the Buckingham method is the most readily used.

The major novelty is to use the extended Lie Symmetry group to study possible structures in 2D turbulence. We generate these as similar structures that are rescaled, rotated and translated. This is achieved starting from a reference solution given in terms of invariant coordinates, whose transformation from original Cartesian coordinates is known at each point on the Lie group path. The reference solution is then stepped along the Lie group path, and a similar solution in Cartesian coordinates is constructed there using the coordinate transformations. In this way, an ensemble of similar structures may be generated. A simple example of only one step is given.

Interaction of the structures in the ensemble must be added separately. If each structure is bounded, we might treat the system as an interacting ensemble of line vortices as an approximate description of the global behaviour. Internally, each vortex will have the same structure. If however each similar flow extends to infinity, then the interaction will be of a ‘sweeping’ kind. That is the motion generated at a neighbouring vortex by a larger and stronger vortex will be convection due to the larger flow. These ideas require elaboration in subsequent work.

1Arbitrary Measures of the Physical World

Abusus non tollit usum (Abuse does not exclude use)

.

1.1 Similarity

It is worth recalling the intuitive notion of Similarity before we begin. We learn about similar triangles in school. Such triangles can be displaced, rotated and dilated (positively or negatively) in order to achieve congruence and thereby be reduced to congruent identity. The dilation is restricted so that each side of the triangle is multiplied by the same factor. This geometric operation can be expressed algebraically by appealing to the theorem of Pythagoras.

In a right-angled triangle of sides , the Pythagorean theorem gives the largest side c as . Any other right-angled triangle is similar to the first triangle in the geometric sense if each side is related to the corresponding side by the same numerical factor. These similar right-angled triangles are said to have the same ‘shape’, independent of rescaling, rotation or translation. This imprecise notion may be henceforth defined operationally as an invariant, subject to these various operations. That is, the ‘shape’ of a right-angled triangle is contained algebraically in the invariance of the Pythagorean theorem under a uniform scale change, plus rigid rotations and displacements.

More complicated geometrical structures also have a ‘shape’ invariant under operations appropriate to their form. A general triangle satisfies the ‘cosine law’ in the form , if is the angle included between the sides . The labelling of the sides is arbitrary so that this law holds for any side in terms of the other two sides and their included angle (a triangle is a zero sum of three vectors and the included angle is interior between two sides). The expression remains true under the same rescaling of all the sides for a given angle. Hence, the shape is invariant under the same operations as above once three sides, or one side and two angles, are specified at any given scale (the sine law helps here).

In analytic geometry, examples abound. If are the Cartesian coordinates of a point on an ellipse of major axis and minor axis , then . The origin is evidently at the centre of the ellipse. An ellipse has the ‘shape’ invariant in its new form , if . That is, once again the invariance is under a uniform spatial rescaling in addition to rigidity.

In all of these geometric examples, the scaling factor might be a function of time, which implies that the sides are ‘rubberized’. In this case, the shape invariance would be between differently expanded (or contracted) versions of the original figure rather than between different figures. In addition, there might be a rigid rotation of the figure, but in any case it would remain similar to itself. This kind of shape invariance is referred to as ‘Self-Similar’.

For example, if the sides of a triangle were to be expanded in time according to the power law , then we could write

1.1

where the subscript o denotes initial quantities, and p