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Scale Invariance: Self-Similarity of the Physical World

ISBN: 978-3-527-41335-5
304 pages
June 2015
Scale Invariance: Self-Similarity of the Physical World (3527413359) cover image

Description

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.
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Table of Contents

Preface XI

Acknowledgments XIII

Introduction XV

1 Arbitrary Measures of the PhysicalWorld 1

1.1 Similarity 1

1.2 Dimensional Similarity 3

1.3 Physical Equations and the ‘Pi’ Theorem 6

1.4 Applications of the PiTheorem 10

1.4.1 Plane Pendulum 11

1.4.2 Pipe Flow of a Fluid 16

1.4.3 Steady Motion of a Rigid Object in Viscous ‘Fluid’ 18

1.4.4 Diffusion and Self-Similarity 20

1.4.5 ShipWave Drag 26

1.4.6 Adiabatic Gas Flow 28

1.4.7 Time-Dependent Adiabatic Flow 30

1.4.8 Point Explosion in a Gaseous Medium 33

1.4.9 Applications in Fundamental Physics 35

1.4.10 Drag on a Flexible Object in Steady Motion 41

1.4.11 Dimensional Analysis of Mammals 42

1.4.12 Trees 47

References 51

2 Lie Groups and Scaling Symmetry 53

2.1 The Rescaling Group 53

2.1.1 Rescaling Physical Objects 55

2.1.2 Reconciliation with the Buckingham PiTheorem 59

2.1.3 Rescaling and Self-Similarity as a Lie Algebra 60

2.1.4 Practical Lie Self-Similarity 63

2.2 Familiar Physical Examples 68

2.2.1 Line Vortex Diffusion: Reprise 69

2.2.2 Burgers’ Equation 71

2.3 Less Familiar Examples 77

2.3.1 Self-Gravitating Collisionless Particles: The Boltzmann-Poisson Problem 77

References 84

3 Poincaré Group Plus Rescaling Group 87

3.1 Galilean Space-Time 87

3.2 Minkowski Space-Time 96

3.2.1 Self-Similar Lorentz Boost 96

3.2.2 Self-Similar Boost/Rotation 102

3.3 Kinematic General Relativity 108

References 119

4 Instructive Classic Problems 121

4.1 Introduction 121

4.2 Ideal Fluid Flow Past aWedge: Self-Similarity of the ‘Second Kind’ 121

4.3 Boundary Layer on a Flat Plate: the Blasius Problem 126

4.4 Adiabatic Self-Similarity in the Diffusion Equation 133

4.5 Waves in a Uniformly Rotating Fluid 140

References 146

5 Variations on Lie Self-Similarity 147

5.1 Variations on the Boltzmann–Poisson System 147

5.1.1 Infinite Self-Gravitating Collisionless Spheres 147

5.1.2 Finite Self-Gravitating Collisionless Spheres 155

5.1.3 Other Approaches to Finite Spheres 159

5.2 Hydrodynamic Examples 164

5.2.1 General Navier–StokesTheory 164

5.2.2 Modified Couette Flow 166

5.2.3 Flow at Large Scale inside a LaminarWake 170

5.3 Axi-Symmetric Ideal Magnetohydrodynamics 178

5.3.1 Incomplete Self-Similarity as Separable Multi-variable Self-Similarity 182

5.3.2 Isothermal Collapse 185

References 187

6 Explorations 189

6.1 Anisotropic Self-Similarity 189

6.1.1 Anisotropic Similarity 192

6.2 Mathematical Variations 193

6.3 Periodicity and Similarity 198

6.3.1 Log Periodicity and Self-Similarity: Diffusion Equation 203

References 207

7 Renormalization Group and Noether Invariants 209

7.1 Hybrid Lie Self-Similarity/Renormalization Group 209

7.1.1 Renormalizing More Complicated Equations 216

7.1.2 Schrödinger: Adiabatic and Fractal 219

7.1.3 Noether Invariants and Self-Similarity 223

References 229

8 Scaling in Hydrodynamical Turbulence 231

8.1 General Introduction 231

8.2 Homogeneous, Isotropic, Decaying Turbulence 232

8.2.1 Third-Order Correlation Negligible 236

8.2.2 Renormalization and Homogeneous, Isotropic, Turbulence 240

8.3 Dimensional Phenomenology of Stationary Turbulence 242

8.4 Structure in 2D Turbulence 246

8.4.1 Similarity of Time-Dependent 2D Vortical Fluid Flow 248

8.4.2 Similarity in Physically Steady, Inviscid Vortical Fluid Flow 258

References 264

Epilogue 267

Appendix: Examples from the literature 269

Index 273

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Author Information

Richard N. (Dick) Henriksen has been a full professor of astrophysics at Queen's University, Ontario, since 1978 and is a recipient of their research excellence award. In addition, he has been a senior visitor at Stanford, a Humboldt Fellow in Germany and Engineur/chercheur at CEA Saclay in France. He was one of the two founders of the Canadian Institute for Theoretical Astrophysics in 1984-85, which resides at the University of Toronto and has become one of the world's premier centres for Astrophysics. He has published more than 125 research papers, many of which employ scaling and relativistic concepts. He is author of the recent Wiley undergraduate textbook entitled "Practical Relativity", Professor Henriksen has extensive lecturing experience, having lectured at all graduate and undergraduate levels in physics, as well as presenting many professional colloquia.
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