
Contents
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1. Introduction 1. Introduction
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1.1 The Discovery and Invention of Materials 1.1 The Discovery and Invention of Materials
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1.2 Different Phases; Different Properties 1.2 Different Phases; Different Properties
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1.2.1 Broken Symmetries and Order Parameters 1.2.1 Broken Symmetries and Order Parameters
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1.2.2 Dynamics and Equilibrium 1.2.2 Dynamics and Equilibrium
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1.2.3 Phase Transitions 1.2.3 Phase Transitions
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1.2.4 The First Mean Field Theory 1.2.4 The First Mean Field Theory
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1.2.5 Maxwell's Improvement 1.2.5 Maxwell's Improvement
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1.3 Fluctuations 1.3 Fluctuations
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1.3.1 Fluctuations I: Boiling 1.3.1 Fluctuations I: Boiling
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1.3.2 Fluctuations II: Critical Opalescence 1.3.2 Fluctuations II: Critical Opalescence
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1.4 Ornstein and Zernike 1.4 Ornstein and Zernike
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1.5 Outline of Essay 1.5 Outline of Essay
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2. The Ising Model 2. The Ising Model
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2.1 Definition 2.1 Definition
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2.2 The Extended Singularity Theorem 2.2 The Extended Singularity Theorem
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3. More Is the Same 3. More Is the Same
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3.1 One Spin 3.1 One Spin
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3.2 Many Spins; Mean Fields 3.2 Many Spins; Mean Fields
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3.3 Mean Field Results 3.3 Mean Field Results
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3.4 Representing Critical Behavior by Power Laws 3.4 Representing Critical Behavior by Power Laws
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4. The Year 1937: A Revolution Begins 4. The Year 1937: A Revolution Begins
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4.1 Landau's Generalization 4.1 Landau's Generalization
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4.2 Summary of Mean Field Theories 4.2 Summary of Mean Field Theories
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4.3 Away from Corresponding States—Toward Universality 4.3 Away from Corresponding States—Toward Universality
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4.4 Statistical Confusion: A Meeting in the Netherlands 4.4 Statistical Confusion: A Meeting in the Netherlands
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5. Beyond (or Beside) Mean Field Theory 5. Beyond (or Beside) Mean Field Theory
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5.1 Experimental Facts 5.1 Experimental Facts
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5.2 Theoretical Facts 5.2 Theoretical Facts
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5.3 Spatial Structures 5.3 Spatial Structures
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6. New Foci; New Ideas 6. New Foci; New Ideas
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6.1 Bureau of Standards Conference 6.1 Bureau of Standards Conference
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6.2 Correlation Function Calculations 6.2 Correlation Function Calculations
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6.3 Widom Scaling 6.3 Widom Scaling
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6.4 Less Is the Same: Block Transforms and Scaling 6.4 Less Is the Same: Block Transforms and Scaling
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7. The Wilson Revolution 7. The Wilson Revolution
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7.1 Physical Space; Fourier Space 7.1 Physical Space; Fourier Space
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7.2 Wilson's Contribution 7.2 Wilson's Contribution
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7.3 Building upon the Revolution 7.3 Building upon the Revolution
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7.3.1 The ε-expansion 7.3.1 The ε-expansion
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7.4 Different Kinds of Fixed Points 7.4 Different Kinds of Fixed Points
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8. New Concepts 8. New Concepts
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8.1 Different Scalings: Relevant, Irrelevant, Marginal 8.1 Different Scalings: Relevant, Irrelevant, Marginal
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8.2 Universality Classes 8.2 Universality Classes
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8.3 New Kinds of Answers 8.3 New Kinds of Answers
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8.4 Flows and Flow Diagrams 8.4 Flows and Flow Diagrams
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8.5 The Renormalization Group Is Not a Group 8.5 The Renormalization Group Is Not a Group
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8.6 A Calculational Method Defines Many Worlds 8.6 A Calculational Method Defines Many Worlds
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8.7 Extended Singularities Revisited 8.7 Extended Singularities Revisited
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Acknowledgment Acknowledgment
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References References
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4 Theories of Matter: Infinities and Renormalization
Get accessLeo P. Kadanoff is a theoretical physicist and applied mathematician who has contributed widely to research in the properties of matter, the development of urban areas, statistical models of physical systems, and the development of chaos in simple mechanical and fluid systems. His best-known contribution was in the development of the concepts of "scale invariance" and "universality" as they are applied to phase transitions. More recently, he has been involved in the understanding of singularities in fluid flow.
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Published:05 September 2013
Cite
Abstract
This chapter examines the theory underlying the science of materials, evaluates the progress in the understanding of the thermodynamic phases of matter, and discusses condensed matter physics and the idea that changes in phase involve the presence of a mathematical singularity. It also argues that the understanding of the behavior of systems at and near phase transitions requires a synthesis between standard statistical mechanical uses of probabilities and concepts from dynamical systems theory.
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