Randomness And Realism: Encounters With Randomness In The Scientific Search For Physical Reality

Description

Randomness is an active element relevant to all scientific activities. The book explores the way in which randomness suffuses the human experience, starting with everyday chance events, followed by developments into modern probability theory, statistical mechanics, scientific data analysis, quantum mechanics, and quantum gravity. An accessible introduction to these theories is provided as a basis for going into deeper topics. Fowler unveils the influence of randomness in the two pillars of science, measurement and theory. Some emphasis is placed on the need and methods for optimal characterization of uncertainty. An example of the cost of neglecting this is the St. Petersburg Paradox, a theoretical game of chance with an infinite expected payoff value. The role of randomness in quantum mechanics reveals another particularly interesting finding: that in order for the physical universe to function as it does and permit conscious beings within it to enjoy sanity, irreducible randomness is necessary at the quantum level. The book employs a certain level of mathematics to describe physical reality in a more precise way that avoids the tendency of nonmathematical descriptions to be occasionally misleading. Thus, it is most readily digested by young students who have taken at least a class in introductory calculus, or professional scientists and engineers curious about the book's topics as a result of hearing about them in popular media. Readers not inclined to savor equations should be able to skip certain technical sections without losing the general flow of ideas. Still, it is hoped that even readers who usually avoid equations will give those within these pages a chance, as they may be surprised at how potentially foreboding concepts fall into line when one makes a legitimate attempt to follow a succession of mathematical implications. Table of Contents Preface Chapter 1 Nothing Ventured, Nothing Gained 1.1 Beginnings 1.2 Mathematical Formulation of Probability 1.3 Intuition Regarding Probability 1.4 Independent Events 1.5 Different Interpretations of Probability 1.6 Sucker Bets 1.7 Russian Roulette 1.8 The St. Petersburg Paradox 1.9 The Standard Deviation and Statistical Significance 1.10 The Origin of Fluctuations 1.11 Microstates, Macrostates, and Entropy 1.12 The Law of Averages 1.13 Lotteries and Spreads 1.14 Evolution Happens Chapter 2 Classical Mathematical Probability 2.1 Quantifying Randomness and Probability 2.2 Probability Mass Distributions 2.3 Probability Density Functions 2.4 The Gaussian Distribution 2.5 Gaussian Relatives 2.6 Products and Ratios of Gaussian Random Variables 2.7 The Poisson Distribution 2.8 Bayes’ Theorem 2.9 Population Mixtures 2.10 Correlated Random Variables 2.11 Sample Statistics 2.12 Summary Chapter 3 Classical Statistical Physics 3.1 Probability Distributions Become Relevant to Physics 3.2 Thermodynamic Foundations of Thermal Physics 3.3 Statistical Mechanics 3.4 Relation Between Clausius Entropy and Boltzmann Entropy 3.5 Entropy of Some Probability Distributions 3.6 Brownian Motion 3.7 Reconciling Newtonian Determinism With Random Walks Through Phase Space 3.8 Carrying Statistical Mechanics into Quantum Mechanics Chapter 4 Scientific Data Analysis 4.1 The Foundation of Science: Quantified Measurements 4.2 Putting Measurements to Work 4.3 Hypothesis Testing 4.4 Hypothesis Testing Example 1: Matching Sources in Two Catalogs With Gaussian Errors 4.5 Hypothesis Testing Example 2: Matching Sources With Non-Gaussian/Variable Errors 4.6 Monte Carlo Simulations 4.7 Systematic Errors 4.8 The Parameter Refinement Theorem 4.9 Curve Fitting 4.10 Random-Walk Interpolation 4.11 Summary Chapter 5 Quantum Mechanics 5.1 Interpreting Symbolic Models 5.2 Clouds on the Classical Horizon 5.3 The Language of Quantum Mechanics 5.4 The Discovery of Quantized Energy 5.5 Gradual Acceptance of Quantization 5.6 The Schrödinger Wave Equations 5.7 Wave Packets 5.8 The Heisenberg Uncertainty Principle 5.9 The Born Interpretation of Wave Mechanics 5.10 Probability Amplitude, Quantum Probability, and Interference Between Coherent States 5.11 Quantum Entanglement and Nonlocality 5.12 Hidden-Variable Theories and Bell’s Inequalities 5.13 Nonlocal Effects and Information Transfer 5.14 “Instantaneous” Nonlocal Effects 5.15 Wave Function “Collapse” and Various Alternatives 5.16 The Status of Nonlocal Hidden-Variable Theories 5.17 Summary Chapter 6 The Quest for Quantum Gravity 6.1 Fields and Field Quantization 6.2 Essential Features of General Relativity 6.3 Reasons Why a Quantum Gravity Theory Is Needed 6.4 Miscellaneous Approaches 6.5 Canonical Quantum Gravity 6.6 String Theories 6.7 Loop Quantum Gravity and Causal Dynamical Triangulations 6.8 Spacetime Phase Transitions, Chaotic Automata, and Block Universes 6.9 Embeddings and Intrinsic vs. Extrinsic Curvature 6.10 Summary Epilogue Appendix A Moments of a Distribution Appendix B Functions of Random Variables Appendix C M Things Taken K at a Time With Replacement Appendix D Chi-Square Minimization Appendix E Chi-Square Distributions Appendix F Quantiles, Central Probabilities, and Tails of Distributions Appendix G Generating Correlated Pseudorandom Numbers Appendix H The Planck Parameters Appendix I Estimating the Mean of a Poisson Population From a Sample Appendix J Bell’s Theorem and Bell’s Inequalities Appendix K The Linear Harmonic Oscillator References Index