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Explaining Low Entropy EXPLAINING LOW ENTROPY AN EXAMINATION OF THE EFFICACY OF COSMOLOGICAL EXPLANATIONS FOR THE SECOND LAW OF THERMODYNAMICS Miles Alexander Dawborne Thesis Revision 1 MA Philosophical Foundations of Physics Columbia University September 2014 Miles Dawborne Explaining Low Entropy To my mum, who has ever supported my disparate endeavors. MA Thesis 2014 2 of 75 Miles Dawborne Explaining Low Entropy 1 ABSTRACT Given a low-entropy past state, statistical mechanics purportedly explains the Second Law of Thermodynamics. Modern cosmology furnishes this account with a low-entropy early universe. However, what explains this past state? One approach simply postulates this initial condition as a contingent fact or law. Alternatively, one can seek a dynamical solution. This paper assesses the explanatory efficacy some popular proposals. In particular, Carroll's multiverse hypothesis is critiqued and shown to be subject to similar problems faced by Boltzmann. MA Thesis 2014 3 of 75 Miles Dawborne Explaining Low Entropy 2 TABLE OF CONTENTS 1 Abstract ................................................................................................................................................. 3 3 Introduction .......................................................................................................................................... 7 4 The Second Law of Thermodynamics ................................................................................................. 12 4.1 Carnot's engine ........................................................................................................................... 12 4.2 Clausius entropy .......................................................................................................................... 13 4.3 The limits of thermodynamics .................................................................................................... 13 5 Statistical Mechanics ........................................................................................................................... 16 5.1 Boltzmann ................................................................................................................................... 17 5.2 H-theorem ................................................................................................................................... 17 5.3 Problems with H-theorem .......................................................................................................... 19 5.4 Combinatorics ............................................................................................................................. 20 5.4.1 Microstates and macrostates.............................................................................................. 20 5.4.2 Boltzmann entropy ............................................................................................................. 21 5.4.3 Boltzmann dynamics ........................................................................................................... 23 5.4.4 The ergodic argument ......................................................................................................... 24 5.4.5 The typicality argument ...................................................................................................... 25 5.4.6 Approach to equilibrium ..................................................................................................... 25 5.5 Poincaré’s recurrence theorem .................................................................................................. 26 5.6 Loschmidt's reversibility paradox ............................................................................................... 27 6 The Low-Entropy Problem .................................................................................................................. 28 6.1 Two hypotheses .......................................................................................................................... 28 6.2 Boltzmann Brains ........................................................................................................................ 29 7 Modern Cosmology ............................................................................................................................. 30 7.1 The Big Bang ................................................................................................................................ 30 7.2 Inflation ....................................................................................................................................... 31 7.3 Low-entropy early universe ........................................................................................................ 33 8 The Past Hypothesis ............................................................................................................................ 34 8.1 Evidence ...................................................................................................................................... 36 9 Objections to the Past Hypothesis ...................................................................................................... 38 9.1 Early universe equilibrium .......................................................................................................... 38 9.2 The expanding universe .............................................................................................................. 39 MA Thesis 2014 4 of 75 Miles Dawborne Explaining Low Entropy 9.3 Relaxation time ........................................................................................................................... 40 9.4 Introducing gravity ...................................................................................................................... 40 9.4.1 General Relativity ................................................................................................................ 40 9.4.2 Extensivity ........................................................................................................................... 41 9.4.3 Dynamics ............................................................................................................................. 42 9.5 Subsystems ................................................................................................................................. 43 9.6 Lower entropy ............................................................................................................................. 44 9.7 Is the Past Hypothesis necessary? .............................................................................................. 45 10 Alternative hypotheses ................................................................................................................... 46 10.1 Carroll's multiverse ..................................................................................................................... 46 10.2 Problems with the multiverse ..................................................................................................... 48 10.2.1 Boltzmann Brains (again) .................................................................................................... 48 10.2.2 Unfalsifiable ........................................................................................................................ 49 10.2.3 Unnecessary ........................................................................................................................ 49 10.3 Cyclic universe ............................................................................................................................. 50 11 Scientific Explanation ...................................................................................................................... 52 11.1 Probability ................................................................................................................................... 52 11.2 Best System Account ................................................................................................................... 54 11.3 Is an explanation required? ........................................................................................................ 55 12 Conclusion ....................................................................................................................................... 58 13 Appendix ......................................................................................................................................... 60 13.1 Carnot engine .............................................................................................................................. 60 13.2 Clausius entropy .......................................................................................................................... 61 13.3 Thermodynamic entropy calculations ........................................................................................ 63 13.4 The Maxwell–Boltzmann distribution ......................................................................................... 64 13.5 H-theorem ................................................................................................................................... 64 13.6 µ-space ........................................................................................................................................ 66 13.7 Phase-space ...............................................................................................................................
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