## Physics and Chance: Philosophical Issues in the Foundations of Statistical MechanicsStatistical mechanics is one of the crucial fundamental theories of physics, and in his new book Lawrence Sklar, one of the pre-eminent philosophers of physics, offers a comprehensive, non-technical introduction to that theory and to attempts to understand its foundational elements. Among the topics treated in detail are: probability and statistical explanation, the basic issues in both equilibrium and non-equilibrium statistical mechanics, the role of cosmology, the reduction of thermodynamics to statistical mechanics, and the alleged foundation of the very notion of time asymmetry in the entropic asymmetry of systems in time. The book emphasises the interaction of scientific and philosophical modes of reasoning, and in this way will interest all philosophers of science as well as those in physics and chemistry concerned with philosophical questions. The book could also be read by an informed general reader interested in the foundations of modern science. |

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### Contents

Introduction | 1 |

II The structure of this book | 7 |

2 Statistical explanation | 8 |

3 The equilibrium problem | 9 |

5 Cosmology and statistical mechanics | 10 |

7 The direction of time | 11 |

Historical sketch | 14 |

I Thermodynamics | 15 |

IV Further readings | 194 |

Describing nonequilibrium | 196 |

II General features of the ensemble approach | 199 |

2 Initial ensembles and dynamical laws | 202 |

III Approaches to the derivation of kinetic behavior | 207 |

2 The master equation approach | 210 |

3 The approach using coarsegraining and a Markov process assumption | 212 |

IV General features of the rationalization program for the nonequilibrium theory | 215 |

2 Conservation and irreversibility | 16 |

3 Formal thermostatics | 20 |

4 Extending thermodynamics | 22 |

II Kinetic theory | 28 |

2 Maxwell | 30 |

3 Boltzmann | 32 |

4 Objections to kinetic theory | 34 |

5 The probabilistic interpretation of the theory | 37 |

6 The origins of the ensemble approach and of ergodic theory | 44 |

III Gibbs statistical mechanics | 48 |

2 The thermodynamic analogies | 51 |

3 The theory of nonequilibrium ensembles | 53 |

IV The critical exposition of the theory of P and T Ehrenfest | 59 |

1 The Ehrenfests on the Boltzmannian theory | 60 |

2 The Ehrenfests on Gibbs statistical mechanics | 67 |

V Subsequent developments | 71 |

2 Rationalizing the equilibrium theory | 76 |

3 The theory of nonequilibrium | 81 |

4 Rationalizing the nonequilibrium theory | 86 |

VI Further readings | 88 |

Probability | 90 |

I Formal aspects of probability | 91 |

2 Some consequences of the basic postulates and definitions | 93 |

3 Some formal aspects of probability in statistical mechanics | 95 |

II Interpretations of probability | 96 |

1 Frequency proportion and the long run | 97 |

2 Probability as a disposition | 99 |

3 Probability as a theoretical term | 102 |

4 Objective randomness | 108 |

5 Subjectivist accounts of probability | 110 |

6 Logical theories of probability | 117 |

III Probability in statistical mechanics | 120 |

IV Further readings | 127 |

Statistical explanation | 128 |

2 Explanation as subsumption under generality | 131 |

3 Subsumption causation and mechanism and explanation | 140 |

II Statistical explanation in statistical mechanics | 148 |

III Further readings | 154 |

Equilibrium theory | 156 |

2 The Ergodic Hypothesis and its critique | 159 |

3 Khinchins contribution | 162 |

II The Development of contemporary ergodic theory | 164 |

2 Sufficient conditions for ergodicity | 167 |

3 The KAM Theorem and the limits of ergodicity | 169 |

III Ergodicity and the rationalization of equilibrium statistical mechanics | 175 |

1 Ensemble probabilities time probabilities and measured quantities | 176 |

2 The uniqueness of the invariant probability measure | 179 |

3 The set of measure zero problem | 182 |

4 Ergodicity and equilibrium theory in the broader nonequilibrium context | 188 |

5 The Objective Bayesian approach to equilibrium theory | 190 |

V Further readings | 217 |

Rationalizing nonequilibrium theory | 219 |

2 Computer modeling of dynamical systems | 222 |

II Rationalizing three approaches to the kinetic equation | 224 |

2 The generalized master equation | 228 |

3 Beyond ergodicity | 232 |

4 Representations obtained by nonunitary transformations | 242 |

5 Macroscopic chaos | 244 |

III Interpretations of irreversibility | 246 |

2 Interventionist approaches | 250 |

3 Jaynes subjective probability approach | 255 |

4 The mainstream approach to irreversibility and its fundamental problems | 260 |

5 Krylovs program | 262 |

6 Prigogines invocation of singular distributions for initial ensembles | 269 |

7 Conflicting rationalizations | 277 |

IV The statistical explanation of nonequilibrium behavior | 279 |

1 Probabilities as features of collections of systems | 281 |

2 Probabilities as features of states of individual systems | 288 |

3 Initial conditions and symmetrybreaking | 293 |

V Further readings | 295 |

Cosmology and irreversibility | 297 |

2 Big Bang cosmologies | 300 |

3 Expansion and entropy | 303 |

4 Radiation asymmetry and cosmology | 305 |

II Conditions at the initial singularity | 307 |

2 Accounting for the initial lowentropy state | 309 |

III Branch systems | 318 |

2 What cosmology and branch systems cant do | 319 |

IV Further readings | 331 |

The reduction of thermodynamics to statistical mechanics | 333 |

2 Conceptbridging and identification | 337 |

3 The problem of radically autonomous concepts | 341 |

II The case of thermodynamics and statistical mechanics | 345 |

2 Connecting the concepts of the two theories | 348 |

III Problematic aspects of the reduction | 361 |

2 The emergence of thermal features | 367 |

IV Further readings | 373 |

The direction of time | 375 |

II Asymmetry of time or asymmetries in time? | 378 |

1 Symmetries of laws and symmetries of spacetime | 379 |

2 Entropic asymmetry and the asymmetry of time | 382 |

III What is the structure of the Boltzmann thesis? | 385 |

2 What is the nature of the proposed entropic theory of the intuitive asymmetries? | 387 |

3 Sketches of some entropic accounts | 396 |

4 Our inner awareness of time order | 404 |

IV Further readings | 411 |

The current state of major questions | 413 |

421 | |

429 | |

### Common terms and phrases

approach to equilibrium appropriate argument assumption behavior Boltzmann equation branch systems canonical ensemble causal Chapter characterize claim coarse-grained entropy components concepts constraints correlation cosmology derive direction dynamic evolution dynamical laws Ehrenfests energy entropic asymmetry entropic increase ergodic Ergodic Hypothesis ergodic theory evolve existence explanatory fact finite frequency function fundamental future Gibbs given idealization individual systems infer initial conditions initial ensemble interaction intuitive invariant irreversibility KAM Theorem kind kinetic equation kinetic theory large number lawlike low entropy macroscopic master equation measure zero micro-canonical micro-states molecular molecules nature non-equilibrium theory notion original outcome parameters particles phase average phase points phase space phenomena physical posit possible postulates Principle of Indifference probabilistic probability distribution probability measure problem quantities quantum mechanics question radiation random rationalization reducing theory relation relative rerandomization sense space-time statistical explanation statistical mechanics structure temperature Theorem thermodynamic limit time-asymmetry tion trajectories universe velocity

### Popular passages

Page 440 - is the James B. and Grace J. Nelson Professor of Philosophy at the University of Michigan. He is the author of Space, Time, and Spacetime