Contents4StatisticalMechanics 14.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Systems, Ensembles, and Distribution Functions . . . . . . . . . . . . . . . . 34.2.1 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2.2 Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2.3 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.3 Liouville’s Theorem and the Evolution of the Distribution Function ..... 104.4 Statistical Equilibrium .............................. 124.4.1 Canonical Ensemble and Distribution . . . . . . . . . . . . . . . . . . 124.4.2 General Equilibrium Ensemble and Distribution; Gibbs Ensemble; GrandCanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4.3 Bose-Einstein and Fermi-Dirac Distributions . . . . . . . . . . . . . . 174.4.4 Equipartition Theorem for Quadratic, Classical Degrees of Freedom . 194.5 The Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 The Ergodic Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 Entropy and Evolution Into Statistical Equilibrium .............. 234.7.1 Entropy and the Second Law of Thermodynamics . . . . . . . . . . . 234.7.2 What Causes the Entropy to Increase? . . . . . . . . . . . . . . . . . 254.8 Entropy Per Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.9 Bose-Einstein Condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.10T2 Statistical Mechanics in the Presence of Gravity . . . . . . . . . . . . . 404.10.1T2 Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.10.2T2 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.10.3 The Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.10.4T2 Structure Formation in the Expanding Universe: Violent Relax-ation and Phase Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 484.11T2 Entropy and Information . . . . . . . . . . . . . . . . . . . . . . . . . . 494.11.1T2 Information Gained When Measuring the State of a System in aMicrocanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 494.11.2T2 Information in Communication Theory . . . . . . . . . . . . . . 504.11.3 T2 Examples of Information Content . . . . . . . . . . . . . . . . . 524.11.4T2 Some Properties of Information . . . . . . . . . . . . . . . . . . 53iii4.11.5T2 Capacity of Communication Channels; Erasing Information fromComputer Memories . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Chapter 4Statistical MechanicsVersion 1104.3.K by Kip, October 14, 2011.Please send comments, suggestions, and errata via email to [email protected] or on paper toKip T horne, 350-17 Caltech, Pasadena CA 91125Box 4.1Reader’s Guide• Relativity enters into portions of this chapter solely via the relativistic energiesand momenta of high-speed particles (Sec. 1.10). We presume that all readers arefamiliar with at least this much relativity and accordingly, we do not provide aNewtonian track through this chapter. We will make occasional additional sideremarks for the benefit of relativistic readers, but a Newtonian reader’s failure tounderstand them will not compromise mastering all of this chapter’s material.• This chapter relies in crucial ways on Secs. 3.2, 3.3, 3.5.1, 3.5.2, 3.5.5, and 3.6, andBox 3.3 of Chap. 3.• Chapter 5 is an extension of this chapter.TounderstanditandportionsofChap.6,one must master the fundamental concepts of statistical mechanics in this chapter(Secs. 4.2–4.8).• Other chapters do not depend strongly on this one.4.1 OverviewWhile kinetic theory (Chap. 3) gives a powerful description of some statistical features ofmatter, other features are outside its realm and must be treated using the more sophisticatedtools of statistical mechanics. Examples are:(i) Correlations:Kinetictheory’sdistributionfunctionN tells us, on average, how manyparticles will occupy a giv en phase-space volume, but it says nothing about whether the12particles like to clump, or avoid each other. It is therefore inadequate to describe, e.g., thedistributions of galaxies and stars, which cluster under their mutual gravitational attraction(Sec. 4.10.1), or that of electrons in a plasma, which are mutually repulsive and thus arespatially anti-correlated (Sec. 21.6).(ii) Fluctuations:Inexperimentstomeasureaveryweakmechanicalforce(e.g.testsofthe equivalence principle and searches for gravitational waves), one typically monitors themotion of a pendulum’s test mass, on which the force acts. Molecules of gas hitting thetest mass also make it move. Kinetic theory predicts how many molecules will hit in onemillisecond, on average, and how strong is the resulting pressure acting in all directions; butkinetic theory’s distribution function N cannot tell us the probability that in one millisecondmore molecules will hit one side of the test mass than the other, mimicking the force to bemeasured. The probability distribution for fluctuations is an essential tool for analyzingthe noise in this and any other physical experiment, and it falls in the domain of statisticalmechanics, not kinetic theory.(iii) Strongly interacting particles:Asshouldbefamiliar,thethermalmotionsofanioniccrystal are best described not in terms of individual atoms (as in the “Einstein theory”), butinstead by decomposing the atoms’ motion into normal modes (phonons; “Debye theory”).The thermal excitation of phonons is governed by statistical mechanics.(iv) Microscopic origin of thermodynamic laws:Thelawsofclassicalthermodynamicscanbe (and often are) derived from a few elementary, macroscopic postulates without any ref-erence to the microscopic, atomic nature of matter. …
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