Contents5StatisticalThermodynamics 15.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Microcanonical Ensemble and the Energy Representation of Thermodynamics 35.2.1 Extensive and Intensive Variables; Fundamental Potential ...... 35.2.2 Intensive Variables Identified Using Measuring Devices; First Law ofThermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2.3 Euler’s Equation and Form of the Fundament al Potential ....... 85.2.4 Everything Deducible from First Law; Maxwell Relations . . . . . . . 85.2.5 Mechanism of Entropy Increase When Energy is Injected . . . . . . . 95.2.6 Representations of Thermodynamics . . . . . . . . . . . . . . . . . . 105.3 Grand Canonical Ensemble and the Grand Potential Representation of Ther-modynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3.1 The Grand Potential Representation, and Computation of Thermody-namic Properties as a Grand Canonical Sum . . . . . . . . . . . . . . 115.3.2 Nonrelativistic van der Waals Gas . . . . . . . . . . . . . . . . . . . . 155.4 Canonical Ensemble and the Physical-Free-Energy Representation of Ther-modynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4.1 Experimental Meaning of Physical Free Energy . . . . . . . . . . . . 225.5 The Gibbs Representation of Thermodynamics; Phase Transitions and Chem-ical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.5.1 Minimum Principles for Gibbs and Other Fundamental Thermody-namic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5.2 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5.3 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.6 Fluctuations of Systems in Statistical Equilibrium ............... 365.7T2 Renormalization Group Methods for The Ising Model of a FerromagneticPhase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.8T2 Monte Carlo Methods for the Ising Model . . . . . . . . . . . . . . . . 50iChapter 5Statistical ThermodynamicsVersion 1105.2.K, 12 October 2011Please send comments, suggestions, and errata via email to [email protected], or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 5.1Reader’s Guide• Relativity enters into portions of this chapter solely via the relativistic energies andmomenta of high-speed particles (Sec. 1.10.)• This chapter relies in crucial ways on Secs. 3.2 and 3.3 of Chap. 3 and on Secs. 4.2–4.8 of Chap. 4.• Portions of Chap. 6 rely on Sec. 5.6 of this chapter. Portions of Part V (FluidMechanics) rely on elementary thermodynamic concepts and equations of statetreated in this chapter, but most readers will already have met these in a courseon elementary thermodynamics.• Other chapters do not depend strongly on this one.5.1 OverviewIn Chap. 4, we introduced the concept of statistical equilibrium and studied, briefly, someof the properties of equilibrated systems. In this chapter we shall develop the theory ofstatistical equilibrium in a more thorough way. The title of this chapter, “Statistical Ther-modynamics,” emphasizes two aspects of the theory of statistical equilibrium. The termthermodynamics is an ancient one that predates statistical mechanics. It refers to a study ofthe macroscopic properties of systems that are in or near equilibrium, such as their energyand entropy. Despite paying no attention to the microphysics, classical thermodynamics is12averypowerfultheoryforderivinggeneralrelationships between macroscopic prop erties.Microphysics influences the macroscopic worldinastatisticalmannerandso,inthelatenineteenth century, Willard Gibbs and others developed statistical mechanics and showedthat it provides a powerful conceptual underpinning for classical thermodynamics. Theresulting synthesis, statistical thermodynamics,addsgreaterpowertothermodynamicsbyaugmenting to it the statistical tools of ensembles and distribution functions.In our study of statistical thermodynamics we shall restrict attention to an ensemble oflarge systems that are in statistical equilibrium.By“large”ismeantasystemthatcanbebroken into a large number Nssof subsystems that are all macroscopically identical to thefull system except for having 1/Nssas many particles, 1/Nssas much volume, 1/Nssas muchenergy, 1/Nssas much entropy, ... .(Notethatthisconstrainstheenergyofinteractionbetween the subsystems to be negligible.) Examples are o ne kilogram of plasma in the centerof the sun and a one kilogram sapphire crystal.The equilibrium thermodynamic properties of any type of large system (e.g. a monatomicgas) can be derived using any one of the statistical equilibrium ensembles of the last chapter(microcanonical, canonical, grand canonical,Gibbs). Forexample,eachoftheseensembleswill predict the same equation of state P =(N/V )kBT for an ideal monatomic gas, eventhough in one ensemble each system’s number of particles N is precisely fixed, while inanother ensemble N can fluctuate so that strictly speaking one should write the equationof state as P =(¯N/V )kBT with¯N the ensemble average of N.(Hereandthroughoutthischapter, for compactness we use bars rather thanbracketstodenoteensembleaverages,i.e.¯N rather than !N")The equations of state are the same to very high accuracy because the fractional fluctua-tions of N are so extremely small, ∆N/N ∼ 1/√¯N;cf.Ex.5.9. Althoughthethermodynamicproperties are independent of the equilibrium ensemble, specific properties are often derivedmost quickly, and the most insight usually accrues, from that ensemble which most closelymatches the physical situation being studied.In Secs. 5.2–5.5, we shall use the microcanonical, grand canonical, canonical and Gibbsensembles to derive many useful results from statistical thermodynamics: fundamental po-tentials expressed as statistical sums over microstates, variants of the first law of thermody-namics, equations of state, Maxwell relations, Euler’s equation, ... . Table 5.1 summarizesthe most important of those statistical-equilibrium results and some generalizations of them.Readers may wish to delay studying this table until they have read further into the chapter.As we saw in Chap. 4, when systems are out of statistical equilibrium, their evolutiontoward equilibrium is driven by the law of entropy increase—the second law of thermo-dynamics. In Sec. 5.5
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