Contents5StatisticalThermodynamics 15.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Microcanonical Ensemble and the Energy Representation of Thermodynamics 35.2.1 Extensive Variables and Fundamental Potential . . .......... 35.2.2 Intensive Variables Identified Using Measuring Devices; First Law ofThermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2.3 Euler’s Equation and Form of Fundamental Potential ......... 75.2.4 Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2.5 Mechanim of Entropy Increase When Energy is Injected . . . . . . . 95.2.6 Representations of Thermodynamics . . . . . . . . . . . . . . . . . . 95.3 Canonical Ensemble and the Physical-Free-Energy Representation of Ther-modynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3.1 Experimental Meaning of Physical Free Energy . . . . . . . . . . . . 125.4 The Gibbs Representation of Thermodynamics; Phase Transitions and Chem-ical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4.1 Minimum Principles for Gibbs and Other Fundamental Thermody-namic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4.2 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4.3 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5 Fluctuations of Systems in Statistical Equilibrium ............... 265.6T2 The Ising Model and Renormalization Group Methods . . . . . . . . . 335.7T2 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 39iChapter 5Statistical ThermodynamicsVersion 1005.1.K, 22 October 2008Please 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 (Box 1.4.)• This chapter relies in crucial w ays on Secs. 2.2 and 2.3 of Chap. 2 and on Secs. 3.2–3.9 of Chap. 3.• Portions of Chap. 5 rely on Sec. 5.5 of this chapter. Portions of Part IV (FluidMechanics) rely on elementary thermodynamic concepts and equations of statetreated in this c hapter and Chap. 4, but most readers will already have met thesein a course on elementary thermodynamics.• Other chapters do not depend strongly on this one.5.1 OverviewIn Chap. 3, 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 term“thermodynamics” is an ancient one that predates statistical mechanics. It refers to a studyof the 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” is meant a system that canbe broken into a large number Nssof subsystems that are all macroscopically identical tothe full system except for having 1/Nssas many particles, 1/Nssas much volume, 1/Nssasmuch energy, 1/Nssas much entropy, ....(Notethatthisconstrainstheenergyofinteractionbetween the subsystems to be negligible.) Examples are one 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.3.7.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 Sec. 3.8 we used the grand canonical ensemble, and in Secs. 5.2, 5.3, and 5.4 we shalluse t he microcanonical, canonical and Gibbs ensembles to derive many useful results fromboth classical and statistical thermodynamics: equations of state, Maxwell relations, Euler’sequation, sum-over-states methods for computing fundamental potentials, applications offundamental potentials, ... . Table 5.1 summarizes those statistical-equilibrium results andsome generalizations of them. Readers may wishtodelaystudyingthistableuntiltheyhaveread further into the chapter.As we saw in Chap. 3, 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. 5.4 we formulate the fundamental potential (Gibbs potential) for anout-of-equilibrium ensemble that interacts with a heat and volume bath, we discover a sim-ple relationship between that fundamental potential and the entropy of system plus bath,and from that relationship we learn that the second law, in this case, is equivalent to alaw of decrease of the Gibbs potential. As an application, we learn how chemical potentialsdrive
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