Contents4 Statistical Thermodynamics 14.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Microcanonical Ensemble and the Energy Representation of Thermodynamics 34.2.1 Extensive Variables and Fundamental Potential . . . . . . . . . . . . 34.2.2 Intensive Variables Identified Using Measuring D evices; First Law ofThermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2.3 Euler’s Equation and Form of Fundamental Potential . . . . . . . . . 74.2.4 Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.5 Mechanim of Entropy Increase When Energy is Injected . . . . . . . 94.2.6 Representations of Thermodynamics . . . . . . . . . . . . . . . . . . 94.3 Canonical Ensemble and the Physical-Free-Energy Representation of Ther-modynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3.1 Experimental Meaning of Physical Free Energy . . . . . . . . . . . . 124.4 The Gibbs Representation of Thermodynamics; Phase Tra nsitions and Chem-ical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.1 Minimum Principles for Gibbs and Other Fundamental Thermody-namic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.2 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.3 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Fluctuations of Systems in Statistical Equilibrium . . . . . . . . . . . . . . . 264.6T2 The Ising Model and Renormalization Group Methods . . . . . . . . . 334.7T2 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 39iChapter 4Statistical ThermodynamicsVersion 0804.1.K, 22 October 2008Please send comments, suggestions, and errata via email to [email protected], o r onpaper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125Box 4.1Reader’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 ways 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. 4.5 of this chapter. Portions of Part IV (FluidMechanics) rely on elementary thermodynamic concepts and equations of statetreated in this chapter 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.4.1 Ove rviewIn Chap. 3, we introduced the concept of statistical equilibrium and studied, briefly, someof the properties of equilibrated systems. In this chapter we shall develop t he 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 is12a very powerful theory for deriving general relationships between macroscopic properties.Microphysics influences the macroscopic world in a statistical manner and so, in the lat enineteenth century, Willard Gibbs a nd others developed statistical m echanics and showedthat it provides a powerful conceptual underpinning for classical thermo dynamics. Theresulting synthesis, statistical thermodynamics, adds greater power to thermodynamics byaugmenting 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, . . . . (Note that this constrains the energy of interactionbetween 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 t he statistical equilibrium ensembles of the last chapter(microcanonical, canonical, gr and canonical, Gibbs). For example, each of these ensembleswill 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. (Here and throughout t hischapter, for compactness we use bars r ather than brackets to denote ensemble averages, i.e.¯N rather than hNi)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. Although the thermodynamicproperties are independent of the equilibrium ensemble, specific properties are often derivedmost quickly, and the most insight usually accrues, fro m that ensemble which most closelymatches the physical situation being studied.In Sec. 3.8 we used the grand canonical ensemble, and in Secs. 4.2, 4.3, and 4.4 we shalluse the 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, ... . Ta ble 4.1 summarizes those statistical-equilibrium results a ndsome generalizations of them. Readers may wish to delay studying this table until they haveread 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. 4.4 we formulate the fundamental potential (Gibbs potential) for anout-of-equilibrium ensemble that interacts with a heat and volume bat h, 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,
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