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**Unformatted text preview: **Chapter 4 Statistical Thermodynamics Version 0203.2, 23 Oct 02 Please send comments, suggestions, and errata via email to [email protected] and [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125 4.1 Overview In Chap. 3, we introduced the concept of statistical equilibrium and studied, briefly, some of the properties of equilibrated systems. In this chapter we shall develop the theory of statistical 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 study of the macroscopic attributes of systems that are in or near equilibrium, such as their energy and entropy. Despite paying no attention to the microphysics, classical thermody- namics is a very powerful theory for deriving general relationships between these attributes. However, microphysics influences macroscopic properties in a statistical manner and so, in the late nineteenth century, Willard Gibbs and others developed statistical mechanics and showed that it provides a powerful conceptual underpinning for classical thermodynamics. The resultant synthesis, statistical thermodynamics , adds greater power to thermodynamics by augmenting to it the statistical tools of ensembles and distribution functions. In our study of statistical thermodynamics we shall restrict attention to an ensemble of large systems that are in statistical equilibrium. By “large” is meant a system that can be broken into a large number N ss of subsystems that are all macroscopically identical to the full system except for having 1 /N ss as many particles, 1 /N ss as much volume, 1 /N ss as much energy, 1 /N ss as much entropy, . . . . (Note that this constrains the energy of interaction between the subsystems to be negligible.) Examples are one kilogram of plasma in the center of the sun and a one kilogram sapphire crystal. The equilibrium thermodynamics of any type of large system (e.g. a monatomic gas) can be derived using any one of the statistical equilibrium ensembles of the last chapter (microcanonical, canonical, grand canonical, Gibbs). For example, each of these ensembles will predict the same equation of state P = ( N/V ) kT for an ideal monatomic gas, even though in one ensemble each system’s number of particles N is precisely fixed, while in 1 2 another ensemble N can fluctuate so that strictly speaking one should write the equation of state as P = ( ¯ N/V ) kT with ¯ N the ensemble average of N . (Here and throughout this chapter, for compactness we use bars rather than brackets to denote ensemble averages, i.e....

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