Chapter 4Statistical ThermodynamicsVersion 0404.1.K, 20 October 2004Please send comments, suggestions, and errata via email to [email protected], or onpaper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125I have decided to change notation for the relativistic chemical potential from µRto ˜µ. I apologize that I did not do so in Chapters 2 and 3; I will do so henceforth.— Kip4.1 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 attributes of systems that are in or near equilibrium, such as theirenergy 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, inthe late nineteenth century, Willard Gibbs and others developed statistical mechanics andshowed that it provides a powerful conceptual underpinning for classical thermodynamics.The resultant synthesis, statistical thermodynamics, adds greater power to thermodynamicsby augmenting 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.12The 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 ensembleswill predict the same equation of state P = (N/V )kT 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 )kT with¯N the ensemble average of N. (Here and throughout thischapter, for compactness we use bars rather 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. 3.8. Although the thermodynamicproperties 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. 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, ... . Table 4.2 summarizes those statistical-equilibrium results andsome 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 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 chemical reactions and also drive phase transitions. In Sec. 4.5 we study spontaneousfluctuations of a system away from equilibrium, when it is coupled to a heat and particlebath, and discover how the fundamental potential (in this case Gibbs potential) can be usedto compute the probabilities of such fluctuations. These out-of-equilibrium aspects of statis-tical mechanics (evolution toward equilibrium and fluctuations away from equilibrium) aresummarized in Table 4.2, not just for heat and volume baths, but for a wide variety of otherbaths. Again, readers may wish to delay studying the table until they have read further intothe chapter.Although the conceptual basis of statistical thermodynamics should be quite clear, de-riving quantitative results for real systems can be formidably difficult. In a macroscopicsample, there is a huge number of possible microscopic arrangements (quantum states) andthese all have to be taken into consideration via statistical sums if we want to understand themacroscopic properties of the most frequently occuring configurations. Direct summationover states is hopelessly impractical for real systems. However, in recent years a number ofpowerful approximation techniques have been devised for performing the statistical sums. InSecs. 4.6 and 4.7 we give the reader the flavor of two of these techniques: the renormalization3group and Monte Carlo methods. We illustrate and compare these techniques by using themto study a phase transition in a simple model for Ferromagnetism called the Ising model.4.2 Microcanonical Ensemble and the Energy Repre-sentation of ThermodynamicsConsider a microcanonical ensemble of large, closed systems that have attained statisticalequilibrium. We can describe the ensemble macroscopically using a set of thermodynamicvariables. As we saw in Chap. 3, these variables can be divided into two classes: extensivevariables which double if one doubles the size (volume, mass, . . .) of the system, and intensivevariables whose magnitudes are independent
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