ContentsII STATISTICAL PHYSICS ii3KineticTheory 13.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Phase Space and D istribution Function . . . . . . . . . . . . . . . . . . . . . 33.2.1 [N] Newtonian Number density in phase space, N ........... 33.2.2 [N] Distribution function f(x, v,t)forParticlesinaPlasma. . . . . . 43.2.3T2 [R] Relativistic Number Density in Phase Space, N ....... 43.2.4 [N & R] Distribution function Iν/ν3for Photons. . . . . . . . . . . . 93.2.5 [N & R] Mean Occupation Number, η .................. 113.3 [N & R] Thermal-Equilibrium Distribution Functions . . . .......... 153.4 Macroscopic Properties of Matter as Integrals over Momentum Space . . . . 193.4.1 [N] Newtonian Particle Density n,FluxS,andStressTensorT .... 193.4.2T2 [R] Relativistic Number-Flux 4-Vector#S and Stress-Energy Ten-sor T .................................... 203.5 Isotropic Distribution Functions and Equations of State . . . . . . . . . . . . 223.5.1 [N] Newtonian Density, Pressure, Energy Density and Equation of State 223.5.2 [N] Equations of State for a Nonrelativistic Hydrogen Gas . . . . . . 233.5.3T2 [R] Relativistic Density, Pressure, Energy Density and Equationof State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5.4T2 [R] Equation of State for a Relativistic Degenerate Hydrogen Gas 283.5.5 [N & R] Equation of State for Radiation . . . . . . . . . . . . . . . . 293.6 [N & R] Evolution of the Distribution Function: Liouville’s Theorem, theCollisionless Boltzmann Equation, and the Boltzmann Transport Equation . 333.7 [N] Transport Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.7.1 Problem to be Analyzed: Diffusive Heat Conduction Inside a Star . . 413.7.2 Order-of-Magnitude Analysis . . . . . . . . . . . . . . . . . . . . . . 423.7.3 Analysis Via the Boltzmann Transport Equation . . . . . . . . . . . . 43iPart IISTATISTICAL PHYSICSiiStatistical PhysicsVersion 1003.1.K, 12 October 2008In this first Part of the book we shall study aspects of classical statistical physics that ev-ery physicist should know but are not usually treated in elementary thermodynamics courses.Our study will lay the microphysical (particle-scale) foundations for the continuum physicsof Parts II—VI; and it will elucidate the intimate connections between relativistic statisticalphysics and Newtonian theory, and b etween quantum statistical physics and classical the-ory. (The quantum-classical connection is of practical importance; even for fully classicalsystems, a quantum viewpoint can be computationally powerful; see, e.g., Chap. 22.) Asin Chap. 1, our treatment will be so organized that readers who wish to restrict themselvesto Newtonian theory can easily do so. Throughout, we presume the reader is familiar withelementary thermodynamics, but not with other aspects of statistical physics.In Chap. 2 we will study kinetic theory —thesimplestofallformalismsforanalyzingsystems of huge n umbers of particles (e.g., molecules of air, or neutrons diffusing throughanuclearreactor,orphotonsproducedinthebig-bang origin of the Universe). In kinetictheory the key concept is the “distribution function” or “number density of particles inphase space”, N;i.e.,thenumberofparticlesperunit3-dimensionalvolumeofordinaryspace and per unit 3-dimensional volume of momentum space. Despite first appearances, Nturns out to be a geometric, frame-independent entity. This N and the frame-independentlaws it obeys provide us with a means for computing, from microphysics, the macroscopicquantities of continuum physics: mass density, thermal energy density, pressure, equationsof state, thermal and electrical conductivities, viscosities, diffusion coefficients, ... .In Chap. 3 we will develop the foundations of statistical mechanics.Hereourstatisticalstudy will be more sophisticated than in kinetic theory: w e shall deal with “ensembles” ofphysical systems. Each ensemble is a (conceptual) collection of a huge number of physicalsystems that are identical in the sense thattheyhavethesamedegreesoffreedom,butdifferent in that their degrees of freedom may be in different states. For example, thesystems in an ensemble might be balloons that are each filled with 1023air molecules so eachis describable by 3 ×1023spatial coordinates (the x, y, z of all the molecules) and 3 × 1023momentum coordinates (the px, py, pzof all the molecules). The state of one of the balloonsis fully described, then, by 6 ×1023numbers. W e introduce a distribution function N whichis a function of these 6 × 1023different coordinates, i.e., it is defined in a phase space with6 × 1023dimensions. This distribution function tells us ho w many systems (balloons) inour ensemble lie in a unit volume of that phase space. Using this distribution function wewill study such issues as the statistical meaning of entropy, the relationship between entropyiiiivand information, the statistical origin of the second law of thermodynamics, the statisticalmeaning of “thermal equilibrium”, and the evolution of ensembles into thermal equilibrium.Our applications will include derivations of the Fermi-Dirac distribution for fermions inthermal equilibrium and the Bose-Einstein distribution for bosons, a study of Bose-Einsteincondensation in a dilute gas, and explorations of the meaning and role of entropy in gases,in black holes and in the universe as a whole.In Chap. 4 we will use the tools of statistical mechanics to study statistical thermodynam-ics,i.e.ensemblesofsystemsthatareinornearthermal equilibrium (also called statisticalequilibrium). Using statistical mechanics, we shall derive the laws of thermodynamics, andwe shall learn how to use thermodynamic and statistical mechanical tools, hand in hand,to study not only equilibria, but also the probabilities for random, spontaneous fluctuationsaway from equilibrium. Among the applications we shall study are: (i) chemical and particlereactions such as ionization equilibrium in a hot gas, and electron-positron pair formation inastillhottergas;and(ii)phasetransitions,suchasthefreezing,melting,vaporizationandcondensation of water. We shall focus special attention on a Ferromagnetic phase transitionin which the magnetic moments of atoms spontaneously align with each other as iron iscooled, using it to illustrate two elegant and powerful techniques of statistical physics: therenormalization group, and Monte Carlo methods.In Chap.
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