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CALTECH PH 136A - STATISTICAL PHYSICS

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ContentsI STATISTICAL PHYSICS ii2 Kinetic Theory 12.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Phase Space and Distribution Function . . . . . . . . . . . . . . . . . . . . . 22.2.1 [N] Newtonian Number density in phase space, N . . . . . . . . . . . 22.2.2 [N] Distribution function f (x, v, t) for Particles in a Plasma. . . . . . 42.2.3 [R] Relativistic Number D ensity in Phase Space, N . . . . . . . . . . 42.2.4 [R & N] Distribution function Iν/ν3for photons. . . . . . . . . . . . . 92.2.5 [N & R] Mean Occupation Number, η . . . . . . . . . . . . . . . . . . 102.3 [N & R] Thermal-Equilibrium Distribution Functions . . . . . . . . . . . . . 142.4 Macroscopic Properties of Matter as Integrals over Momentum Space . . . . 182.4.1 [N] Newtonian Particle Density n, Flux S, and Stress Tensor T . . . . 182.4.2 [R] Relativistic Number-Flux 4-Vector~S and Stress-Energy Tensor T 192.5 Isotropic Distribution Functions and Equations of Sta te . . . . . . . . . . . . 212.5.1 [N] Newtonian Density, Pressure, Energy Density and Equation of State 212.5.2 [N] Equations of State for a Nonrelativistic Hydrogen Gas . . . . . . 232.5.3 [R] Relativistic Density, Pressure, Energy Density and Equation of State 262.5.4 [R] Equation of State for a Relativistic Degenerate Hydrogen Gas . . 2 72.5.5 [R] Equation of State for Radiation . . . . . . . . . . . . . . . . . . . 282.6 [N & R] Evolution of the Distribution Function: Liouville’s Theorem, theCollisionless Boltzmann Equation, and the Boltzmann Transport Equation . 322.7 [N] Transport Coefficient s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7.1 Problem to be Analyzed: D iffusive Heat Conduction Inside a Star . . 402.7.2 Order-of-Magnitude Analysis . . . . . . . . . . . . . . . . . . . . . . 412.7.3 Analysis Via the Boltzmann Transport Equation . . . . . . . . . . . . 42iPart ISTATISTICAL PHYSICSiiStatistical PhysicsVersion 0802.1.K, 8 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 between quantum statistical physics a nd 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 t heory 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 — the simplest of all formalisms for analyzingsystems of huge numbers of pa rt icles (e.g., molecules of air, or neutrons diffusing througha nuclear reactor, or photons produced in the big-bang origin of the Universe). In kinetictheory the key concept is the “distribution function” or “number density of particles inphase space”, N; i.e., the number of particles per unit 3-dimensional volume of ordinaryspace 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 fra me-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. Here our statisticalstudy will b e more sophisticated than in kinetic theory: we shall deal with “ensembles” ofphysical systems. Each ensemble is a (conceptual) collection of a huge number of physicalsystems that are identical in the sense that they have the same degrees of freedom, 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. We 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 how many systems (balloons) inour ensemble lie in a unit volume of that phase space. Using this distribution function wewill study such issues a s the statistical meaning of entropy, the relationship between entropyiiiivand information, the statistical orig in 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 fo r fermions inthermal equilibrium and the Bose-Einstein distribution for bosons, a study of Bose-Einsteincondensation in a dilute gas, and explorations o f the meaning and r ole 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 therm odynam-ics, i.e. ensembles of systems that are in or near thermal equilibrium (also called statisticalequilibrium). Using statistical mechanics, we shall derive the laws of thermodynamics, andwe shall learn how to use thermo dynamic and statistical mechanical tools, hand in hand,to study not only equilibria, but also the probabilities for random, spo ntaneous 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 for matio n ina still hotter gas; and (ii) phase transitions, such as the freezing, melting, vaporization andcondensation 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


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CALTECH PH 136A - STATISTICAL PHYSICS

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