Contents8.7Hydrodynamics of Binary Mixtures3118.7.1Entropy Production in Binary Mixtures312?8.7.2Fick'sLawforDiffusion3158.7.3ThermalDiffusion 3178.8Thermoelectricity3188.8.1The Peltier Effect 3188.8.2The SeebeckEffect3208.8.3ThomsonHeat3218.9322Superfluid Hydrodynamics8.9.1Superfluid Hydrodynamic Equations3228.9.2Sound Modes3268.10Problems3299Transport Coefficients3339.1Introduction3339.2ElementaryTransportTheory3349.2.1TransportofMolecularProperties3389.2.2The Rate of Reaction 3399.3The BoltzmannEquation3419.3.1DerivationoftheBoltzmannEquation3429.4Linearized Boltzmann Equations for Mixtures3439.4.1KineticEquationsforaTwo-ComponentGas3449.4.2CollisionOperators3469.5348Coefficientof Self-Diffusion9.5.1348Derivationof theDiffusionEquation9.5.2349EigenfrequenciesoftheLorentz-BoltzmannEquation9.6Coefficients of Viscosity and Thermal Conductivity3519.6.1DerivationoftheHydrodynamicEquations3519.6.2EigenfrequenciesoftheBoltzmannEquation3559.6.3ShearViscosityandThermalConductivity3589.7Computationof TransportCoefficients3599.7.1SoninePolynomials3599.7.2Diffusion Coefficient 3609.7.3361Thermal Conductivity9.7.4ShearViscosity3639.8Beyond the Boltzmann Equation3659.9Problems36610Nonequilibrium Phase Transitions36910.1Introduction36910.2370Near Equilibrium Stability Criteria10.3The Chemically Reacting Systems37210.3.1TheBrusselator-ANonlinearChemical Model37310.3.2BoundaryConditions37410.3.3StabilityAnalysis 375
Contents IX 8.7 Hydrodynamics of Binary Mixtures 311 8.7.1 Entropy Production in Binary Mixtures 312 8.7.2 Fick’s Law for Diffusion 315 8.7.3 Thermal Diffusion 317 8.8 Thermoelectricity 318 8.8.1 The Peltier Effect 318 8.8.2 The Seebeck Effect 320 8.8.3 Thomson Heat 321 8.9 Superfluid Hydrodynamics 322 8.9.1 Superfluid Hydrodynamic Equations 322 8.9.2 Sound Modes 326 8.10 Problems 329 9 Transport Coefficients 333 9.1 Introduction 333 9.2 Elementary Transport Theory 334 9.2.1 Transport of Molecular Properties 338 9.2.2 The Rate of Reaction 339 9.3 The Boltzmann Equation 341 9.3.1 Derivation of the Boltzmann Equation 342 9.4 Linearized Boltzmann Equations for Mixtures 343 9.4.1 Kinetic Equations for a Two-Component Gas 344 9.4.2 Collision Operators 346 9.5 Coefficient of Self-Diffusion 348 9.5.1 Derivation of the Diffusion Equation 348 9.5.2 Eigenfrequencies of the Lorentz–Boltzmann Equation 349 9.6 Coefficients of Viscosity and Thermal Conductivity 351 9.6.1 Derivation of the Hydrodynamic Equations 351 9.6.2 Eigenfrequencies of the Boltzmann Equation 355 9.6.3 Shear Viscosity and Thermal Conductivity 358 9.7 Computation of Transport Coefficients 359 9.7.1 Sonine Polynomials 359 9.7.2 Diffusion Coefficient 360 9.7.3 Thermal Conductivity 361 9.7.4 Shear Viscosity 363 9.8 Beyond the Boltzmann Equation 365 9.9 Problems 366 10 Nonequilibrium Phase Transitions 369 10.1 Introduction 369 10.2 Near Equilibrium Stability Criteria 370 10.3 The Chemically Reacting Systems 372 10.3.1 The Brusselator – A Nonlinear Chemical Model 373 10.3.2 Boundary Conditions 374 10.3.3 Stability Analysis 375
Contents10.3.4Chemical Crystals37710.4The Rayleigh-Benard Instability37810.4.1Hydrodynamic Equations and Boundary Conditions37910.4.2Linear StabilityAnalysis 38210.5Problems385AppendixAProbabilityandStochasticProcesses387A.1Probability387A.1.1Definitionof Probability387A.1.2ProbabilityDistribution FunctionsS389A.1.3Binomial Distributions393A.1.4Central LimitTheorem and theLawof LargeNumbersS400A.2Stochastic Processes402A.2.1MarkovChains402A.2.2405The Master EquationA.2.3Probability Densityfor Classical Phase Space409A.2.4QuantumProbabilityDensityOperator412A.3Problems415AppendixB ExactDifferentials417AppendixC Ergodicity 421AppendixDNumberRepresentation425D.1Symmetrized and Antisymmetrized States425D.1.1Free Particles426D.1.2Particle ina Box426D.1.3N-Particle Eigenstates427D.1.4Symmetrized Momentum Eigenstates for Bose-EinsteinParticles427D.1.5Antisymmetrized Momentum Eigenstates for Fermi-DiracParticles428D.1.6Partition Functionsand Expectation Values429D.2TheNumberRepresentation431D.2.1TheNumberRepresentationforBosons431D.2.2TheNumberRepresentationforFermions434D.2.3ThermodynamicAverages ofQuantumOperators435AppendixEScatteringTheory 437E.1437Classical Dynamics ofthe Scattering ProcessE.2TheScatteringCross Section440E.3442QuantumDynamics of Low-Energy Scattering
X Contents 10.3.4 Chemical Crystals 377 10.4 The Rayleigh–Bénard Instability 378 10.4.1 Hydrodynamic Equations and Boundary Conditions 379 10.4.2 Linear Stability Analysis 382 10.5 Problems 385 Appendix A Probability and Stochastic Processes 387 A.1 Probability 387 A.1.1 Definition of Probability 387 A.1.2 Probability Distribution Functions 389 A.1.3 Binomial Distributions 393 A.1.4 Central Limit Theorem and the Law of Large Numbers 400 A.2 Stochastic Processes 402 A.2.1 Markov Chains 402 A.2.2 The Master Equation 405 A.2.3 Probability Density for Classical Phase Space 409 A.2.4 Quantum Probability Density Operator 412 A.3 Problems 415 Appendix B Exact Differentials 417 Appendix C Ergodicity 421 Appendix D Number Representation 425 D.1 Symmetrized and Antisymmetrized States 425 D.1.1 Free Particles 426 D.1.2 Particle in a Box 426 D.1.3 N-Particle Eigenstates 427 D.1.4 Symmetrized Momentum Eigenstates for Bose–Einstein Particles 427 D.1.5 Antisymmetrized Momentum Eigenstates for Fermi–Dirac Particles 428 D.1.6 Partition Functions and Expectation Values 429 D.2 The Number Representation 431 D.2.1 The Number Representation for Bosons 431 D.2.2 The Number Representation for Fermions 434 D.2.3 Thermodynamic Averages of Quantum Operators 435 Appendix E Scattering Theory 437 E.1 Classical Dynamics of the Scattering Process 437 E.2 The Scattering Cross Section 440 E.3 Quantum Dynamics of Low-Energy Scattering 442
ContentsxIAppendixFUseful Mathand Problem Solutions/445F.1UsefulMathematics445F.2447SolutionsforOdd-NumberedProblems453References459Index
Contents XI Appendix F Useful Math and Problem Solutions 445 F.1 Useful Mathematics 445 F.2 Solutions for Odd-Numbered Problems 447 References 453 Index 459
IxIIPrefacetotheFourthEditionA Modern Course in Statistical Physics has gone through several editions.Thefirst edition was published in 1980 by University of Texas Press.It was well re-ceived because it contained apresentation of statistical physics that synthesizedthe best of theamerican and european"schools"of statistical physics at thattime.In1997,the rights toA Modern Course in Statistical Physics were transferred toJohn Wiley&Sons and thesecond edition waspublished.Thesecondedition wasa much expanded version ofthe first edition, and as we subsequently realized, wastoo longto be used easily as a textbook although it served as a great reference onstatistical physics.In2004,Wiley-VCHVerlagassumed rightstothesecond edition,and in 2007 wedecided to produce a shortened edition (thethird)that wasexplicitly written as a textbook.Thethird edition appeared in 2009.Statistical physics is afast moving subject and many newdevelopmentshaveoccurred in thelast ten years.Therefore,in order tokeep the book"modern',wedecided that it wastime to adjust the focus of thebook to include more applica-tions in biology, chemistry and condensed matter physics.The core material ofthe book has not changed, so previous editions are still extremely useful.Howev-er, the new fourth edition, which is slightly longer than the third edition, changessomeofitsfocus to resonate withmodern research topics.The first edition acknowledged the support and encouragement of Ilya Pri-gogine, who directed the Center for Statistical Mechanics at the U.T. Austin from1968 to 2003.He had an incredible depth of knowledge in many fields of scienceand helped make U.T. Austin an exciting place to be. The second edition was ded-icated to Ilya Prigogine“for his encouragement and support, and because he haschanged our view of the world"The second edition also acknowledged anothergreat scientist, Nico van Kampen, whosebeautiful lectures on stochastic process-es,and critically humorous view of everything,werean inspiration and spurredmy interest statistical physics. Although both of these great people are now gone,I thankthemboth.The world exists and is stable because of a few symmetries at the microscopiclevel. Statistical physics explains how thermodynamics, and the incredible com-plexity of the world around us, emerges from those symmetries.This book at-tempts to tell the story of how that happens.L.E.ReichlAustin,Texas January2016
XIII Preface to the Fourth Edition A Modern Course in Statistical Physics has gone through several editions. The first edition was published in 1980 by University of Texas Press. It was well received because it contained a presentation of statistical physics that synthesized the best of the american and european “schools” of statistical physics at that time. In 1997, the rights to A Modern Course in Statistical Physics were transferred to John Wiley & Sons and the second edition was published. The second edition was a much expanded version of the first edition, and as we subsequently realized, was too long to be used easily as a textbook although it served as a great reference on statistical physics. In 2004, Wiley-VCH Verlag assumed rights to the second edition, and in 2007 we decided to produce a shortened edition (the third) that was explicitly written as a textbook. The third edition appeared in 2009. Statistical physics is a fast moving subject and many new developments have occurred in the last ten years. Therefore, in order to keep the book “modern”, we decided that it was time to adjust the focus of the book to include more applications in biology, chemistry and condensed matter physics. The core material of the book has not changed, so previous editions are still extremely useful. However, the new fourth edition, which is slightly longer than the third edition, changes some of its focus to resonate with modern research topics. The first edition acknowledged the support and encouragement of Ilya Prigogine, who directed the Center for Statistical Mechanics at the U.T. Austin from 1968 to 2003. He had an incredible depth of knowledge in many fields of science and helped make U.T. Austin an exciting place to be. The second edition was dedicated to Ilya Prigogine “for his encouragement and support, and because he has changed our view of the world.” The second edition also acknowledged another great scientist, Nico van Kampen, whose beautiful lectures on stochastic processes, and critically humorous view of everything, were an inspiration and spurred my interest statistical physics. Although both of these great people are now gone, I thank them both. The world exists and is stable because of a few symmetries at the microscopic level. Statistical physics explains how thermodynamics, and the incredible complexity of the world around us, emerges from those symmetries. This book attempts to tell the story of how that happens. Austin, Texas January 2016 L. E. Reichl
一1IntroductionThermodynamics, which is a macroscopic theory ofmatter, emerges from thesymmetries of nature at the microscopic level and provides a universal theoryofmatter at the macroscopiclevel.Quantities that cannotbe destroyed atthe mi-croscopiclevel,due to symmetries and their resulting conservation laws,giveriseto the state variables upon which the theory of thermodynamics is built.Statistical physics provides themicroscopic foundations of thermodynamics.At the microscopic level, many-body systems have a huge number of states avail-able tothem and are continually sampling large subsets of these states.Thetaskofstatisticalphysicsistodeterminethemacroscopic(measurable)behaviorofmany-body systems, given some knowledge of properties of the underlying mi-croscopic states,andto recoverthethermodynamicbehavior ofsuch systems.The field of statistical physics has expanded dramatically during the last halfcentury.New results in quantum fluids, nonlinear chemical physics, critical phe-nomena,transport theory,and biophysics have revolutionized the subject,andyet these results are rarely presented in a form that students who have little back-ground in statistical physics can appreciate orunderstand.Thisbook attempts toincorporate many of these subjects into a basic course on statistical physics. It in-cludes, in a unified and integrated manner, thefoundations of statistical physicsand develops fromthem most of thetools needed to understand the conceptsunderlying modern research in the above fields.There is a tendency in many books to focus on equilibrium statistical mechan-ics and derive thermodynamics as a consequence. As a result, students do not getthe experience oftraversing the vast world ofthermodynamics and do not under-standhowtoapplyittosystemswhich aretoocomplicatedtobeanalyzedusingthe methods of statistical mechanics. We will begin in Chapter 2, by deriving theequations of state for some simple systems starting from ourknowledgeofthemicroscopicstatesofthosesystems (themicrocanonicalensemble).Thiswillgivesome intuition about the complexity ofmicroscopic behaviorunderlyingthe verysimple equations of state that emerge in those systems.In Chapter 3, we provide a thorough grounding in thermodynamics.We reviewthefoundations of thermodynamics and thermodynamic stability theory and de-vote a large part of the chapter to a variety of applications which do not involvephasetransitions,suchasheatengines,the cooling ofgases,mixing,osmosisAModern Coursein StatisticalPhysics,4.EditionLindaEReich@ 2016 WILEY-VCH Verlag GmbH & Co. KGaA.Published 2016 by WILEY-VCH Verlag GmbH& Co. KGaA
1 1 Introduction Thermodynamics, which is a macroscopic theory of matter, emerges from the symmetries of nature at the microscopic level and provides a universal theory of matter at the macroscopic level. Quantities that cannot be destroyed at the microscopic level, due to symmetries and their resulting conservation laws, give rise to the state variables upon which the theory of thermodynamics is built. Statistical physics provides the microscopic foundations of thermodynamics. At the microscopic level, many-body systems have a huge number of states available to them and are continually sampling large subsets of these states. The task of statistical physics is to determine the macroscopic (measurable) behavior of many-body systems, given some knowledge of properties of the underlying microscopic states, and to recover the thermodynamic behavior of such systems. The field of statistical physics has expanded dramatically during the last halfcentury. New results in quantum fluids, nonlinear chemical physics, critical phenomena, transport theory, and biophysics have revolutionized the subject, and yet these results are rarely presented in a form that students who have little background in statistical physics can appreciate or understand. This book attempts to incorporate many of these subjects into a basic course on statistical physics. It includes, in a unified and integrated manner, the foundations of statistical physics and develops from them most of the tools needed to understand the concepts underlying modern research in the above fields. There is a tendency in many books to focus on equilibrium statistical mechanics and derive thermodynamics as a consequence. As a result, students do not get the experience of traversing the vast world of thermodynamics and do not understand how to apply it to systems which are too complicated to be analyzed using the methods of statistical mechanics. We will begin in Chapter 2, by deriving the equations of state for some simple systems starting from our knowledge of the microscopic states of those systems (the microcanonical ensemble). This will give some intuition about the complexity of microscopic behavior underlying the very simple equations of state that emerge in those systems. In Chapter 3, we provide a thorough grounding in thermodynamics. We review the foundations of thermodynamics and thermodynamic stability theory and devote a large part of the chapter to a variety of applications which do not involve phase transitions, such as heat engines, the cooling of gases, mixing, osmosis, A Modern Course in Statistical Physics, 4. Edition. Linda E. Reichl. © 2016WILEY-VCH Verlag GmbH & Co.KGaA. Published 2016 byWILEY-VCH Verlag GmbH & Co.KGaA