LindaE.ReichlAModernCourseinStatisticalPhysics4threvisedandupdatededitionWILEY-VCHVerlagGmbH&Co.KGaA
Linda E. Reichl A Modern Course in Statistical Physics 4th revised and updated edition
二ContentsPrefacetotheFourthEditionxill1Introduction12Complexity and Entropy52.1Introduction52.25Counting Microscopic States2.3Probability92.4Multiplicity and Entropy of Macroscopic Physical States112.5Multiplicity and Entropy of a Spin System122.5.1Multiplicityofa SpinSystem122.5.2Entropyof SpinSystem 132.616EntropicTension ina Polymer2.71:18Multiplicity and Entropy of an Einstein Solid2.7.1MultiplicityofanEinstein Solid182.7.2Entropyof the Einstein Solid 192.820Multiplicity and Entropy of an Ideal Gas2.8.1Multiplicity of an Ideal Gas202.8.2EntropyofanIdealGas222.9Problems233Thermodynamics273.1Introduction 273.229EnergyConservation3.3Entropy303.3.130CarnotEngine3.3.2The Third Law343.4Fundamental Equationof Thermodynamics353.5ThermodynamicPotentials383.5.1Internal Energy393.5.2Enthalpy403.5.342Helmholtz Free Energy3.5.4Gibbs Free Energy43
V Contents Preface to the Fourth Edition XIII 1 Introduction 1 2 Complexity and Entropy 5 2.1 Introduction 5 2.2 Counting Microscopic States 5 2.3 Probability 9 2.4 Multiplicity and Entropy of Macroscopic Physical States 11 2.5 Multiplicity and Entropy of a Spin System 12 2.5.1 Multiplicity of a Spin System 12 2.5.2 Entropy of Spin System 13 2.6 Entropic Tension in a Polymer 16 2.7 Multiplicity and Entropy of an Einstein Solid 18 2.7.1 Multiplicity of an Einstein Solid 18 2.7.2 Entropy of the Einstein Solid 19 2.8 Multiplicity and Entropy of an Ideal Gas 20 2.8.1 Multiplicity of an Ideal Gas 20 2.8.2 Entropy of an Ideal Gas 22 2.9 Problems 23 3 Thermodynamics 27 3.1 Introduction 27 3.2 Energy Conservation 29 3.3 Entropy 30 3.3.1 Carnot Engine 30 3.3.2 The Third Law 34 3.4 Fundamental Equation of Thermodynamics 35 3.5 Thermodynamic Potentials 38 3.5.1 Internal Energy 39 3.5.2 Enthalpy 40 3.5.3 Helmholtz Free Energy 42 3.5.4 Gibbs Free Energy 43
Contents3.5.5Grand Potential 453.6Response Functions463.6.1Thermal Response Functions (Heat Capacity)463.6.2Mechanical Response Functions493.751Stabilityof theEquilibrium State3.7.151Conditions forLocal Equilibrium in a PVT System3.7.2ConditionsforLocal StabilityinaPVTSystem523.7.356ImplicationsoftheStabilityRequirementsfortheFreeEnergies3.7.4Correlations BetweenFluctuations583.8CoolingandLiquefactionofGases613.9OsmoticPressureinDiluteSolutions643.10The Thermodynamics of Chemical Reactions673.10.1TheAffinity683.11The Thermodynamics of Electrolytes5743.11.1Batteries and the Nernst Equation 753.11.277Cell Potentials and the Nernst Equation3.12Problems784The Thermodynamics of Phase Transitions874.1Introduction874.2Coexistence of Phases: Gibbs Phase Rule884.3ClassificationofPhaseTransitions894.4ClassicalPurePVTSystems914.4.1Phase Diagrams914.4.2CoexistenceCurves:Clausius-ClapeyronEquation924.4.3Liquid-Vapor Coexistence Region954.4.4100Thevan derWaals Equation4.4.5Steam Engines -The Rankine Cycle1024.5BinaryMixtures 1054.5.1Equilibrium Conditions1064.6The Helium Liquids 1084.6.1Liquid He1094.6.2Liquid He31124.6.3113Liquid He3-He+ Mixtures4.7Superconductors 1144.8y116Ginzburg-Landau Theory4.8.1117Continuous Phase Transitions4.8.2First-OrderTransitions1204.8.3Some Applications of Ginzburg-Landau Theory1214.9Critical Exponents1234.9.1Definition of Critical Exponents1244.9.2The Critical Exponents for Pure PVT Systems1244.9.3The Critical Exponents for the Curie Point1264.9.4The Critical Exponents for Mean Field Theories1284.10Problems130
VI Contents 3.5.5 Grand Potential 45 3.6 Response Functions 46 3.6.1 Thermal Response Functions (Heat Capacity) 46 3.6.2 Mechanical Response Functions 49 3.7 Stability of the Equilibrium State 51 3.7.1 Conditions for Local Equilibrium in a PVT System 51 3.7.2 Conditions for Local Stability in a PVT System 52 3.7.3 Implications of the Stability Requirements for the Free Energies 56 3.7.4 Correlations Between Fluctuations 58 3.8 Cooling and Liquefaction of Gases 61 3.9 Osmotic Pressure in Dilute Solutions 64 3.10 The Thermodynamics of Chemical Reactions 67 3.10.1 The Affinity 68 3.11 The Thermodynamics of Electrolytes 74 3.11.1 Batteries and the Nernst Equation 75 3.11.2 Cell Potentials and the Nernst Equation 77 3.12 Problems 78 4 The Thermodynamics of Phase Transitions 87 4.1 Introduction 87 4.2 Coexistence of Phases: Gibbs Phase Rule 88 4.3 Classification of Phase Transitions 89 4.4 Classical Pure PVT Systems 91 4.4.1 Phase Diagrams 91 4.4.2 Coexistence Curves: Clausius–Clapeyron Equation 92 4.4.3 Liquid–Vapor Coexistence Region 95 4.4.4 The van der Waals Equation 100 4.4.5 Steam Engines – The Rankine Cycle 102 4.5 Binary Mixtures 105 4.5.1 Equilibrium Conditions 106 4.6 The Helium Liquids 108 4.6.1 Liquid He4 109 4.6.2 Liquid He3 112 4.6.3 Liquid He3-He4 Mixtures 113 4.7 Superconductors 114 4.8 Ginzburg–Landau Theory 116 4.8.1 Continuous Phase Transitions 117 4.8.2 First-Order Transitions 120 4.8.3 Some Applications of Ginzburg–Landau Theory 121 4.9 Critical Exponents 123 4.9.1 Definition of Critical Exponents 124 4.9.2 The Critical Exponents for Pure PVT Systems 124 4.9.3 The Critical Exponents for the Curie Point 126 4.9.4 The Critical Exponents for Mean Field Theories 128 4.10 Problems 130
Contents5Equilibrium Statistical MechanicsI-Canonical Ensemble1355.1 Introduction1355.2ProbabilityDensityOperator-CanonicalEnsemble1375.2.1EnergyFluctuations 1385.3Semiclassical Ideal Gas of Indistinguishable Particles1395.3.1Approximations to the Partition Functionfor Semiclassical IdealGases1395.3.2Maxwell-BoltzmannDistribution1435.4InteractingClassicalFluids1455.4.1Density Correlations and the Radial Distribution Function1465.4.2MagnetizationDensityCorrelations1485.5HeatCapacityofaDebyeSolid1495.6153Order-DisorderTransitionsonSpinLattices5.6.1Exact Solutionfor a One-Dimensional Lattice1545.6.2MeanFieldTheoryforad-Dimensional Lattice1565.6.35159MeanFieldTheoryof SpatialCorrelationFunctions5.6.4Exact Solution to IsingLattice for d=21605.7162Scaling5.7.1Homogeneous Functions1625.7.2WidomScaling1635.7.3166KadanoffScaling5.8MicroscopicCalculation of Critical Exponents1695.8.1General Theory1695.8.2e172Application to Triangular Lattice5.8.3The S4Model1755.9Problems1776Equilibrium Statistical Mechanics II-GrandCanonical Ensemble1836.1Introduction1836.2The Grand Canonical Ensemble1846.2.1ParticleNumberFluctuations1856.2.2Ideal ClassicalGas 1866.3AdsorptionIsotherms1876.4191Virial Expansion for Interacting Classical Fluids6.4.1VirialExpansionandClusterFunctions1916.4.2194The Second Virial Coefficient, B2(T)6.5Blackbody Radiation1976.6IdealQuantumGases2006.7Ideal Bose-EinsteinGas2026.7.1206Bose-EinsteinCondensation6.7.2Experimental Observation of Bose-Einstein Condensation2086.8210Bogoliubov Mean Field Theory6.9IdealFermi-DiracGas2146.10Magnetic Susceptibility of an Ideal Fermi Gas2206.10.1Paramagnetism221
Contents VII 5 Equilibrium Statistical Mechanics I – Canonical Ensemble 135 5.1 Introduction 135 5.2 Probability Density Operator – Canonical Ensemble 137 5.2.1 Energy Fluctuations 138 5.3 Semiclassical Ideal Gas of Indistinguishable Particles 139 5.3.1 Approximations to the Partition Function for Semiclassical Ideal Gases 139 5.3.2 Maxwell–Boltzmann Distribution 143 5.4 Interacting Classical Fluids 145 5.4.1 Density Correlations and the Radial Distribution Function 146 5.4.2 Magnetization Density Correlations 148 5.5 Heat Capacity of a Debye Solid 149 5.6 Order–Disorder Transitions on Spin Lattices 153 5.6.1 Exact Solution for a One-Dimensional Lattice 154 5.6.2 Mean Field Theory for a d-Dimensional Lattice 156 5.6.3 Mean Field Theory of Spatial Correlation Functions 159 5.6.4 Exact Solution to Ising Lattice for d = 2 160 5.7 Scaling 162 5.7.1 Homogeneous Functions 162 5.7.2 Widom Scaling 163 5.7.3 Kadanoff Scaling 166 5.8 Microscopic Calculation of Critical Exponents 169 5.8.1 General Theory 169 5.8.2 Application to Triangular Lattice 172 5.8.3 The S4 Model 175 5.9 Problems 177 6 Equilibrium Statistical Mechanics II – Grand Canonical Ensemble 183 6.1 Introduction 183 6.2 The Grand Canonical Ensemble 184 6.2.1 Particle Number Fluctuations 185 6.2.2 Ideal Classical Gas 186 6.3 Adsorption Isotherms 187 6.4 Virial Expansion for Interacting Classical Fluids 191 6.4.1 Virial Expansion and Cluster Functions 191 6.4.2 The Second Virial Coefficient, B2(T) 194 6.5 Blackbody Radiation 197 6.6 Ideal Quantum Gases 200 6.7 Ideal Bose–Einstein Gas 202 6.7.1 Bose–Einstein Condensation 206 6.7.2 Experimental Observation of Bose–Einstein Condensation 208 6.8 Bogoliubov Mean Field Theory 210 6.9 Ideal Fermi–Dirac Gas 214 6.10 Magnetic Susceptibility of an Ideal Fermi Gas 220 6.10.1 Paramagnetism 221
VIIContents6.10.2Diamagnetism2226.11MomentumCondensationinanInteractingFermiFluid2246.12231Problems7235Brownian Motionand Fluctuation-Dissipation7.1Introduction 2357.2236BrownianMotion7.2.1237Langevin Equation7.2.2238Correlation Function and Spectral Density7.3TheFokker-PlanckEquation2407.3.1ProbabilityFlowinPhase Space2427.3.2ProbabilityFlow for Brownian Particle2437.3.3The StrongFriction Limit2457.4Dynamic Equilibrium Fluctuations2507.4.1Regressionof Fluctuations 2527.4.2253Wiener-KhintchineTheorem7.5Linear Response Theory255and theFluctuation-Dissipation Theorem7.5.1The ResponseMatrix2557.5.2Causality2577.5.3The Fluctuation-Dissipation Theorem2607.5.4PowerAbsorption:2627.6Microscopic Linear ResponseTheory2647.6.1264Density Operator Perturbed by External Field7.6.2The Electric Conductance2657.6.3PowerAbsorption2707.7272Thermal NoiseintheElectronCurrent7.8Problems2738Hydrodynamics52778.1Introduction 2778.2Navier-Stokes Hydrodynamic Equations2788.2.1BalanceEquations2788.2.2Entropy SourceandEntropyCurrent2838.2.3Transport Coefficients2868.3LinearizedHydrodynamic Equations2898.3.1LinearizationoftheHydrodynamicEquations2898.3.2TransverseHydrodynamic Modes2938.3.3Longitudinal Hydrodynamic Modes2948.3.4Dynamic CorrelationFunctionand Spectral Density2968.4Light Scattering2978.4.1Scattered Electric Field 2998.4.2Intensityof Scattered Light 3018.5Friction on a Brownian particle3038.6Brownian Motion with Memory307
VIII Contents 6.10.2 Diamagnetism 222 6.11 Momentum Condensation in an Interacting Fermi Fluid 224 6.12 Problems 231 7 Brownian Motion and Fluctuation–Dissipation 235 7.1 Introduction 235 7.2 Brownian Motion 236 7.2.1 Langevin Equation 237 7.2.2 Correlation Function and Spectral Density 238 7.3 The Fokker–Planck Equation 240 7.3.1 Probability Flow in Phase Space 242 7.3.2 Probability Flow for Brownian Particle 243 7.3.3 The Strong Friction Limit 245 7.4 Dynamic Equilibrium Fluctuations 250 7.4.1 Regression of Fluctuations 252 7.4.2 Wiener–Khintchine Theorem 253 7.5 Linear Response Theory and the Fluctuation–Dissipation Theorem 255 7.5.1 The Response Matrix 255 7.5.2 Causality 257 7.5.3 The Fluctuation–Dissipation Theorem 260 7.5.4 Power Absorption 262 7.6 Microscopic Linear Response Theory 264 7.6.1 Density Operator Perturbed by External Field 264 7.6.2 The Electric Conductance 265 7.6.3 Power Absorption 270 7.7 Thermal Noise in the Electron Current 272 7.8 Problems 273 8 Hydrodynamics 277 8.1 Introduction 277 8.2 Navier–Stokes Hydrodynamic Equations 278 8.2.1 Balance Equations 278 8.2.2 Entropy Source and Entropy Current 283 8.2.3 Transport Coefficients 286 8.3 Linearized Hydrodynamic Equations 289 8.3.1 Linearization of the Hydrodynamic Equations 289 8.3.2 Transverse Hydrodynamic Modes 293 8.3.3 Longitudinal Hydrodynamic Modes 294 8.3.4 Dynamic Correlation Function and Spectral Density 296 8.4 Light Scattering 297 8.4.1 Scattered Electric Field 299 8.4.2 Intensity of Scattered Light 301 8.5 Friction on a Brownian particle 303 8.6 Brownian Motion with Memory 307