EQUILIBRIUM AND NON-EQUILIBRIUM STATISTICAL THERMODYNAMICS
CONTENTS THERMOSTATICS THERMODYNAMIC EQUILIBRIUM MICROSCOPIC AND MACROSCOPIC DESCRIPTIONS WALLS WORK, HEAT, INTERNAL ENERGY DE?NITION OF THERMAL EQUILIBRIUM POSTULATE OF MAXIMUM ENTROPY INTERNAL CONSTRAINTS PRINCIPLE OF MAXIMUM ENTROPY INTENSIVE VARIABLES: TEMPERATURE, PRESSURE, CHEMICAL POTENTIAL QUASI-STATIC AND REVERSIBLE PROCESSES MAXIMUM WORK AND HEAT ENGINES THERMODYNAMIC POTENTIALS THERMODYNAMIC POTENTIALS AND MASSIEU FUNCTIONS SPECI?C HEATS GIBBS–DUHEM RELATION STABILITY CONDITIONS CONCAVITY OF ENTROPY AND CONVEXITY OF ENERGY STABILITY CONDITIONS AND THEIR CONSEQUENCES THIRD LAW OF THERMODYNAMICS STATEMENT OF THE THIRD LAW APPLICATION TO METASTABLE STATES LOW TEMPERATURE BEHAVIOUR OF SPECI?C HEATS EXERCISES MASSIEU FUNCTIONS INTERNAL VARIABLE IN EQUILIBRIUM RELATIONS BETWEEN THERMODYNAMIC COEF?CIENTS CONTACT BETWEEN TWO SYSTEMS EQUATION OF STATE FOR A ?UID PROBLEMS REVERSIBLE AND IRREVERSIBLE FREE EXPANSIONS OF AN IDEAL GAS VAN DER WAALS EQUATION OF STATE EQUATION OF STATE FOR A SOLID SPECI?C HEATS OF A ROD SURFACE TENSION OF A SOAP ?LM JOULE–THOMSON PROCESS ADIABATIC DEMAGNETIZATION OF A PARAMAGNETIC SALT FURTHER READING STATISTICAL ENTROPY AND BOLTZMANN DISTRIBUTION QUANTUM DESCRIPTION TIME EVOLUTION IN QUANTUM MECHANICS THE DENSITY OPERATORS AND THEIR TIME EVOLUTION QUANTUM PHASE SPACE (P, V, E) RELATION FOR A MONO-ATOMIC IDEAL GAS CLASSICAL DESCRIPTION LIOUVILLE’S THEOREM DENSITY IN PHASE SPACE STATISTICAL ENTROPY ENTROPY OF A PROBABILITY DISTRIBUTION STATISTICAL ENTROPY OF A MIXED QUANTUM STATE TIME EVOLUTION OF THE STATISTICAL ENTROPY BOLTZMANN DISTRIBUTION POSTULATE OF MAXIMUM OF STATISTICAL ENTROPY EQUILIBRIUM DISTRIBUTION LEGENDRE TRANSFORMATION CANONICAL AND GRAND CANONICAL ENSEMBLES THERMODYNAMICS REVISITED HEAT AND WORK: ?RST LAW ENTROPY AND TEMPERATURE: SECOND LAW ENTROPY OF MIXING PRESSURE AND CHEMICAL POTENTIAL IRREVERSIBILITY AND THE GROWTH OF ENTROPY MICROSCOPIC REVERSIBILITY AND MACROSCOPIC IRREVERSIBILITY PHYSICAL BASIS OF IRREVERSIBILITY LOSS OF INFORMATION AND THE GROWTH OF ENTROPY DENSITY OF STATES AND THE DIMENSION OF SPACE LIOUVILLE THEOREM AND CONTINUITY EQUATION LOADED DICE AND STATISTICAL ENTROPY ENTROPY OF A COMPOSITE SYSTEM HEAT EXCHANGES BETWEEN SYSTEM AND RESERVOIR GALILEAN TRANSFORMATION FLUCTUATION-RESPONSE THEOREM PHASE SPACE VOLUME FOR N FREE PARTICLES 2 7 10 ENTROPY OF MIXING AND OSMOTIC PRESSURE CANONICAL AND GRAND CANONICAL ENSEMBLES: APPLICATIONS SIMPLE EXAMPLES IN THE CANONICAL ENSEMBLE MEAN VALUES AND ?UCTUATIONS PARTITION FUNCTION AND THERMODYNAMICS OF AN IDEAL GAS PARAMAGNETISM FERROMAGNETISM AND THE ISING MODEL THERMODYNAMIC LIMIT CLASSICAL STATISTICAL MECHANICS CLASSICAL LIMIT MAXWELL DISTRIBUTION EQUIPARTITION THEOREM SPECI?C HEAT OF A DIATOMIC IDEAL GAS QUANTUM OSCILLATORS AND ROTATORS QUALITATIVE DISCUSSION PARTITION FUNCTION OF A DIATOMIC MOLECULE FROM IDEAL GASES TO LIQUIDS PAIR CORRELATION FUNCTION MEASUREMENT OF THE PAIR CORRELATION FUNCTION PRESSURE AND ENERGY CHEMICAL POTENTIAL BASIC FORMULAE COEXISTENCE OF PHASES EQUILIBRIUM CONDITION AT CONSTANT PRESSURE EQUILIBRIUM AND STABILITY CONDITIONS AT CONSTANT µ CHEMICAL REACTIONS GRAND CANONICAL ENSEMBLE GRAND PARTITION FUNCTION MONO-ATOMIC IDEAL GAS THERMODYNAMICS AND ?UCTUATIONS DENSITY OF STATES EQUATION OF STATE FOR THE EINSTEIN MODEL OF A SOLID SPECI?C HEAT OF A FERROMAGNETIC CRYSTAL NUCLEAR SPECI?C HEAT OF A METAL SOLID AND LIQUID VAPOUR PRESSURES ELECTRON TRAPPING IN A SOLID ONE-DIMENSIONAL ISING MODEL NEGATIVE TEMPERATURES DIATOMIC MOLECULES MODELS OF A BOUNDARY SURFACE DEBYE–H¨UCKEL APPROXIMATION THIN METALLIC ?LM BEYOND THE IDEAL GAS: ?RST TERM OF VIRIAL EXPANSION THEORY OF NUCLEATION CRITICAL PHENOMENA ISING MODEL REVISITED SOME EXACT RESULTS FOR THE ISING MODEL CORRELATION FUNCTIONS BROKEN SYMMETRY CRITICAL EXPONENTS MEAN ?ELD THEORY A CONVEXITY INEQUALITY FUNDAMENTAL EQUATION OF MEAN ?ELD THEORY BROKEN SYMMETRY AND CRITICAL EXPONENTS LANDAU’S THEORY LANDAU FUNCTIONAL BROKEN CONTINUOUS SYMMETRY GINZBURG–LANDAU HAMILTONIAN BEYOND LANDAU’S THEORY GINZBURG CRITERION RENORMALIZATION GROUP: GENERAL THEORY SPIN BLOCKS CRITICAL EXPONENTS AND SCALING TRANSFORMATIONS CRITICAL MANIFOLD AND ?XED POINTS LIMIT DISTRIBUTIONS AND CORRELATION FUNCTIONS MAGNETIZATION AND FREE ENERGY RENORMALIZATION GROUP: EXAMPLES GAUSSIAN ?XED POINT NON-GAUSSIAN ?XED POINT CRITICAL EXPONENTS TO ORDER ? SCALING OPERATORS AND ANOMALOUS DIMENSIONS HIGH TEMPERATURE EXPANSION AND KRAMERS–WANNIER DUALITY ENERGY–ENERGY CORRELATIONS IN THE ISING MODEL MEAN ?ELD CRITICAL EXPONENTS FOR T < TC ACCURACY OF THE VARIATIONAL METHOD SHAPE AND ENERGY OF AN ISING WALL THE GINZBURG–LANDAU THEORY OF SUPERCONDUCTIVITY MEAN ?ELD CORRELATION FUNCTION IN ?R-SPACE RENORMALIZATION OF THE GAUSSIAN MODEL 4 6 10 SCALING ?ELDS AT THE GAUSSIAN ?XED POINT 4 6 11 CRITICAL EXPONENTS TO ORDER ? FOR N ?= 4 6 12 IRRELEVANT EXPONENTS 4 6 13 ENERGY–ENERGY CORRELATIONS 4 6 14 ‘DERIVATION’ OF THE GINZBURG–LANDAU HAMILTONIAN FROM THE ISING MODEL QUANTUM STATISTICS BOSE–EINSTEIN AND FERMI–DIRAC DISTRIBUTIONS CLASSICAL LIMIT: MAXWELL–BOLTZMANN STATISTICS CHEMICAL POTENTIAL AND RELATIVITY IDEAL FERMI GAS IDEAL FERMI GAS AT ZERO TEMPERATURE IDEAL FERMI GAS AT LOW TEMPERATURE CORRECTIONS TO THE IDEAL FERMI GAS BLACK BODY RADIATION ELECTROMAGNETIC RADIATION IN THERMAL EQUILIBRIUM DEBYE MODEL SIMPLE MODEL OF VIBRATIONS IN SOLIDS DEBYE APPROXIMATION CALCULATION OF THERMODYNAMIC FUNCTIONS IDEAL BOSE GAS WITH A ?XED NUMBER OF PARTICLES BOSE–EINSTEIN CONDENSATION THERMODYNAMICS OF THE CONDENSED PHASE THE MAXWELL–BOLTZMANN PARTITION FUNCTION EQUILIBRIUM RADIUS OF A NEUTRON STAR TWO-DIMENSIONAL FERMI GAS NON-DEGENERATE FERMI GAS TWO-DIMENSIONAL BOSE GAS PHONONS AND MAGNONS PHOTON–ELECTRON–POSITRON EQUILIBRIUM IN A STAR PAULI PARAMAGNETISM LANDAU DIAMAGNETISM WHITE DWARF STARS QUARK–GLUON PLASMA BOSE–EINSTEIN CONDENSATES OF ATOMIC GASES SUPER?UIDITY FOR HARDCORE BOSONS IRREVERSIBLE PROCESSES: MACROSCOPIC THEORY FLUX, AF?NITIES, TRANSPORT COEF?CIENTS CONSERVATION LAWS LOCAL EQUATION OF STATE AF?NITIES AND TRANSPORT COEF?CIENTS EXAMPLES DISSIPATION AND ENTROPY PRODUCTION COUPLING BETWEEN THERMAL AND PARTICLE DIFFUSION ELECTRODYNAMICS HYDRODYNAMICS OF SIMPLE ?UIDS CONSERVATION LAWS IN A SIMPLE ?UID DERIVATION OF CURRENT DENSITIES TRANSPORT COEF?CIENTS AND THE NAVIER–STOKES EQUATION CONTINUITY EQUATION FOR THE DENSITY OF PARTICLES DIFFUSION EQUATION AND RANDOM WALK RELATION BETWEEN VISCOSITY AND DIFFUSION DERIVATION OF THE ENERGY CURRENT LORD KELVIN’S MODEL OF EARTH COOLING ENTROPY CURRENT IN HYDRODYNAMICS HYDRODYNAMICS OF THE PERFECT ?UID THERMOELECTRIC EFFECTS ISOMERIZATION REACTIONS NUMERICAL SIMULATIONS MARKOV CHAINS, CONVERGENCE AND DETAILED BALANCE CLASSICAL MONTE CARLO IMPLEMENTATION MEASUREMENTS AUTOCORRELATION, THERMALIZATION AND ERROR BARS CRITICAL SLOWING DOWN AND CLUSTER ALGORITHMS QUANTUM MONTE CARLO: BOSONS FORMULATION AND IMPLEMENTATION QUANTUM SPIN-1/2 MODELS QUANTUM MONTE CARLO: FERMIONS FINITE SIZE SCALING RANDOM NUMBER GENERATORS DETERMINATION OF THE CRITICAL EXPONENT ? FINITE SIZE SCALING IN IN?NITE GEOMETRIES BOSONS ON A SINGLE SITE TWO-DIMENSIONAL ISING MODEL: METROPOLIS TWO-DIMENSIONAL ISING MODEL: GLAUBER TWO-DIMENSIONAL CLOCK MODEL TWO-DIMENSIONAL XY MODEL: KOSTERLITZ–THOULESS TRANSITION TWO-DIMENSIONAL XY MODEL: SUPER?UIDITY AND CRITICAL VELOCITY SIMPLE QUANTUM MODEL: SINGLE SPIN IN TRANSVERSE ?ELD ONE-DIMENSIONAL ISING MODEL IN TRANSVERSE ?ELD: QUANTUM PHASE TRANSITION QUANTUM ANHARMONIC OSCILLATOR: PATH INTEGRALS 7 10 FURTHER READING IRREVERSIBLE PROCESSES: KINETIC THEORY GENERALITIES, ELEMENTARY THEORY OF TRANSPORT COEF?CIENTS DISTRIBUTION FUNCTION CROSS SECTION, COLLISION TIME, MEAN FREE PATH TRANSPORT COEF?CIENTS IN THE MEAN FREE PATH APPROXIMATION BOLTZMANN–LORENTZ MODEL SPATIO-TEMPORAL EVOLUTION OF THE DISTRIBUTION FUNCTION BASIC EQUATIONS OF THE BOLTZMANN–LORENTZ MODEL CONSERVATION LAWS AND CONTINUITY EQUATIONS LINEARIZATION: CHAPMAN–ENSKOG APPROXIMATION CURRENTS AND TRANSPORT COEF?CIENTS BOLTZMANN EQUATION COLLISION TERM H-THEOREM TRANSPORT COEF?CIENTS FROM THE BOLTZMANN EQUATION LINEARIZATION OF THE BOLTZMANN EQUATION VARIATIONAL METHOD CALCULATION OF THE VISCOSITY TIME DISTRIBUTION OF COLLISIONS SYMMETRIES OF AN INTEGRAL POSITIVITY CONDITIONS CALCULATION OF THE COLLISION TIME EQUILIBRIUM DISTRIBUTION FROM THE BOLTZMANN EQUATION THERMAL DIFFUSION IN THE BOLTZMANN–LORENTZ MODEL ELECTRON GAS IN THE BOLTZMANN–LORENTZ MODEL PHOTON DIFFUSION AND ENERGY TRANSPORT IN THE SUN MOMENTUM TRANSFER IN A SHEAR ?OW ELECTRICAL CONDUCTIVITY IN A MAGNETIC ?ELD AND QUANTUM HALL EFFECT SPECI?C HEAT AND TWO-?UID MODEL FOR HELIUM II LANDAU THEORY OF FERMI LIQUIDS CALCULATION OF THE COEF?CIENT OF THERMAL CONDUCTIVITY TOPICS IN NON-EQUILIBRIUM STATISTICAL MECHANICS LINEAR RESPONSE: CLASSICAL THEORY DYNAMICAL SUSCEPTIBILITY NYQUIST THEOREM ANALYTICITY PROPERTIES SPIN DIFFUSION LINEAR RESPONSE: QUANTUM THEORY QUANTUM ?UCTUATION RESPONSE THEOREM QUANTUM KUBO FUNCTION FLUCTUATION-DISSIPATION THEOREM SYMMETRY PROPERTIES AND DISSIPATION SUM RULES PROJECTION METHOD AND MEMORY EFFECTS PHENOMENOLOGICAL INTRODUCTION TO MEMORY EFFECTS PROJECTORS LANGEVIN–MORI EQUATION BROWNIAN MOTION: QUALITATIVE DESCRIPTION BROWNIAN MOTION: THE M/M ?0 LIMIT LANGEVIN EQUATION DE?NITIONS AND ?RST PROPERTIES ORNSTEIN–UHLENBECK PROCESS FOKKER–PLANCK EQUATION DERIVATION OF FOKKER–PLANCK FROM LANGEVIN EQUATION EQUILIBRIUM AND CONVERGENCE TO EQUILIBRIUM SPACE-DEPENDENT DIFFUSION COEF?CIENT NUMERICAL INTEGRATION LINEAR RESPONSE: FORCED HARMONIC OSCILLATOR FORCE ON A BROWNIAN PARTICLE GREEN–KUBO FORMULA MORI’S SCALAR PRODUCT SYMMETRY PROPERTIES OF ??? I J DISSIPATION PROOF OF THE F-SUM RULE IN QUANTUM MECHANICS DIFFUSION OF A BROWNIAN PARTICLE STRONG FRICTION LIMIT: HARMONIC OSCILLATOR 9 7 10 GREEN’S FUNCTION METHOD 9 7 11 MOMENTS OF THE FOKKER–PLANCK EQUATION 9 7 12 BACKWARD VELOCITY 9 7 13 NUMERICAL INTEGRATION OF THE LANGEVIN EQUATION 9 7 14 METASTABLE STATES AND ESCAPE TIMES INELASTIC LIGHT SCATTERING FROM A SUSPENSION OF PARTICLES LIGHT SCATTERING BY A SIMPLE ?UID EXACTLY SOLVABLE MODEL OF A BROWNIAN PARTICLE ITˆO VERSUS STRATONOVITCH DILEMMA KRAMERS EQUATION THERMOSTATICS THE GOAL OF THIS ?RST CHAPTER IS TO GIVE A PRESENTATION OF THERMODYNAMICS, DUE TO H CALLEN, WHICH WILL ALLOW US TO MAKE THE MOST DIRECT CONNECTION WITH THE STATISTICAL APPROACH OF THE FOLLOWING CHAPTER INSTEAD OF INTRODUCING ENTROPY BY STARTING WITH THE SECOND LAW, FOR EXAMPLE WITH THE KELVIN STATEMENT ‘THERE EXISTS NO TRANSFORMA- TION WHOSE SOLE EFFECT IS TO EXTRACT A QUANTITY OF HEAT FROM A RESERVOIR AND CONVERT IT ENTIRELY TO WORK’, CALLEN ASSUMES, IN PRINCIPLE, THE EXISTENCE OF AN ENTROPY FUNCTION AND ITS FUNDAMENTAL PROPERTY: THE PRINCIPLE OF MAXIMUM ENTROPY SUCH A PRESEN- TATION LEADS TO A CONCISE DISCUSSION OF THE FOUNDATIONS OF THERMODYNAMICS (AT THE COST OF SOME ABSTRACTION) AND HAS THE ADVANTAGE OF ALLOWING DIRECT COMPARISON WITH THE STATISTICAL ENTROPY THAT WE SHALL INTRODUCE IN CHAPTER 2 CLEARLY, IT IS NOT POSSI- BLE IN ONE CHAPTER TO GIVE AN EXHAUSTIVE ACCOUNT OF THERMODYNAMICS; THE READER IS, INSTEAD, REFERRED TO CLASSIC BOOKS ON THE SUBJECT FOR FURTHER DETAILS 1 1 THERMODYNAMIC EQUILIBRIUM 1 1 1 MICROSCOPIC AND MACROSCOPIC DESCRIPTIONS THE AIM OF STATISTICAL THERMODYNAMICS IS TO DESCRIBE THE BEHAVIOUR OF MACROSCOPIC SYSTEMS CONTAINING OF THE ORDER OF N ?1023 PARTICLES 1 AN EXAMPLE OF SUCH A MACROSCOPIC SYSTEM IS A MOLE OF GAS IN A CONTAINER UNDER STANDARD CONDITIONS OF TEMPERATURE AND PRESSURE 2 THIS GAS HAS 6 × 1023 MOLECULES3 IN INCESSANT MO- TION, CONTINUALLY COLLIDING WITH EACH OTHER AND WITH THE WALLS OF THE CONTAINER TO A ?RST APPROXIMATION, WHICH WILL BE JUSTI?ED IN CHAPTER 2, WE MAY CONSIDER THESE MOLECULES AS CLASSICAL OBJECTS ONE CAN, THEREFORE, ASK THE USUAL QUESTION OF CLASSICAL MECHANICS: GIVEN THE INITIAL POSITIONS AND VELOCITIES (OR MOMENTA) OF THE 1 WITH SOME PRECAUTIONS, ONE CAN APPLY THERMODYNAMICS TO MESOSCOPIC SYSTEMS, I E INTERMEDIATE BETWEEN MICRO- AND MACROSCOPIC, FOR EXAMPLE SYSTEM SIZE OF THE ORDER OF 1 M 2 THE READER WILL ALLOW US TO TALK ABOUT TEMPERATURE AND PRESSURE EVEN THOUGH THESE CONCEPTS WILL NOT BE DE?NED UNTIL LATER FOR THE MOMENT INTUITIVE NOTIONS OF THESE CONCEPTS ARE SUF?CIENT 3 IN THE CASE OF A GAS, WE USE THE TERM ‘MOLECULES’ INSTEAD OF THE GENERIC TERM ‘PARTICLES’