Fundamentals of statistical and thermal physics ::- F. REIF Professor of Physics University of California, Berkeley

Contents:-

Preface vii

1 Introduction to statistical methods 1

RANDOM WALK AND BINOMIAL DISTRIBUTION

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Elementary statistical concepts and examples 4

The simple random walk problem in one dimension 7

General discussion of mean values 11

Calculation of mean values for the random walk problem 18

Probability distribution for large N 17

Gaussian probability distributions 21

GENERAL DISCUSSION OF THE RANDOM WALK

17

1-8

19

*1'10

*l'll

Probability distributions involving several variables 25

Comments on continuous probability distributions 27

General calculation of mean values for the random walk 82

Calculation of the probability distribution 35

Probability distribution for large N 87

2 Statistical description of systems of particles 47

STATISTICAL FORMULATION OF THE MECHANICAL PROBLEM

2'1

2'2

2'3

2'4

2'5

Specification of the state of a system 48

Statistical ensemble 52

Basic postulates 58

Probability calculations 60

Behavior of the density of states 61

INTERACTION BETWEEN MACROSCOPIC SYSTEMS

2- 6 Thermal interaction 66

2-7 Mechanical interaction 68

2- 8 General interaction 73

2-9 Quasi-static processes ’74

2- 10 Quasi-static work done by pressure 76‘

2-11 Exact and “inexact” differentials 78

3 Statistical thermodynamics 87

IRREVERSIBILITY AND THE ATTAINMENT OF EQUILIBRIUM

3-1 Equilibrium conditions and constraints 87

3-2 Reversible and irreversible processes .91

THERMAL INTERACTION BETWEEN MACROSCOPIC SYSTEMS

3-3 Distribution of energy between systems in equilibrium .94

3 - 4 The approach to thermal equilibrium 100

3 ~ 5 Temperature 1 02

3 - 6 Heat reservoirs _ 1 06‘

3-7 Sharpness of the probability distribution 108

GENERAL INTERACTION BETWEEN MACROSCOPIC SYSTEMS

3-8 Dependence of the density of states on the external parameters 112

3 - 9 Equilibrium between interacting systems 114

3 - 10 Properties of the entropy 117

SUMMARY OF FUNDAMENTAL RESULTS

3- 11 Thermodynamic laws and basic statistical relations 122

3 - 12 Statistical calculation of thermodynamic quantities 124

4' Macroscopic parameters and their measurement 128

4-1 Work and internal energy 128

42 Heat 131

4-3 Absolute temperature 133

44 Heat capacity and specific heat 13.9

4-5 Entropy 142

4- 6 Consequences of the absolute definition of entropy 145

4-7 Extensive and intensive parameters 148

5 Simple applications of macroscopic thermodynamics 152

PROPERTIES OF IDEAL CASES

5- 1 Equation of state and internal energy 153

5-2 Specific heats 156‘

5 - 3 Adiabatic expansion or compression 158,

5-4 Entropy 160

GENERAL RELATIONS FOR A HOMOGENEOUS SUBSTANCE

5 - 5 Derivation of general relations 161

5 ' 6 Summary of Maxwell relations and thermodynamic functions 164

5 - 7 Specific heats 166

5-8 E’ntropy and internal energy 171

FREE EXPANSION AND THROTTLING PROCESSES

59 Free expansion of a gas 175

5- 10 Throttling (or Joule-Thomson) process 178

HEAT ENGINES AND REFRIGERATORS

5 - 1 1 Heat engines 184

5 - 12 Refrigerators 1 90

Basic methods and results of statistical mechanics 201

ENSEMBLES REPRESENTATIVE OF SITUATIONS OF PHYSICAL INTEREST

6-1 Isolated system 201

6- 2 System in contact with a heat reservoir 202

6-3 Simple applications of the canonical distribution 206

6-4 System with specified mean energy 211

6-5 Calculation of mean values in a canonical ensemble 212

6-6 Connection with thermodynamics 214

APPROXIMATION METHODS

6-7 E'nsembles used as approximations 219 ' *6-8 Mathematical approximation methods 221

GENERALIZATIONS AND ALTERNATIVE APPROACHES

*6-9 Grand canonical and other ensembles 225

*6- 10 Alternative derivation of the canonical distribution 229

Simple applications of statistical mechanics 237

GENERAL METHOD OF APPROACH

7-1 Partition functions and their properties 237

IDEAL MONATOMIC GAS

7-2 Calculation of thermodynamic quantities 23.9

7 ' 3 Gibbs paradox 243

7-4 Validity of the classical approximation 246‘

THE EQUIPARTITION THEOREM

7-5 Proof of the theorem 248

76 Simple applications 250

77 Specific heats of solids 253

PARAMAGNETI SM

78 General calculation of magnetization 257

KINETIC THEORY OF DILUTE GASES IN EQUILIBRIUM

7'9 Maxwell velocity distribution 262

7 - 10 Related velocity distributions and mean values 265

7 ' 11 Number of molecules striking a surface 269

7 - 12 E’jfusion 273

7 ' 13 Pressure and momentum transfer 278

8 Equilibrium between phases or chemical species 288

GENERAL EQUILIBRIUM CONDITIONS

8-1 Isolated system 289

8 ~ 2 System in contact with a reservoir at constant temperature 291

83 System in contact with a reservoir at constant temperature and

pressure 294

84 Stability conditions for a homogeneous substance 296

EQUI LIBRIUM BETWEEN PHA SES

8-5 Equilibrium conditions and the Clausius-Clapeyron equation 301

86 Phase transformations and the equation of state 306

SYSTEMS WITH SEVERAL COMPONENTS; CHEMICAL EQUILIBRIUM

87 General relations for a system with several components 312

88 Alternative discussion of equilibrium between phases 315

89 General conditions for chemical equilibrium 317

8- 10 Chemical equilibrium between ideal gases 319

9 Quantum statistics of ideal gases 331

MAXWELL-BOLTZMANN, BOSE-EINSTEIN, AND FERMI-DIRAC STATISTICS

9-1 Identical particles and symmetry requirements 331

9-2 Formulation of the statistical problem 335

9-3 The quantum distribution functions 338

9 ~ 4 Maxwell-Boltzmann statistics 343

9'5 Photon statistics 345

9'6 Bose-Einstein statistics 346‘

9-7 Fermi-Dirac statistics 350

98 Quantum statistics in the classical limit 351

IDEAL GAS IN THE CLASSICAL LIMIT 9-9 Quantum states of a single particle 353

9- 10 Evaluation of the partition function 360

9- 11 Physical implications of the quantum-mechanical enumeration of

states 363

*9- 12 Partition functions of polyatomic molecules 367

BLACK-BODY RADIATION

9- 13 Electromagnetic radiation in thermal equilibrium inside an

enclosure 373

9- 14 Nature of the radiation inside an arbitrary enclosure 378

9- 15 Radiation emitted by a body at temperature T 381

CONDUCTION ELECTRONS IN METALS

9 - 16 Consequences of the Fermi-Dirac distribution 388

*9- 17 Quantitative calculation of the electronic specific heat 393

10 Systems of interacting particles 404

SOLIDS

10 -1 Lattice vibrations and normal modes 407

10 - 2 Debye approximation 411

NONIDEAL CLASSICAL CAS

103 Calculation of the partition function for low densities 418

10-4 Equation of state and virial coefiicients 433

10 - 5 Alternative derivation of the van der Waals equation 436‘

FERROMAGNETI SM

10-6 Interaction between spins 438

10 - 7 Weiss molecular-field approximation 430

1 1 Magnetism and low temperatures 438

11 - 1 Magnetic work 439

11 - 2 Magnetic cooling 445

11-3 Measurement of very low absolute temperatures 453

1 1 - 4 Superconductivity 455

12 Elementary kinetic theory of transport processes 461

12-1 Collision time 463

12-2 Collision time and scattering cross section 467

12 - 3 Viscosity 471

12 - 4 Thermal conductivity 478

12 - 5 Self-difiusion 483

12 - 6 Electrical Conductivity 488

Transport theory using the relaxation time approximation 494

13 -1 Transport processes and distribution functions 494

13-2 Boltzmann equation in the absence of collisions 4.98

13 - 3 Path integral formulation 502

13 - 4 Example: calculation of electrical conductivity 504

135 Example: calculation of viscosity 507

13 - 6 Boltzmann difierential equation formulation 508

13-7 Equivalence of the two formulations 510

13-8 Examples of the Boltzmann equation method 511

Near-exactformulation of transport theory 516

141 Description of two-particle collisions 516

14-2 Scattering cross sections and symmetry properties 520

14-3 Derivation of the Boltzmann equation 523

14-4 Equation of change for mean values 525

14-5 Conservation equations and hydrodynamics 529

146 Example: simple discussion of electrical conductivity 531

14-7 Approximation methods for solving the Boltzmann equation 534

148 Example: calculation of the coeflicient of viscosity 53.9

Irreversible processes and fluctuations 548

TRANSITION PROBABILITIES AND MASTER EQUATION

15-1 Isolated system 548

15 - 2 System in contact with a heat reservoir 551

15 - 3 Magnetic resonance 553

15-4 Dynamic nuclear polarization; Overhauser efi'ect 556‘

SIMPLE DISCUSSION OF BROWNIAN MOTION

15-5 Langevin equation 560

15- 6 Calculation of the mean-square displacement 565

DETAILED ANALYSIS OF BROWNIAN MOTION

15-7 Relation between dissipation and the fluctuating force 567

15.8 Correlation functions and the friction constant 570

*15-9 Calculation of the mean-square velocity increment 574

*15- 10 Velocity correlation function and mean-square displacement 575

CALCULATION OF PROBABILITY DISTRIBUTIONS

*15- 11 The Fokker-Planck equation 577

*15- 12 Solution of the Fokker-Planck equation 580

FOURIER ANALYSIS OF RANDOM FUNCTIONS

15- 13 Fourier analysis 582

15- 14 Ensemble and time averages 583

15- 15 Wiener-Khintchine relations 585

15- 16 Nyquist’s theorem 587

15 - 17 Nyquist’s theorem and equilibrium conditions 589

GENERAL DISCUSSION OF IRREVERSIBLE PROCESSES

15- 18 Fluctuations and Onsager relations 5.94

Appendices 605

A.1 Review of elementary sums 605

A2 Evaluation of the integral [:0 e‘” dx 606

A3 Evaluation of the integral [0” e‘zx" dx 607

AA Evaluation of integrals of the form A” e‘azzx" dx 608

A.5 The error function 609

A.6 Stirling’s formula 61 0

A.7 The Dirac delta function 614

A.8 The inequality ln :1: g x — 1 618

A.9 Relations between partial derivatives of several variables 61 9

A.10 The method of Lagrange multipliers 6‘20

A.11 Evaluation of the integral [0” (e’ — 1)-1a3 dx 622

A.12 The H theorem and the approach to equilibrium 6‘24

A.13 Liouville’s theorem in classical mechanics 6‘96

Numerical constants 629

Bibliography 631

Answers to selected problems 637

Index 643

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