Table Of ContentContents
1 From Microscopic to Macroscopic Behavior 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Some qualitative observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Doing work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Quality of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Some simple simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Work, heating, and the first law of thermodynamics . . . . . . . . . . . . . . . . . 14
1.7 Measuring the pressure and temperature . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 *The fundamental need for a statistical approach . . . . . . . . . . . . . . . . . . . 18
1.9 *Time and ensemble averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.10 *Models of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.10.1 The ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.10.2 Interparticle potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.10.3 Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.11 Importance of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Thermodynamic Concepts 26
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Pressure Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Some Thermodynamic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
i
CONTENTS ii
2.8 The First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.9 Energy Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.10 Heat Capacities and Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.11 Adiabatic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.12 The Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.13 The Thermodynamic Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.14 The Second Law and Heat Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.15 Entropy Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.16 Equivalence of Thermodynamic and Ideal Gas Scale Temperatures . . . . . . . . . 60
2.17 The Thermodynamic Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.18 The Fundamental Thermodynamic Relation . . . . . . . . . . . . . . . . . . . . . . 62
2.19 The Entropy of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.20 The Third Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.21 Free Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Appendix 2B: Mathematics of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 70
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 Concepts of Probability 82
3.1 Probability in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 The rules of probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Mean values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4 The meaning of probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.1 Information and uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.2 *Bayesianinference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.5 Bernoulli processes and the binomial distribution . . . . . . . . . . . . . . . . . . . 99
3.6 Continuous probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.7 The Gaussian distribution as a limit of the binomial distribution . . . . . . . . . . 111
3.8 The central limit theorem or why is thermodynamics possible? . . . . . . . . . . . 113
3.9 The Poisson distribution and should you fly in airplanes? . . . . . . . . . . . . . . 116
3.10 *Traffic flow and the exponential distribution . . . . . . . . . . . . . . . . . . . . . 117
3.11 *Are all probability distributions Gaussian? . . . . . . . . . . . . . . . . . . . . . . 119
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
CONTENTS iii
4 Statistical Mechanics 138
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.2 A simple example of a thermal interaction . . . . . . . . . . . . . . . . . . . . . . . 140
4.3 Counting microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3.1 Noninteracting spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3.2 *One-dimensional Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3.3 A particle in a one-dimensional box . . . . . . . . . . . . . . . . . . . . . . 151
4.3.4 One-dimensional harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 153
4.3.5 One particle in a two-dimensional box . . . . . . . . . . . . . . . . . . . . . 154
4.3.6 One particle in a three-dimensional box . . . . . . . . . . . . . . . . . . . . 156
4.3.7 Two noninteracting identical particles and the semiclassical limit . . . . . . 156
4.4 The number of states of N noninteracting particles: Semiclassical limit . . . . . . . 158
4.5 The microcanonical ensemble (fixed E, V, and N) . . . . . . . . . . . . . . . . . . . 160
4.6 Systems in contact with a heat bath: The canonical ensemble (fixed T, V, and N) 165
4.7 Connection between statistical mechanics and thermodynamics . . . . . . . . . . . 170
4.8 Simple applications of the canonical ensemble . . . . . . . . . . . . . . . . . . . . . 172
4.9 A simple thermometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.10 Simulations of the microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . . 177
4.11 Simulations of the canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.12 Grand canonical ensemble (fixed T, V, and µ) . . . . . . . . . . . . . . . . . . . . . 179
4.13 Entropy and disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Appendix 4A: The Volume of a Hypersphere . . . . . . . . . . . . . . . . . . . . . . . . 183
Appendix 4B: Fluctuations in the Canonical Ensemble . . . . . . . . . . . . . . . . . . . 184
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5 Magnetic Systems 190
5.1 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.2 Thermodynamics of magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.3 The Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.4 The Ising Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.4.1 Exact enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.4.2 Spin-spin correlation function . . . . . . . . . . . . . . . . . . . . . . . . . 199
∗
5.4.3 Simulations of the Ising chain . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.4.4 *Transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.4.5 Absence of a phase transition in one dimension . . . . . . . . . . . . . . . . 205
5.5 The Two-Dimensional Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
CONTENTS iv
5.5.1 Onsager solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
5.5.2 Computer simulation of the two-dimensional Ising model . . . . . . . . . . 211
5.6 Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.7 *Infinite-range interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
6 Noninteracting Particle Systems 230
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.2 The Classical Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.3 Classical Systems and the Equipartition Theorem . . . . . . . . . . . . . . . . . . . 238
6.4 Maxwell Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
6.5 Occupation Numbers and Bose and Fermi Statistics . . . . . . . . . . . . . . . . . 243
6.6 Distribution Functions of Ideal Bose and Fermi Gases . . . . . . . . . . . . . . . . 245
6.7 Single Particle Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.7.1 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.7.2 Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.8 The Equation of State for a Noninteracting Classical Gas . . . . . . . . . . . . . . 252
6.9 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.10 Noninteracting Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.10.1 Ground-state properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6.10.2 Low temperature thermodynamic properties. . . . . . . . . . . . . . . . . . 263
6.11 Bose Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.12 The Heat Capacity of a Crystalline Solid. . . . . . . . . . . . . . . . . . . . . . . . 272
6.12.1 The Einstein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
6.12.2 Debye theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Appendix 6A: Low Temperature Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 275
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
7 Thermodynamic Relations and Processes 288
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
7.2 Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
7.3 Applications of the Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . 291
7.3.1 Internal energy of an ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . 291
7.3.2 Relation between the specific heats . . . . . . . . . . . . . . . . . . . . . . . 291
7.4 Applications to Irreversible Processes. . . . . . . . . . . . . . . . . . . . . . . . . . 292
7.4.1 The Joule or free expansion process . . . . . . . . . . . . . . . . . . . . . . 293
CONTENTS v
7.4.2 Joule-Thomson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
7.5 Equilibrium Between Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
7.5.1 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7.5.2 Clausius-Clapeyron equation . . . . . . . . . . . . . . . . . . . . . . . . . . 298
7.5.3 Simple phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
7.5.4 Pressure dependence of the melting point . . . . . . . . . . . . . . . . . . . 301
7.5.5 Pressure dependence of the boiling point. . . . . . . . . . . . . . . . . . . . 302
7.5.6 The vapor pressure curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
7.6 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
8 Classical Gases and Liquids 306
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
8.2 The Free Energy of an Interacting System . . . . . . . . . . . . . . . . . . . . . . . 306
8.3 Second Virial Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
8.4 Cumulant Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
8.5 High Temperature Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.6 Density Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
8.7 Radial Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
8.7.1 Relation of thermodynamic functions to g(r) . . . . . . . . . . . . . . . . . 326
8.7.2 Relation of g(r) to static structure function S(k) . . . . . . . . . . . . . . . 327
8.7.3 Variable number of particles. . . . . . . . . . . . . . . . . . . . . . . . . . . 329
8.7.4 Density expansion of g(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
8.8 Computer Simulation of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
8.9 PerturbationTheory of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
8.9.1 The van der Waals Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 334
8.9.2 Chandler-Weeks-Andersen theory . . . . . . . . . . . . . . . . . . . . . . . . 335
8.10 *The Ornstein-Zernicke Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
8.11 *IntegralEquations for g(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.12 *Coulomb Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
8.12.1 Debye-Hu¨ckel Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
8.12.2 Linearized Debye-Hu¨ckel approximation . . . . . . . . . . . . . . . . . . . . 341
8.12.3 Diagrammatic Expansion for Charged Particles . . . . . . . . . . . . . . . . 342
8.13 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Appendix 8A: The third virial coefficient for hard spheres . . . . . . . . . . . . . . . . . 344
8.14 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
CONTENTS vi
9 Critical Phenomena 350
9.1 A Geometrical Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
9.2 Renormalization Group for Percolation . . . . . . . . . . . . . . . . . . . . . . . . . 354
9.3 The Liquid-Gas Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
9.4 Bethe Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
9.5 Landau Theory of Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.6 Other Models of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
9.7 Universality and Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
9.8 The Renormalization Group and the 1D Ising Model . . . . . . . . . . . . . . . . . 372
9.9 The Renormalization Group and the Two-Dimensional Ising Model . . . . . . . . . 376
9.10 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
9.11 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
10 Introduction to Many-Body Perturbation Theory 387
10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
10.2 Occupation Number Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 388
10.3 Operators in the Second Quantization Formalism . . . . . . . . . . . . . . . . . . . 389
10.4 Weakly Interacting Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
A Useful Formulae 397
A.1 Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
A.2 SI derived units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
A.3 Conversionfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
A.4 Mathematical Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
A.5 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
A.6 Euler-Maclaurinformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
A.7 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
A.8 Stirling’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
A.9 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
A.10Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
A.11Fermi integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
A.12Bose integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Chapter 1
From Microscopic to Macroscopic
Behavior
(cid:13)c2006 by Harvey Gould and Jan Tobochnik
28 August 2006
The goal of this introductory chapter is to explore the fundamental differences between micro-
scopic and macroscopic systems and the connections between classical mechanics and statistical
mechanics. We note that bouncing balls come to rest and hot objects cool, and discuss how the
behavior of macroscopic objects is related to the behavior of their microscopic constituents. Com-
puter simulations will be introduced to demonstrate the relation of microscopic and macroscopic
behavior.
1.1 Introduction
Our goal is to understand the properties of macroscopic systems, that is, systems of many elec-
trons, atoms, molecules, photons, or other constituents. Examples of familiar macroscopic objects
include systems such as the air in your room, a glass of water, a copper coin, and a rubber band
(examples of a gas, liquid, solid, and polymer, respectively). Less familiar macroscopic systems
are superconductors, cell membranes, the brain, and the galaxies.
We will find that the type of questions we ask about macroscopic systems differ in important
ways from the questions we ask about microscopic systems. An example of a question about a
microscopic system is “What is the shape of the trajectory of the Earth in the solar system?”
In contrast, have you ever wondered about the trajectory of a particular molecule in the air of
your room? Why not? Is it relevant that these molecules are not visible to the eye? Examples of
questions that we might ask about macroscopic systems include the following:
1. How does the pressure of a gas depend on the temperature and the volume of its container?
2. How does a refrigerator work? What is its maximum efficiency?
1
CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 2
3. How much energy do we need to add to a kettle of water to change it to steam?
4. Why are the properties of water different from those of steam, even though water and steam
consist of the same type of molecules?
5. How are the molecules arranged in a liquid?
6. How and why does water freeze into a particular crystalline structure?
7. Why does iron lose its magnetism above a certain temperature?
8. Why does helium condense into a superfluid phase at very low temperatures? Why do some
materials exhibit zero resistance to electrical current at sufficiently low temperatures?
9. How fast does a river current have to be before its flow changes from laminar to turbulent?
10. What will the weather be tomorrow?
The above questions can be roughly classified into three groups. Questions 1–3 are concerned
with macroscopic properties such as pressure, volume, and temperature and questions related to
heating and work. These questions are relevant to thermodynamics which provides a framework
for relating the macroscopic properties of a system to one another. Thermodynamics is concerned
onlywithmacroscopicquantitiesandignoresthemicroscopicvariablesthatcharacterizeindividual
molecules. For example, we will find that understanding the maximum efficiency of a refrigerator
doesnotrequireaknowledgeoftheparticularliquidusedasthecoolant. Manyoftheapplications
of thermodynamics are to thermal engines, for example, the internal combustion engine and the
steam turbine.
Questions 4–8 relate to understanding the behavior of macroscopic systems starting from the
atomic nature of matter. For example, we know that water consists of molecules of hydrogen
and oxygen. We also know that the laws of classical and quantum mechanics determine the
behavior of molecules at the microscopic level. The goal of statistical mechanics is to begin with
the microscopic laws of physics that govern the behavior of the constituents of the system and
deduce the properties of the system as a whole. Statistical mechanics is the bridge between the
microscopic and macroscopic worlds.
Thermodynamics and statistical mechanics assume that the macroscopic properties of the
system do not change with time on the average. Thermodynamics describes the change of a
macroscopic system from one equilibrium state to another. Questions 9 and 10 concern macro-
scopic phenomena that change with time. Related areas are nonequilibrium thermodynamics and
fluid mechanics from the macroscopic point of view and nonequilibrium statistical mechanics from
the microscopic point of view. Although there has been progress in our understanding of nonequi-
librium phenomena such as turbulent flow and hurricanes, our understanding of nonequilibrium
phenomena is much less advanced than our understanding of equilibrium systems. Because un-
derstanding the properties of macroscopic systems that are independent of time is easier, we will
focus our attention on equilibrium systems and consider questions such as those in Questions 1–8.
CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 3
1.2 Some qualitative observations
We begin our discussion of macroscopic systems by considering a glass of water. We know that if
we place a glass of hot water into a cool room, the hot water cools until its temperature equals
that of the room. This simple observation illustrates two important properties associated with
macroscopic systems – the importance of temperature and the arrow of time. Temperature is
familiar because it is associated with the physiological sensation of hot and cold and is important
in our everyday experience. We will find that temperature is a subtle concept.
Thedirectionorarrowoftimeisanevenmoresubtleconcept. Haveyoueverobservedaglass
of water at room temperature spontaneously become hotter? Why not? What other phenomena
exhibit a direction of time? Time has a direction as is expressed by the nursery rhyme:
Humpty Dumpty sat on a wall
Humpty Dumpty had a great fall
All the king’s horses and all the king’s men
Couldn’t put Humpty Dumpty back together again.
Is there a a direction of time for a single particle? Newton’s second law for a single particle,
F=dp/dt,impliesthatthemotionofparticlesistime reversal invariant,thatis,Newton’ssecond
law looks the same if the time t is replaced by −t and the momentum p by −p. There is no
directionoftimeatthemicroscopiclevel. Yetifwedropabasketballontoafloor, weknowthatit
will bounce and eventually come to rest. Nobody has observed a ball at rest spontaneously begin
to bounce, and then bounce higher and higher. So based on simple everyday observations, we can
conclude that the behavior of macroscopic bodies and single particles is very different.
Unlike generations of about a century or so ago, we know that macroscopic systems such as a
glass of water and a basketball consist of many molecules. Although the intermolecular forces in
water produce a complicated trajectory for each molecule, the observable properties of water are
easy to describe. Moreover, if we prepare two glasses of water under similar conditions, we would
find that the observable properties of the water in each glass are indistinguishable, even though
the motion of the individual particles in the two glasses would be very different.
Becausethemacroscopicbehaviorofwatermustberelatedinsomewaytothetrajectoriesofits
constituentmolecules,weconcludethattheremustbearelationbetweenthenotionoftemperature
andmechanics. Forthisreason,aswediscussthebehaviorofthemacroscopicpropertiesofaglass
of water and a basketball, it will be useful to discuss the relation of these properties to the motion
of their constituent molecules.
Forexample, ifwetakeintoaccountthatthebouncingballandthefloorconsistofmolecules,
then we know that the total energy of the ball and the floor is conserved as the ball bounces
and eventually comes to rest. What is the cause of the ball eventually coming to rest? You
might be tempted to say the cause is “friction,” but friction is just a name for an effective or
phenomenological force. At the microscopic level we know that the fundamental forces associated
with mass, charge, and the nucleus conserve the total energy. So if we take into account the
molecules of the ball and the floor, their total energy is conserved. Conservation of energy does
not explain why the inverse process does not occur, because such a process also would conserve
the total energy. So a more fundamental explanation is that the ball comes to rest consistent with
conservation of the total energy and consistent with some other principle of physics. We will learn
CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 4
that this principle is associated with an increase in the entropy of the system. For now, entropy is
only a name, and it is important only to understand that energy conservation is not sufficient to
understand the behavior of macroscopic systems. (As for most concepts in physics, the meaning
of entropy in the context of thermodynamics and statistical mechanics is very different than the
way entropy is used by nonscientists.)
For now, the nature of entropy is vague, because we do not have an entropy meter like we do
for energy and temperature. What is important at this stage is to understand why the concept of
energy is not sufficient to describe the behavior of macroscopic systems.
By thinking about the constituent molecules, we can gain some insight into the nature of
entropy. Let us consider the ball bouncing on the floor again. Initially, the energy of the ball
is associated with the motion of its center of mass, that is, the energy is associated with one
degree of freedom. However, after some time, the energy becomes associated with many degrees
of freedom associated with the individual molecules of the ball and the floor. If we were to bounce
the ball on the floor many times, the ball and the floor would each feel warm to our hands. So we
can hypothesize that energy has been transferred from one degree of freedom to many degrees of
freedom at the same time that the total energy has been conserved. Hence, we conclude that the
entropy is a measure of how the energy is distributed over the degrees of freedom.
What other quantities are associated with macroscopic systems besides temperature, energy,
andentropy? Wearealreadyfamiliarwithsomeofthesequantities. Forexample, wecanmeasure
the air pressure in a basketball and its volume. More complicated quantities are the thermal
conductivity of a solid and the viscosity of oil. How are these macroscopic quantities related to
each other and to the motion of the individual constituent molecules? The answers to questions
such as these and the meaning of temperature and entropy will take us through many chapters.
1.3 Doing work
We already have observed that hot objects cool, and cool objects do not spontaneously become
hot; bouncing balls come to rest, and a stationary ball does not spontaneously begin to bounce.
Andalthoughthetotalenergymustbeconservedinanyprocess,thedistributionofenergychanges
in an irreversible manner. We also have concluded that a new concept, the entropy, needs to be
introduced to explain the direction of change of the distribution of energy.
Now let us take a purely macroscopic viewpoint and discuss how we can arrive at a similar
qualitative conclusion about the asymmetry of nature. This viewpoint was especially important
historically because of the lack of a microscopic theory of matter in the 19th century when the
laws of thermodynamics were being developed.
Consider the conversion of stored energy into heating a house or a glass of water. The stored
energycouldbeintheformofwood,coal,oranimalandvegetableoilsforexample. Weknowthat
this conversion is easy to do using simple methods, for example, an open fireplace. We also know
that if we rub our hands together, they will become warmer. In fact, there is no theoretical limit1
to the efficiency at which we can convert stored energy to energy used for heating an object.
What about the process of converting stored energy into work? Work like many of the other
concepts that we have mentioned is difficult to define. For now let us say that doing work is
1Ofcourse,theefficiencycannotexceed100%.
