Preface ....................................................... vii
Preface to the second edition ................................... x
I Preliminaries ............................................... 1
1 Introduction ................................................. 2
1.1 What is a mole? ......................................... 3
1.2 The thermodynamic limit ................................. 4
1.3 The ideal gas ........................................... 6
1.4 Combinatorial problems .................................. 7
1.5 Plan of the book ........................................ 9
Exercises ................................................... 12
2 Heat ........................................................ 13
2.1 A definition of heat ................................... 13
2.2 Heat capacity .......................................... 14
Exercises ................................................... 17
3 Probability ................................................. 18
3.1 Discrete probability distributions ..................... 19
3.2 Continuous probability distributions ................... 20
3.3 Linear transformation .................................. 21
3.4 Variance ............................................... 22
3.5 Linear transformation and the variance ................. 23
3.6 Independent variables .................................. 24
3.7 Binomial distribution .................................. 26
Further reading ............................................. 29
Exercises ................................................... 29
4 Temperature and the Boltzmann factor ........................ 32
4.1 Thermal equilibrium .................................... 32
4.2 Thermometers ........................................... 33
4.3 The microstates and macrostates ........................ 35
4.4 A statistical definition of temperature ................ 36
4.5 Ensembles .............................................. 38
4.6 Canonical ensemble ..................................... 38
4.7 Applications of the Boltzmann distribution ............. 42
Further reading ............................................. 46
Exercises ................................................... 46
II Kinetic theory of gases ..................................... 47
5 The Maxwell-Boltzmann distribution .......................... 48
5.1 The velocity distribution .............................. 48
5.2 The speed distribution ................................. 49
5.3 Experimental justification ............................. 51
Exercises ................................................... 54
6 Pressure .................................................... 56
6.1 Molecular distributions ................................ 57
6.2 The ideal gas law ...................................... 58
6.3 Dalton's law ........................................... 60
Exercises ................................................... 61
7 Molecular effusion .......................................... 64
7.1 Flux ................................................... 64
7.2 Effusion ............................................... 66
Exercises ................................................... 69
8 The mean free path and collisions ........................... 70
8.1 The mean collision time ................................ 70
8.2 The collision cross-section ............................ 71
8.3 The mean free path ..................................... 73
Exercises ................................................... 74
III Transport and thermal diffusion ............................ 75
9 Transport properties in gases ............................... 76
9.1 Viscosity .............................................. 76
9.2 Thermal conductivity ................................... 81
9.3 Diffusion .............................................. 83
9.4 More detailed theory ................................... 86
Further reading ............................................. 88
Exercises ................................................... 89
10 The thermal diffusion equation .............................. 90
10.1 Derivation of the thermal diffusion equation ........... 90
10.2 The one-dimensional thermal diffusion equation ......... 91
10.3 The steady state ....................................... 94
10.4 The thermal diffusion equation for a sphere ............ 94
10.5 Newton's law of cooling ................................ 99
10.6 The Prandtl number .................................... 100
10.7 Sources of heat ....................................... 101
10.8 Particle diffusion .................................... 102
Exercises .................................................. 103
IV The first law .............................................. 107
11 Energy ..................................................... 108
11.1 Some definitions ...................................... 108
11.2 The first law of thermodynamics ....................... 110
11.3 Heat capacity ......................................... 112
Exercises .................................................. 115
12 Isothermal and adiabatic processes ......................... 118
12.1 Reversibility ......................................... 118
12.2 Isothermal expansion of an ideal gas .................. 120
12.3 Adiabatic expansion of an ideal gas ................... 121
12.4 Adiabatic atmosphere .................................. 121
Exercises .................................................. 123
V The second law .............................................. 125
13 Heat engines and the second law ............................ 126
13.1 The second law of thermodynamics ...................... 126
13.2 The Carnot engine ..................................... 127
13.3 Carnot's theorem ...................................... 130
13.4 Equivalence of Clausius' and Kelvin's statements ...... 131
13.5 Examples of heat engines .............................. 131
13.6 Heat engines running backwards ........................ 133
13.7 Clausius' theorem ..................................... 134
Further reading ............................................ 137
Exercises .................................................. 137
14 Entropy .................................................... 140
14.1 Definition of entropy ................................. 140
14.2 Irreversible change ................................... 140
14.3 The first law revisited ............................... 142
14.4 The Joule expansion ................................... 144
14.5 The statistical basis for entropy ..................... 146
14.6 The entropy of mixing ................................. 147
14.7 Maxwell's demon ....................................... 149
14.8 Entropy and probability ............................... 150
Exercises .................................................. 153
15 Information theory ......................................... 157
15.1 Information and Shannon entropy ....................... 157
15.2 Information and thermodynamics ........................ 159
15.3 Data compression ...................................... 160
15.4 Quantum information ................................... 162
15.5 Conditional and joint probabilities ................... 165
15.6 Вayes' theorem ........................................ 165
Further reading ............................................ 168
Exercises .................................................. 169
VI Thermodynamics in action ................................... 171
16 Thermodynamic potentials ................................... 172
16.1 Internal energy, U .................................... 172
16.2 Enthalpy, H ........................................... 173
16.3 Helmholtz function, F ................................. 174
16.4 Gibbs function, G ..................................... 175
16.5 Constraints ........................................... 176
16.6 Maxwell's relations ................................... 179
Exercises .................................................. 187
17 Rods, bubbles, and magnets ................................. 191
17.1 Elastic rod ........................................... 191
17.2 Surface tension ....................................... 194
17.3 Electric and magnetic dipoles ......................... 195
17.4 Paramagnetism ......................................... 196
Exercises .................................................. 201
18 The third law .............................................. 203
18.1 Different statements of the third law ................. 203
18.2 Consequences of the third law ......................... 205
Exercises .................................................. 208
VII Statistical mechanics ..................................... 209
19 Equipartition of energy .................................... 210
19.1 Equipartition theorem ................................. 210
19.2 Applications .......................................... 213
19.3 Assumptions made ...................................... 215
19.4 Brownian motion ....................................... 217
Exercises .................................................. 218
20 The partition function ..................................... 219
20.1 Writing down the partition function ................... 220
20.2 Obtaining the functions of state ...................... 221
20.3 The big idea .......................................... 228
20.4 Combining partition functions ......................... 228
Exercises .................................................. 232
21 Statistical mechanics of an ideal gas ...................... 233
21.1 Density of states ..................................... 233
21.2 Quantum concentration ................................. 235
21.3 Distinguishability .................................... 236
21.4 Functions of state of the ideal gas ................... 237
21.5 Gibbs paradox ......................................... 240
21.6 Heat capacity of a diatomic gas ....................... 241
Exercises .................................................. 243
22 The chemical potential ..................................... 244
22.1 A definition of the chemical potential ................ 244
22.2 The meaning of the chemical potential ................. 245
22.3 Grand partition function .............................. 247
22.4 Grand potential ....................................... 248
22.5 Chemical potential as Gibbs function per particle ..... 250
22.6 Many types of particle ................................ 250
22.7 Particle number conservation laws ..................... 251
22.8 Chemical potential and chemical reactions ............. 252
22.9 Osmosis ............................................... 257
Further reading ............................................ 261
Exercises .................................................. 262
23 Photons .................................................... 263
23.1 The classical thermodynamics of electromagnetic
radiation ............................................. 264
23.2 Spectral energy density ............................... 265
23.3 Kirchhoff's law ....................................... 266
23.4 Radiation pressure .................................... 268
23.5 The statistical mechanics of the photon gas ........... 269
23.6 Black-body distribution ............................... 270
23.7 Cosmic microwave background radiation ................. 273
23.8 The Einstein A and В coefficients ..................... 274
Further reading ............................................ 277
Exercises .................................................. 278
24 Phonons .................................................... 279
24.1 The Einstein model .................................... 279
24.2 The Debye model ....................................... 281
24.3 Phonon dispersion ..................................... 284
Further reading ............................................ 287
Exercises .................................................. 287
VIII Beyond the ideal gas ..................................... 289
25 Relativistic gases ......................................... 290
25.1 Relativistic dispersion relation for massive
particles ............................................. 290
25.2 The ultrarelativistic gas ............................. 290
25.3 Adiabatic expansion of an ultrarelativistic gas ....... 293
Exercises .................................................. 295
26 Real gases ................................................. 296
26.1 The van der Waals gas ................................. 296
26.2 The Dieterici equation ................................ 304
26.3 Virial expansion ...................................... 306
26.4 The law of corresponding states ....................... 310
Exercises .................................................. 312
27 Cooling real gases ......................................... 313
27.1 The Joule expansion ................................... 313
27.2 Isothermal expansion .................................. 315
27.3 Joule-Kelvin expansion ................................ 316
27.4 Liquefaction of gases ................................. 318
Exercises .................................................. 320
28 Phase transitions .......................................... 321
28.1 Latent heat ........................................... 321
28.2 Chemical potential and phase changes .................. 324
28.3 The Clausius-Clapeyron equation ....................... 324
28.4 Stability and metastability ........................... 329
28.5 The Gibbs phase rule .................................. 332
28.6 Colligative properties ................................ 334
28.7 Classification of phase transitions ................... 335
28.8 The Ising model 338 Further reading ................... 343
Exercises .................................................. 343
29 Bose-Einstein and Fermi-Dirac distributions ................ 345
29.1 Exchange and symmetry ................................. 345
29.2 Wave functions of identical particles ................. 346
29.3 The statistics of identical particles ................. 349
Further reading ............................................ 353
Exercises .................................................. 354
30 Quantum gases and condensates .............................. 358
30.1 The non-interacting quantum fluid ..................... 358
30.2 The Fermi gas ......................................... 361
30.3 The Bose gas .......................................... 366
30.4 Bose Einstein condensation (ВЕС) ...................... 367
Further reading ............................................ 373
Exercises .................................................. 373
IX Special topics ............................................. 375
31 Sound waves ................................................ 376
31.1 Sound waves under isothermal conditions ............... 377
31.2 Sound waves under adiabatic conditions ................ 377
31.3 Are sound waves in general adiabatic or isothermal? ... 378
31.4 Derivation of the speed of sound within fluids ........ 379
Further reading ............................................ 382
Exercises .................................................. 382
32 Shock waves ................................................ 383
32.1 The Mach number ....................................... 383
32.2 Structure of shock waves .............................. 383
32.3 Shock conservation laws ............................... 385
32.1 32.4 The Rankine-Hugoniot conditions .................. 386
Further reading ............................................ 389
Exercises .................................................. 389
33 Brownian motion and fluctuations ........................... 390
33.1 Brownian motion ....................................... 390
33.2 Johnson noise ......................................... 393
33.3 Fluctuations .......................................... 394
33.4 Fluctuations and the availability ..................... 395
33.5 Linear response ....................................... 397
33.6 Correlation functions ................................. 400
Further reading ............................................ 407
Exercises .................................................. 407
34 Non-equilibrium thermodynamics ............................. 408
34.1 Entropy production .................................... 408
34.2 The kinetic coefficients .............................. 409
34.3 Proof of the Onsager reciprocal relations ............. 410
34.4 Thermoelectricity ..................................... 413
34.5 Time reversal and the arrow of time ................... 417
Further reading ............................................ 419
Exercises .................................................. 419
35 Stars ...................................................... 420
35.1 Gravitational interaction ............................. 421
35.2 Nuclear reactions ..................................... 426
35.3 Heat transfer ......................................... 427
Further reading ............................................ 434
Exercises .................................................. 434
36 Compact objects ............................................ 435
36.1 Electron degeneracy pressure .......................... 435
36.2 White dwarfs .......................................... 437
36.3 Neutron stars ......................................... 438
36.4 Black holes ........................................... 440
36.5 Accretion ............................................. 441
36.6 Black holes and entropy ............................... 442
36.7 Life, the Universe, and entropy ....................... 443
Further reading ............................................ 445
Exercises .................................................. 445
37 Earth's atmosphere ......................................... 446
37.1 Solar energy .......................................... 446
37.2 The temperature profile in the atmosphere ............. 447
37.3 Radiative transfer .................................... 449
37.4 The greenhouse effect ................................. 452
37.5 Global warming ........................................ 456
Further reading ............................................ 460
Exercises .................................................. 460
A Fundamental constants ....................................... 461
В Useful formulae ............................................ 462
С Useful mathematics ......................................... 464
C.l The factorial integral ................................. 464
C.2 The Gaussian integral .................................. 464
C.3 Stirling's formula ..................................... 467
C.4 Riemann zeta function .................................. 469
C.5 The poly logarithm ..................................... 470
C.6 Partial derivatives .................................... 471
С.7 Exact differentials .................................... 472
C.8 Volume of a hypersphere ................................ 473
C.9 Jacobians .............................................. 473
C.10 The Dirac delta function .............................. 475
C.11 Fourier transforms .................................... 475
C.12 Solution of the diffusion equation .................... 476
C.13 Lagrange multipliers .................................. 477
D The electromagnetic spectrum ................................ 479
E Some thermodynamical definitions ............................ 480
F Thermodynamic expansion formulae ............................ 481
G Reduced mass ................................................ 482
H Glossary of main symbols .................................... 483
Bibliography .................................................. 485
Index ......................................................... 489
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