E-Book, Englisch, 562 Seiten
ISBN: 978-1-4831-0337-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark
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Weitere Infos & Material
1;Front Cover;1
2;Course of Theoretical Physics;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface to the third Russian edition;14
6;From the prefaces to previous Russian editions;16
7;Notation;18
8;CHAPTER I. THE FUNDAMENTAL PRINCIPLES OF STATISTICAL PHYSICS;20
8.1;§ 1. Statistical distributions;20
8.2;§ 2. Statistical independence;25
8.3;§ 3. Liouville's theorem;28
8.4;§ 4. The significance of energy;30
8.5;§ 5. The statistical matrix;33
8.6;§ 6. Statistical distributions in quantum statistics;40
8.7;§ 7. Entropy;42
8.8;§ 8. The law of increase of entropy;48
9;CHAPTER II. THERMODYNAMIC QUANTITIES;53
9.1;§ 9. Temperature;53
9.2;§ 10. Macroscopic motion;55
9.3;§ 11. Adiabatic processes;57
9.4;§ 12. Pressure;60
9.5;§ 13. Work and quantity of heat;63
9.6;§ 14. The heat function;66
9.7;§ 15. The free energy and the thermodynamic potential;67
9.8;§ 16. Relations between the derivatives of thermodynamic quantities;70
9.9;§ 17. The thermodynamic scale of temperature;74
9.10;§ 18. The Joule-Thomson process;75
9.11;§ 19. Maximum work;76
9.12;§ 20. Maximum work done by a body in an external medium;78
9.13;§ 21. Thermodynamic inequalities;82
9.14;§ 22. Le Chatelier's principle;84
9.15;§ 23. Nernst's theorem;87
9.16;§ 24. The dependence of the thermodynamic quantities on the number of particles;89
9.17;§ 25. Equilibrium of a body in an external field;92
9.18;§ 26. Rotating bodies;93
9.19;§ 27. Thermodynamic relations in the relativistic region;95
10;CHAPTER III. THE GIBBS DISTRIBUTION;98
10.1;§ 28. The Gibbs distribution;98
10.2;§ 29. The Maxwellian distribution;101
10.3;§ 30. The probability distribution for an oscillator;106
10.4;§ 31. The free energy in the Gibbs distribution;110
10.5;§ 32. Thermodynamic perturbation theory;114
10.6;§ 33. Expansion in powers of h;117
10.7;§ 34. The Gibbs distribution for rotating bodies;123
10.8;§ 35. The Gibbs distribution for a variable number of particles;125
10.9;§ 36. The derivation of the thermodynamic relations from the Gibbs distribution;128
11;CHAPTER IV. IDEAL GASES;130
11.1;§ 37. The Boltzmann distribution;130
11.2;§ 38. The Boltzmann distribution in classical statistics;132
11.3;§ 39. Molecular collisions;134
11.4;§ 40. Ideal gases not in equilibrium;137
11.5;§ 41. The free energy of an ideal Boltzmann gas;139
11.6;§ 42. The equation of state of an ideal gas;140
11.7;§ 43. Ideal gases with constant specific heat;144
11.8;§ 44. The law of equipartition;148
11.9;§ 45. Monatomic ideal gases;151
11.10;§ 46. Monatomic gases. The effect of the electronic angular momentum;154
11.11;§ 47. Diatomic gases with molecules of unlike atoms. Rotation of molecules;156
11.12;§ 48. Diatomic gases with molecules of like atoms. Rotation of molecules;160
11.13;§ 49. Diatomic gases. Vibrations of atoms;162
11.14;§ 50. Diatomic gases. The effect of the electronic angular momentum;165
11.15;§ 51. Polyatomic gases;167
11.16;§ 52. Magnetism of gases;171
12;CHAPTER V. THE FERMI AND BOSE DISTRIBUTIONS;177
12.1;§ 53. Hie Fermi distribution;177
12.2;§ 54. The Bose distribution;178
12.3;§ 55. Fermi and Bose gases not in equilibrium;179
12.4;§ 56. Fermi and Bose gases of elementar)' particles;181
12.5;§ 57. A degenerate electron gas;185
12.6;§ 58. The specific heat of a degenerate electron gas;187
12.7;§ 59. Magnetism of an electron gas. Weak fields;190
12.8;§ 60. Magnetism of an electron gas. Strong fields;194
12.9;§ 61. A relativistic degenerate electron gas;197
12.10;§ 62. A degenerate Bose gas;199
12.11;§ 63. Black-body radiation;202
13;CHAPTER VI. SOLIDS;210
13.1;§ 64. Solids at low temperatures;210
13.2;§ 65. Solids at high temperatures;214
13.3;§ 66. Debye's interpolation formula;217
13.4;§ 67. Thermal expansion of solids;220
13.5;§ 68. Highly anisotropic crystals;222
13.6;§ 69. Crystal lattice vibrations;226
13.7;§ 70. Number density of vibrations;230
13.8;§ 71. Phonons;234
13.9;§ 72. Phonon creation and annihilation operators;237
13.10;§ 73. Negative temperatures;240
14;CHAPTER VII. NON-IDEAL GASES;244
14.1;§ 74. Deviations of gases from the ideal state;244
14.2;§ 75. Expansion in powers of the density;249
14.3;§ 76. Van der Waals' formula;251
14.4;§ 77. Relationship of the virial coeflicient and the scattering amplitude;255
14.5;§ 78. Thermodynamic quantities for a classical plasma;258
14.6;§ 79. The method of correlation functions;262
14.7;§ 80. Thermodynamic quantities for a degenerate plasma;264
15;CHAPTER VIII. PHASE EQUILIBRIUM;270
15.1;§ 81. Conditions of phase equilibrium;270
15.2;§ 82. The Clapeyron–Clausius formula;274
15.3;§ 83. The critical point;276
15.4;§ 84. The law of corresponding states;279
16;CHAPTER IX. SOLUTIONS;282
16.1;§ 85. Systems containing different particles;282
16.2;§ 86. The phase rule;283
16.3;§ 87. Weak solutions;284
16.4;§ 88. Osmotic pressure;286
16.5;§ 89. Solvent phases in contact;287
16.6;§ 90. Equilibrium with respect to the solute;290
16.7;§ 91. Evolution of heat and change of volume on dissolution;293
16.8;§ 92. Solutions of strong electrolytes;296
16.9;§ 93. Mixtures of ideal gases;298
16.10;§ 94. Mixtures of isotopes;300
16.11;§ 95. Vapour pressure over concentrated solutions;302
16.12;§ 96. Thermodynamic inequalities for solutions;305
16.13;§ 97. Equilibrium curves;308
16.14;§ 98. Examples of phase diagrams;314
16.15;§ 99. Intersection of singular curves on the equilibrium surface;319
16.16;§ 100. Gases and liquids;320
17;CHAPTER X. CHEMICAL REACTIONS;324
17.1;§ 101. The condition for chemical equilibrium;324
17.2;§ 102. The law of mass action;325
17.3;§ 103. Heat of reaction;329
17.4;§ 104. Ionisation equilibrium;332
17.5;§ 105. Equilibrium with respect to pair production;334
18;CHAPTER XI PROPERTIES OF MATTER AT VERY HIGH DENSITY;336
18.1;§ 106. The equation of state of matter at high density;336
18.2;§ 107. Equilibrium of bodies of large mass;339
18.3;§ 108. The energy of a gravitating body;346
18.4;§ 109. Equilibrium of a neutron sphere;348
19;CHAPTER XII. FLUCTUATIONS;352
19.1;§ 110. The Gaussian distribution;352
19.2;§111. The Gaussian distribution for more than one variable;354
19.3;§ 112. Fluctuations of the fundamental thermodynamic quantities;357
19.4;§ 113. Fluctuations in an ideal gas;364
19.5;§ 114. Poisson's formula;366
19.6;§ 115. Fluctuations in solutions;368
19.7;§ 116. Spatial correlation of density fluctuations;369
19.8;§ 117. Correlation of density fluctuations in a degenerate gas;373
19.9;§ 118. Correlations of fluctuations in time;378
19.10;§ 119. Time correlations of the fluctuations of more than one variable;382
19.11;§ 120. The symmetry of the kinetic coefficients;384
19.12;§ 121. The dissipative function;387
19.13;§ 122. Spectral resolution of fluctuations;390
19.14;§ 123. The generalised susceptibility;396
19.15;§ 124. The fluctuation-dissipation theorem;403
19.16;§ 125. The fluctuation-dissipation theorem for more than one variable;408
19.17;§ 126. The operator form of the generalised susceptibility;412
19.18;§ 127. Fluctuations in the curvature of long molecules;415
20;CHAPTER XIII. THE SYMMETRY OF CRYSTALS;420
20.1;§128. Symmetry elements of a crystal lattice;420
20.2;§ 129. The Bravais lattice;422
20.3;§ 130. Crystal systems;424
20.4;§ 131. Crystal classes;428
20.5;§ 132. Space groups;430
20.6;§ 133. The reciprocal lattice;432
20.7;§ 134. Irreducible representations of space groups;435
20.8;§ 135. Symmetry under time reversal;441
20.9;§ 136. Symmetry properties of normal vibrations of a crystal lattice;446
20.10;§ 137. Structures periodic in one and two dimensions;451
20.11;§ 138. The correlation function in two-dimensional systems;455
20.12;§ 139. Symmetry with respect to orientation of molecules;457
20.13;§ 140. Nematic and cholesteric liquid crystals;459
20.14;§ 141. Fluctuations in liquid crystals;461
21;CHAPTER XIV. PHASE TRANSITIONS OF THE SECOND KIND AND CRITICAL PHENOMENA;465
21.1;§ 142. Phase transitions of the second kind;465
21.2;§ 143. The discontinuity of specific heat;470
21.3;§ 144. Effect of an external field on a phase transition;475
21.4;§ 145. Change in symmetry in a phase transition of the second kind;478
21.5;§ 146. Fluctuations of the order parameter;490
21.6;§ 147. The effective Hamiltonian;497
21.7;§ 148. Critical indices;502
21.8;§ 149. Scale invariance;508
21.9;§ 150. Isolated and critical points of continuous transition;512
21.10;§ 151. Phase transitions of the second kind in a two-dimensional lattice;517
21.11;§ 152. Van der Waals theory of the critical point;525
21.12;§ 153. Fluctuation theory of the critical point;530
22;CHAPTER XV. SURFACES;536
22.1;§ 154. Surface tension;536
22.2;§ 155. Surface tension of crystals;539
22.3;§ 156. Surface pressure;541
22.4;§ 157. Surface tension of solutions;543
22.5;§ 158. Surface tension of solutions of strong electrolytes;545
22.6;§ 159. Adsorption;546
22.7;§ 160. Wetting;548
22.8;§ 161. The angle of contact;550
22.9;§ 162. Nucleation in phase transitions;552
22.10;§ 163. The impossibility of the existence of phases in one-dimensional systems;556
23;INDEX;558
CHAPTER I THE FUNDAMENTAL PRINCIPLES OF STATISTICAL PHYSICS
Publisher Summary
Statistical physics consists in the study of the special laws that govern the behavior and properties of macroscopic bodies, that is, bodies formed of a very large number of individual particles, such as atoms and molecules. To a considerable extent, the general character of these laws does not depend on the mechanics (classical or quantum), which describes the motion of the individual particles in a body; however, their substantiation demands a different argument in the two cases. When statistical physics is applied to macroscopic bodies, its probabilistic nature is not usually apparent. The reason is that, if any macroscopic body—in external conditions independent of time—is observed over a sufficiently long period of time, it is found that all physical quantities describing the body are practically constant and equal to their mean values and undergo appreciable changes relatively very rarely. This chapter discusses the properties of the statistical distribution function, Liouville’s theorem, the significance of energy and the statistical matrix. § 1. Statistical distributions
Statistical physics, often called for brevity simply statistics, consists in the study of the special laws which govern the behaviour and properties of macroscopic bodies (that is, bodies formed of a very large number of individual particles, such as atoms and molecules). To a considerable extent the general character of these laws does not depend on the mechanics (classical or quantum) which describes the motion of the individual particles in a body, but their substantiation demands a different argument in the two cases. For convenience of exposition we shall begin by assuming that classical mechanics is everywhere valid. In principle, we can obtain complete information concerning the motion of a mechanical system by constructing and integrating the equations of motion of the system, which are equal in number to its degrees of freedom. But if we are concerned with a system which, though it obeys the laws of classical mechanics, has a very large number of degrees of freedom, the actual application of the methods of mechanics involves the necessity of setting up and solving the same number of differential equations, which in general is impracticable. It should be emphasised that, even if we could integrate these equations in a general form, it would be completely impossible to substitute in the general solution the initial conditions for the velocities and coordinates of all the particles. At first sight we might conclude from this that, as the number of particles increases, so also must the complexity and intricacy of the properties of the mechanical system, and that no trace of regularity can be found in the behaviour of a macroscopic body. This is not so, however, and we shall see below that, when the number of particles is very large, new types of regularity appear. These statistical laws resulting from the very presence of a large number of particles forming the body cannot in any way be reduced to purely mechanical laws. One of their distinctive features is that they cease to have meaning when applied to mechanical systems with a small number of degrees of freedom. Thus, although the motion of systems with a very large number of degrees of freedom obeys the same laws of mechanics as that of systems consisting of a small number of particles, the existence of many degrees of freedom results in laws of a different kind. The importance of statistical physics in many other branches of theoretical physics is due to the fact that in Nature we continually encounter macroscopic bodies whose behaviour can not be fully described by the methods of mechanics alone, for the reasons mentioned above, and which obey statistical laws. In proceeding to formulate the fundamental problem of classical statistics, we must first of all define the concept of phase space, which will be constantly used hereafter. Let a given macroscopic mechanical system have s degrees of freedom: that is, let the position of points of the system in space be described by s coordinates, which we denote by qi, the suffix i taking the values 1, 2, …, s. Then the state of the system at a given instant will be defined by the values at that instant of the s coordinates qi and the s corresponding velocities . In statistics it is customary to describe a system by its coordinates and momenta pi, not velocities, since this affords a number of very important advantages. The various states of the system can be represented mathematically by points in phase space (which is, of course, a purely mathematical concept); the coordinates in phase space are the coordinates and momenta of the system considered. Every system has its own phase space, with a number of dimensions equal to twice the number of degrees of freedom. Any point in phase space, corresponding to particular values of the coordinates qi and momenta pi of the system, represents a particular state of the system. The state of the system changes with time, and consequently the point in phase space representing this state (which we shall call simply the phase point of the system) moves along a curve called the phase trajectory. Let us now consider a macroscopic body or system of bodies, and assume that the system is closed, i.e. does not interact with any other bodies. A part of the system, which is very small compared with the whole system but still macroscopic, may be imagined to be separated from the rest; clearly, when the number of particles in the whole system is sufficiently large, the number in a small part of it may still be very large. Such relatively small but still macroscopic parts will be called subsystems. A subsystem is again a mechanical system, but not a closed one; on the contrary, it interacts in various ways with the other parts of the system. Because of the very large number of degrees of freedom of the other parts, these interactions will be very complex and intricate. Thus the state of the subsystem considered will vary with time in a very complex and intricate manner. An exact solution for the behaviour of the subsystem can be obtained only by solving the mechanical problem for the entire closed system, i.e. by setting up and solving all the differential equations of motion with given initial conditions, which, as already mentioned, is an impracticable task. Fortunately, it is just this very complicated manner of variation of the state of subsystems which, though rendering the methods of mechanics inapplicable, allows a different approach to the solution of the problem. A fundamental feature of this approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state. This may be more precisely formulated as follows. Let ?p ?q denote some small “volume” of the phase space of the subsystem, corresponding to coordinates qi and momenta pi lying in short intervals ?qi and ?pi. We can say that, in a sufficiently long time T, the extremely intricate phase trajectory passes many times through each such volume of phase space. Let ?t be the part of the total time T during which the subsystem was in the given volume of phase space ?p ?q.† When the total time T increases indefinitely, the ratio ?t/T tends to some limit =limT?8?t/T. (1.1) (1.1) This quantity may clearly be regarded as the probability that, if the subsystem is observed at an arbitrary instant, it will be found in the given volume of phase space ?p?q. On taking the limit of an infinitesimal phase volume‡ q dp=dq1dq2…dqsdp1dp2…dps, (1.2) (1.2) we can define the probability dw of states represented by points in this volume element, i.e. the probability that the coordinates qi and momenta pi have values in given infinitesimal intervals between qi,pi and qi + dqi, pi + dpi. This probability dw may be written w=?(p1,…,ps, q1,…,qs)dp dq, (1.3) (1.3) where (p1, …, ps, q1, …, qs) is a function of all the coordinates and momenta; we shall usually write for brevity (p, q) or even simply. The function , which represents the “density” of the probability distribution in phase space, is called the statistical distribution function, or simply the distribution function, for the body concerned. This function must obviously satisfy the normalisation condition ? dp dq=1 (1.4) (1.4) (the integral being taken over all phase space), which simply expresses the fact that the sum of the probabilities of all possible states must be unity. The following...
