The first objective of statistical mechanics is to explain the fundamental laws of thermodynamics from first principles based on the atomic structure of matter. This problem was attacked successfully first by MAXWELL and CLAUSIUS in studies on the kinetic theory of gases. It will be treated briefly in Sec. II-A, to gain some understanding and experience before dealing with more general problems. The second objective is then to calculate thermodynamics quantities from the microscopic laws governing the atomic motion. Whenever we try to lay the foundation of thermodynamics on an atomistic theory, we are confronted with a very strange situation. The thermodynamical state of a system is defined uniquely by only a few quantities, such as pressure, volume, energy, temperature, flow velocities, etc. In contrast, the atomistic descrip- tion needs an enormous number of variables to define a state, e. g. , positions and velocities of all the atoms involved in classical mechanics or Schrodinger's wave function of the corresponding N body-problem in quantum mechanics.
Classical mechanics, for instance, can predict the future development only if all the positions and velocities are known, say at time t = O. The number of values needed for this 23 purpose is of the order of 10 . Actually, only a few parameters are at our disposal from thermodynamics. Therefore, from thermodynamics we know almost nothing about the atomistic situation.
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Schweitzer Klassifikation
I. Thermodynamics.- A. The First and the Second Law of Thermodynamics.- 1. The concept of temperature and the equation of state of the ideal gas.- 2. State variables and equations of state.- 3. The first law of thermodynamics.- a) Change of volume.- b) Magnetization.- c) Energy of the ideal gas.- 4. Specific heat.- a) Heating at constant pressure.- b) Adiabatic processes.- c) The velocity of sound.- d) Energy and specific heat for ideal gases.- e) Equations of state for other "ideal" systems.- 5. The second law of thermodynamics.- 6. The Carnot efficiency.- 7. The general Carnot cycle and the definition of absolute temperature.- B. The Entropy.- 8. The entropy as state variable.- 9. The increase of entropy in closed systems.- 10. The entropy of an ideal gas.- a) N identical molecules.- b) A gas consisting of several components.- c) The increase of entropy for irreversible mixing.- C. Some Applications.- 11. The free energy.- 12. Equation of state and integrability.- a) T and V as independent variables.- b) Magnetization.- 13. The van der Waals equation.- a) General.- b) The critical point.- c) Condensation.- 14. The Joule-Thomson effect.- D. The Method of Carnot Cycles.- 15. Vapour pressure (Clausius-Capeyron's equation).- 16. Solutions.- a) The osmotic pressure.- b) The vapour pressure of solutions.- c) The boiling temperature of solutions.- d) The melting temperature of solutions.- 17. Chemical reactions in gases.- 18. Rutger's relation for superconductors.- E. Thermodynamical Functions (Potentials) and the Theory of Equilibrium.- 19. Thermodynamical functions (potentials) and applications to homogeneous phases.- a) Thermodynamical functions (potentials).- b) Homogeneous phases.- 20. Thermodynamical equilibrium.- 21. The chemical potential of the ideal gas.- 22. The vapour pressure of small liquid droplets.- II. Statistical Mechanics.- 23. Introduction.- A. Kinetic Theory of Gases.- 24. Equation of state of ideal gases.- 25. Maxwell's velocity distribution.- 26. Boltzmann's derivation of Maxwell's distribution and the H-theorem.- a) One single encounter.- b) Boltzmann's collision terms A, B.- c) Calculation of A and B.- d) The stationary velocity distribution.- e) Boltzmann's H-theorem.- f) Irreversibility and time reversal.- g) Boltzmann equation and the velocity of sound.- 27. The barometric pressure formula.- a) The mechanical method.- b) Kinetic theory of gases and barometric pressure.- c) Barometric pressure formula and thermodynamics.- 28. The virial theorem.- 29. Rarefied real gases.- B. Basic Concepts of Classical Mechanics.- 30. Hamilton's equations of motion.- a) Variational calculus, Lagrange's and Hamilton's equations of motion.- b) Canonical transformations.- 31. The ? space.- a) Definition of ? space.- b) Liouville's theorem.- c) The ergodic hypothesis.- d) The phase volume ?*.- C. The Microcanonical Ensemble.- 32. Time and ensemble averages.- a) The microcanonical ensemble.- b) Density fluctuations as an example.- c) Irreversibility.- d) The H-theorem.- 33. Some simple applications.- a) The equipartition theorem.- b) Again Maxwell's velocity distribution.- 34. The entropy.- a) The Hamiltonian contains parameters.- b) Adiabatic invariance of ?*.- c) The entropy S = k In ?*.- 35. The division by AM and the reduced phase volume ?(E, V, N).- a) ?*(E, V, N) of the ideal gas.- b) The final definition of phase volume.- c) The entropy of the ideal gas.- d) The volume Vv of a v-dimensional sphere.- D. The Canonical Ensemble.- 36. Two systems in thermal contact.- 37. The canonical ensemble.- a) One system is large and acts as a heat bath.- b) The canonical ensemble.- c) Two simple applications.- 38. Macroscopic systems.- a) The width of the canonical distribution.- b) The partition function (integral over states, Zustandsintegral).- c) The partition function of quantum theory (sum over states, Zustandssumme).- d) The partition function can be replaced by the maximum value of the integrand.- e) The connection between microcanonical and canonical ensemble.- E. Two More Ensembles.- 39. The free enthalpy.- a) Various experimental situations.- b) The mobile wall.- c) Fluctuations of volume.- 40. The grand canonical ensemble (given T, V, ?).- 41. Summary.- III. Quantum Statistics.- 42. Some preliminary results.- 43. Recollection of quantum theory.- a) The Schrodinger equation.- b) Hermitean and unitary operators.- c) Expectation values.- d) Time dependence of expectation values and of operators.- e) Parameter in the Hamilton operator depending on time.- 44. Statistical ensembles in quantum theory.- a) Definition of a statistical ensemble.- b) Time dependence of statistical ensembles.- c) The method of statistical phases.- d) Calculation of the transition probabilities.- 45. The entropy of a closed system.- a) The H-theorem and the microcanonical ensemble.- b) The "phase volume" ?(E, a) and the entropy in quantum theory.- 46. The canonical ensemble.- a) Two systems in thermal contact.- b) Many like systems in thermal contact.- c) The saddle point method.- d) The method of Lagrangean parameters.- IV. Ideal and Real Gases.- A. The Ideal Gas.- 47. The partition function for one particle.- 48. N like noninteracting particles.- 49. Closed system.- 50. System in a heat bath.- 51. The grand canonical ensemble.- 52. The limit of small densities.- 53. The Fermi-Dirac gas.- a) General treatment.- b) Applications to electrons in metals.- 54. The Bose-Einstein gas.- a) General discussion.- b) The condensation of the Bose-Einstein gas.- B. Real Gases and Their Condensation.- 55. The partition function.- 56. The unsaturated vapor.- 57. Condensation.- 58. The liquid phase.- 59. The analogy between the real classical and the perfect Bose-Einstein gas.- 60. Nucleation.- a) General.- b) Crude estimate of the critical supersaturation.- c) Kinetics of droplets formation.- V. Solids.- A. Caloric Properties.- 61. Classical theory.- 62. General quantum theoretical treatment.- 63. The linear chain.- 64. The three-dimensional crystal.- 65. Specific heat of crystals in the harmonic approximation.- B. Order and Disorder Phenomena in Crystals.- 66. Introduction.- 67. The statistical treatment.- 68. Superstructure.- a) Short range order.- b) Long range order.- 69. Precipitation.- 70. Ferromagnetism.- a) General.- b) The partition function of the Ising model.- c) Restriction to a single spin.- d) Connection to Weiss's theory of ferromagnetism.- 71. Bethe's treatment of the Ising model.- 72. Miscellaneous.- a) The matrix method.- b) Negative temperatures.- VI. Fluctuations and Brownian Motion.- A. Entropy and Probability.- 73. The general connection.- a) The statistical definition of entropy.- b) Entropy and probability.- 74. Fluctuations.- B. Brownian Motion.- 75. General statements.- 76. Mobility and diffusion.- 77. Langevin's equation.- 78. The method of Einstein and Hopf.- 79. The approach to Maxwell's distribution.- 80. Random walk.- 81. Correlation of a statistical function.- 82. Fokker-Planck's equation.- a) No external forces.- b) With external forces.- 83. The spectral distribution of a statistical function.- 84. The oscillator with weak, frequency dependent, damping.- a) General.- b) Cavity radiation.- c) Thermal noise in electrical circuits.- 85. The Nyquist theorem.- a) General derivation.- b) A simple model.- 86. The shot noise.- VII. Thermodynamics of Irreversible Processes.- 87. Increase of entropy by irreversible processes.- a) Linear oscillator with friction.- b) Heat exchange.- c) Heat conductivity in a continuous system.- 88. Irreversible processes in statistical mechanics.- 89. Simultaneous change of several macroscopic quantities.- a) Onsager's relations.- b) The derivation by Onsager.- 90. Cox's treatment.- 91. Irreversible exchange of energy and particles.- 92. Some applications.- a) Flow through a large aperture.- b) The diameter of the aperture is small as compared with the mean free path.- c) Helium II.- 93. The justification of Thomson's treatment by the Onsager relation.- a) Thomson's treatment.- b) The equivalence between Thomson's treatment and Onsager's relation.- c) The transport heat.- 94. Thermoelectric effects according to Thomson.- 95. Thermoelectric effects and Onsager relations.- a) The Thomson coefficient.- b) The Peltier heat.- c) The thermoelectric power.- Literature.- Author and Subject Index.
