Statistical Mechanics for Beginners: A Textbook for Undergraduates

by Benguigui, Lucien Gilles

• ISBN13: 9789814299114
• ISBN10: 9814299111
• Format: Hardcover
• Copyright: 2010-09-30
• Publisher: World Scientific Pub Co Inc

Summary

This textbook is for undergraduate students on a basic course in Statistical Mechanics. The prerequisite is thermodynamics. It begins with a study of three situations - the closed system and the systems in thermal contact with a reservoir - in order to formulate the important fundamentals: entropy from Boltzmann formula, partition function and grand partition function. Through the presentation of quantum statistics, Bose statistics and FermiDirac statistics are established, including as a special case the classical situation of MaxellBoltzmann statistics. A series of examples ensue it: the harmonic oscillator, the polymer chain, the two level system, bosons (photons, phonons, and the BoseEinstein condensation) and fermions (electrons in metals and in semiconductors). A compact historical note on influential scientists forms the concluding chapter. The unique presentation starts off with the principles, elucidating the well-developed theory, and only thereafter the application of theory. Calculations on the main steps are detailed, leaving behind minimal gap. The author emphasizes with theory the link between the macroscopic world (thermodynamics) and the microscopic world.

Table of Contents

Dedication	p. v
Preface	p. vii
List of Figures	p. xiii
Physical Constants	p. xv
Introduction	p. 1
Fundamentals	p. 7
The Closed System or the Microcanonical Ensemble	p. 9
The Microcanonical Ensemble	p. 9
Properties of the Entropy	p. 12
An Example	p. 17
The System in Thermal Contact with a Reservoir: The Canonical and Grand Canonical Ensembles	p. 22
The Canonical Ensemble	p. 23
The partition function	p. 23
Energy, entropy and thermodynamic potential	p. 26
A two-level system	p. 29
The ideal gas; equipartition of energy in classical mechanics	p. 30
The Grand Canonical Ensemble	p. 33
The grand partition function	p. 34
The number of particles, energy, entropy and the grand potential	p. 36
An example	p. 38
Summary	p. 41
Fluctuations	p. 42
Final remark	p. 43
Quantum Statistics	p. 44
The Partition Function and the Free Energy	p. 45
For variable N and for fermions	p. 46
For variable N and for bosons	p. 46
For fixed N, and for both fermions and bosons	p. 47
The Energy and the Entropy	p. 49
The Classical Ideal Gas: Maxwell-Boltzmann Statistics	p. 50
Qualitative Behavior of the Chemical Potential and the Derivation of (∂¿/∂T)V,N <0	p. 53
Bosons	p. 53
Fermions	p. 54
The Density of States	p. 55
The Wave Vector	p. 56
The Density of States	p. 57
The Monatomic Ideal Gas	p. 58
The partition function	p. 58
The internal energy, entropy and equation of state	p. 59
The classical limit	p. 60
Some Problems	p. 61
The Quantum Harmonic Oscillator	p. 61
Low temperature limit	p. 64
High temperature limit	p. 65
The Polyatomic Ideal Gas	p. 66
Bosons and Fermions in a Two-Level System	p. 68
The particles are bosons	p. 69
The particles are fermions	p. 72
Classical particles	p. 73
The Magnetic Chain	p. 75
Applications	p. 79
The Gas of Photons: The Black Body Radiation	p. 81
The Energy and the Energy Spectrum	p. 83
The Free Energy and the Entropy	p. 87
The relation with the wave picture	p. 88
Light Emission and Absorption of Solids; Kirchhoff's Law	p. 88
The Black Body Emission	p. 90
The Properties of Photon Gas are Independent of the Shape and the Material of the Cavity	p. 90
Atomic Vibration in Solids: Phonons	p. 92
Atomic Vibration in Solids	p. 92
The Properties of Phonons	p. 97
The Low Temperature Case	p. 97
The High Temperature Case	p. 99
The Debye Formula	p. 100
Resolution of the Differential Equation (7.1) by Means of Trigonometric Functions	p. 104
Derivation of the Expression (7.30) Giving Cv in the Debye Model	p. 105
The Boson Gas at Low Temperature: The Bose-Einstein Condensation	p. 107
The Chemical Potential	p. 108
The Energy, Specific Heat, Free Energy and Entropy	p. 112
Experimental Verfication	p. 114
The Gas of Fermions: Electrons in Metals and in Semiconductors	p. 116
Free Electrons in a Box	p. 116
The Fermi-Dirac function	p. 116
The chemical potential or the Fermi level	p. 120
The energy	p. 123
The specific heat	p. 127
Applications to metals	p. 129
Electrons in Semiconductors	p. 131
A History of Statistical Mechanics	p. 136
Thermodynamics and Statistical Mechanics Before Maxwell and Bolztmann	p. 136
The Kinetic Theory of Maxwell	p. 138
Boltzmann and Irreversibility	p. 139
Gibbs, the Father of Statistical Mechanics	p. 141
Planck and Einstein: Quantum Theory and Statistics	p. 142
The Method of Bose and the Bose-Einstein Condensation	p. 145
The Principle of Pauli and the Statistics of Fermi and Dirac	p. 146
Modern Developments	p. 147
Exercises	p. 148
Index	p. 161
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