Basics of Statistical Physics

Description

Statistics links microscopic and macroscopic phenomena, and requires for this reason a large number of microscopic elements like atoms. The results are values of maximum probability or of averaging. This introduction to statistical physics concentrates on the basic principles and attempts to explain these in simple terms, supplemented by numerous examples. These basic principles include the difference between classical and quantum statistics, a priori probabilities as related to degeneracies, the vital aspect of indistinguishability as compared with distinguishability in classical physics, the differences between conserved and non-conserved elements, the different ways of counting arrangements in the three statistics (Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein), the difference between maximization of the number of arrangements of elements, and averaging in the Darwin-Fowler method. Significant applications to solids, radiation and electrons in metals are treated in separate chapters, as well as Bose-Einstein condensation. In this latest edition, apart from a general revision, the topic of thermal radiation has been expanded with a new section on black bodies and an additional chapter on black holes. Other additions are more examples with applications of statistical mechanics in solid state physics and superconductivity. Throughout the presentation, the introduction carries almost all details for calculations. Contents Preface to Third Edition Preface to Second Edition Preface to First Edition 1 Introduction 1.1 Introductory Remarks 1.2 Thermodynamic Potentials 1.3 Capacity of Heat 1.4 Frequently Used Terms 1.5 Applications and Examples 1.6 Problems without Worked Solutions 2 Statistical Mechanics of an Ideal Gas (Maxwell) 2.1 Introductory Remarks 2.2 Maxwell’s Treatment 2.3 Lagrange’s Method of Multipliers 2.4 Applications 2.4.1 Pressure exerted on the wall of a vessel 2.4.2 Effusion of gas through a hole 2.4.3 Thermionic emission 2.5 Distribution Function for all Directions 2.6 Applications and Examples 2.7 Problems without Worked Solutions 3 The a priori Probability 3.1 Introductory Remarks 3.2 The a priori Probability 3.3 Examples Illustrating Liouville’s Theorem 3.4 Insertion of Physical Conditions 3.5 Applications and Examples 3.6 Problems without Worked Solutions 4 Classical Statistics (Maxwell–Boltzmann) 4.1 Introductory Remarks 4.2 The Number of Arrangements of Elements in Maxwell–Boltzmann Statistics 4.3 Method of Maximum Probability 4.3.1 The case of nonconserved elements 4.3.2 The case of conserved elements 4.3.3 The meaning of μ 4.3.4 Identification of μ with 1/kT 4.3.5 Distribution of particles in the atmosphere 4.3.6 Law of equipartition of energy 4.4 Applications 4.4.1 The monatomic gas 4.4.2 A solid 4.5 Applications and Examples 4.6 Problems without Worked Solutions 5 Entropy 5.1 Introductory Remarks 5.2 The Boltzmann Formula 5.3 Applications and Examples 5.4 Problems without Worked Solutions 6 Quantum Statistics 6.1 Introductory Remarks 6.2 A priori Weighting in Quantum Statistics 6.2.1 Approximate calculation of number of states 6.2.2 Accurate calculation of number of states 6.2.3 Examples 6.3 The Allowed Number of Elements in Quantum States 6.3.1 One element 6.3.2 Two non-interacting elements 6.3.3 More than two elements stuck together 6.4 Counting of Number of Arrangements 6.4.1 Fermi–Dirac statistics 6.4.2 Bose–Einstein statistics 6.5 Quantum Statistics at High Temperatures 6.6 Applications 6.7 Summary 6.8 Applications and Examples 6.9 Problems without Worked Solutions 7 Exact Form of Distribution Functions 7.1 Introductory Remarks 7.2 Fermi–Dirac Occupation Numbers 7.3 Bose–Einstein Occupation Numbers 7.4 Thermodynamical Functions 7.5 Applications and Examples 7.6 Problems without Worked Solutions 8 Application to Radiation (Light Quanta) 8.1 Introductory Remarks 8.2 Planck’s Radiation Law 8.3 Black Body Thermal Radiation 8.4 Applications and Examples 8.5 Problems without Worked Solutions 9 Debye Theory of Specific Heat of Solids 9.1 Introductory Remarks 9.2 The Calculation 9.3 Applications and Examples 9.4 Problems without Worked Solutions 10 Electrons in Metals 10.1 Introductory Remarks 10.2 Evaluation of the Distribution Function 10.2.1 First approximation 10.2.2 Second degree of approximation 10.3 Applications and Examples 10.4 Problems without Worked Solutions 11 Limitations of the Preceding Theory — Improvement with Ensemble Method 11.1 Introductory Remarks 11.2 Ensembles — Three Types 11.2.1 Ensembles and ergodic hypothesis 11.2.2 The ensemble distribution function 11.3 The Canonical Ensemble of a Closed System 11.3.1 Thermodynamics of a closed system in a heat bath 11.4 The Grand Canonical Ensemble 11.5 Ensemble Method of Maximum Probability 11.6 Comments on the Function ρ 11.7 Applications and Examples 11.8 Problems without Worked Solutions 12 Averaging instead of Maximization, and Bose–Einstein Condensation 12.1 Introductory Remarks 12.2 The Darwin–Fowler Method of Mean Values 12.2.1 Mean occupation number nj 12.2.2 Taking subsidiary condition into account 12.3 Classical Statistics 12.4 Quantum Statistics 12.4.1 Fermi–Dirac statistics 12.4.2 Bose–Einstein statistics 12.4.3 Evaluation of the coefficient of ωN in Zω 12.5 Bose–Einstein Condensation 12.5.1 The phenomenon of Bose–Einstein condensation 12.5.2 Derivation of the Bose–Einstein distribution function under condensation conditions 12.6 Applications and Examples 12.7 Problems without Worked Solutions 13 The Boltzmann Transport Equation 13.1 Introductory Remarks 13.2 Distribution Functions 13.3 Solution of the Boltzmann Equation 13.3.1 Solving the Boltzmann equation for two typical cases 13.3.2 Calculation of the current density 13.3.3 Application to metals 13.3.4 Calculation of the relaxation time 13.4 Applications and Examples 13.5 Problems without Worked Solutions 14 Thermal Radiation of Black Holes 14.1 Preliminary Remarks 14.2 Background Geometry 14.3 Rindler Coordinates 14.4 Introduction of Fields 14.5 Thermalization 14.6 Black Hole Evaporation 14.7 Applications and Examples 14.8 Problems without Worked Solutions Bibliography Index