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.
Author(s): Harald J .W. Müller-Kirsten
Edition: 3
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 270
City: Singapore
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
