Statistical Mechanics

by Schwabl, Franz

• ISBN13: 9783540431633
• ISBN10: 3540431632
• Format: Hardcover
• Copyright: 2003-01-01
• Publisher: Springer Verlag

Summary

In addition to a deductive approach to equilibrium statistics and thermodynamics based on a single hypothesis - the form of the microcanonical density matrix - this book treats the most important elements of non-equilibrium phenomena. Intermediate calculations are presented in complete detail. Problems at the end of each chapter help students to consolidate their understanding of the material. Beyond the fundamentals, this text demonstrates the breadth of the field and its great variety of applications. Modern areas such as renormalization group theory, percolation, stochastic equations of motion and their applications to critical dynamics, as well as fundamental considerations of irreversibility, are discussed. The text will be useful for advanced students of physics and other natural sciences; a basic knowledge of quantum mechanics is presumed.

Table of Contents

Basic Principles	1	(24)
Introduction	1	(3)
A Brief Excursion into Probability Theory	4	(5)
Probability Density and Characteristic Functions	4	(3)
The Central Limit Theorem	7	(2)
Ensembles in Classical Statistics	9	(5)
Phase Space and Distribution Functions	9	(2)
The Liouville Equation	11	(3)
Quantum Statistics	14	(2)
The Density Matrix for Pure and Mixed Ensembles	14	(1)
The Von Neumann Equation	15	(1)
Additional remarks	16	(9)
The Binomial and the Poisson Distributions	16	(3)
Mixed Ensembles and the Density Matrix of Subsystems	19	(2)
Problems	21	(4)
Equilibrium Ensembles	25	(50)
Introductory Remarks	25	(1)
Microcanonical Ensembles	26	(9)
Microcanonical Distribution Functions and Density Matrices	26	(4)
The Classical Ideal Gas	30	(3)
Quantum-mechanical Harmonic Oscillators and Spin Systems	33	(2)
Entropy	35	(3)
General Definition	35	(1)
An Extremal Property of the Entropy	36	(1)
Entropy of the Microcanonical Ensemble	37	(1)
Temperature and Pressure	38	(8)
Systems in Contact: the Energy Distribution Function, Definition of the Temperature	38	(3)
On the Widths of the Distribution Functions of Macroscopic Quantities	41	(1)
External Parameters: Pressure	42	(4)
Properties of Some Non-interacting Systems	46	(4)
The Ideal Gas	46	(2)
Non-interacting Quantum Mechanical Harmonic Oscillators and Spins	48	(2)
The Canonical Ensemble	50	(12)
The Density Matrix	50	(2)
Examples: the Maxwell Distribution and the Barometric Pressure Formula	52	(1)
The Entropy of the Canonical Ensemble and Its Extremal Values	53	(1)
The Virial Theorem and the Equipartition Theorem	54	(3)
Thermodynamic Quantities in the Canonical Ensemble	57	(3)
Additional Properties of the Entropy	60	(2)
The Grand Canonical Ensemble	62	(13)
Systems with Particle Exchange	62	(1)
The Grand Canonical Density Matrix	63	(2)
Thermodynamic Quantities	65	(1)
The Grand Partition Function for the Classical Ideal Gas	66	(2)
The Grand Canonical Density Matrix in Second Quantization	68	(1)
Problems	69	(6)
Thermodynamics	75	(90)
Potentials and Laws of Equilibrium Thermodynamics	75	(7)
Definitions	75	(4)
The Legendre Transformation	79	(2)
The Gibbs-Duhem Relation in Homogeneous Systems	81	(1)
Derivatives of Thermodynamic Quantities	82	(7)
Definitions	82	(2)
Integrability and the Maxwell Relations	84	(3)
Jacobians	87	(1)
Examples	88	(1)
Fluctuations and Thermodynamic Inequalities	89	(2)
Fluctuations	89	(1)
Inequalities	90	(1)
Absolute Temperature and Empirical Temperatures	91	(1)
Thermodynamic Processes	92	(11)
Thermodynamic Concepts	93	(2)
The Irreversible Expansion of a Gas; the Gay-Lussac Experiment	95	(2)
The Statistical Foundation of Irreversibility	97	(1)
Reversible Processes	98	(4)
The Adiabatic Equation	102	(1)
The First and Second Laws of Thermodynamics	103	(22)
The First and the Second Law for Reversible and Irreversible Processes	103	(4)
Historical Formulations of the Laws of Thermodynamics and other Remarks	107	(2)
Examples and Supplements to the Second Law	109	(11)
Extremal Properties	120	(3)
Thermodynamic Inequalities Derived from Maximization of the entropy	123	(2)
Cyclic Processes	125	(5)
General Considerations	125	(1)
The Carnot Cycle	126	(2)
General Cyclic Processes	128	(2)
Phases of Single-Component Systems	130	(14)
Phase-Boundary Curves	130	(4)
The Clausius-Clapeyron Equation	134	(5)
The Convexity of the Free Energy and the Concavity of the Free Enthalpy (Gibbs' Free Energy)	139	(2)
The Triple Point	141	(3)
Equilibrium in Multicomponent Systems	144	(21)
Generalization of the Thermodynamic Potentials	144	(2)
Gibbs' Phase Rule and Phase Equilibrium	146	(4)
Chemical Reactions, Thermodynamic Equilibrium and the Law of Mass Action	150	(4)
Vapor-pressure Increase by Other Gases and by Surface Tension	154	(4)
Problems	158	(7)
Ideal Quantum Gases	165	(56)
The Grand Potential	165	(6)
The Classical Limit z = eμ/kT << 1	171	(1)
The Nearly-degenerate Ideal Fermi Gas	172	(14)
Ground State, T = 0 (Degeneracy)	173	(1)
The Limit of Complete Degeneracy	174	(7)
Real Fermions	181	(5)
The Bose-Einstein Condensation	186	(7)
The Photon Gas	193	(9)
Properties of Photons	193	(2)
The Canonical Partition Function	195	(1)
Planck's Radiation Law	196	(4)
Supplemental Remarks	200	(1)
Fluctuations in the Particle Number of Fermions and Bosons	201	(1)
Phonons in Solids	202	(7)
The Harmonic Hamiltonian	202	(3)
Thermodynamic Properties	205	(2)
Anharmonic Effects, the Mie-Gruneisen Equation of State	207	(2)
Phonons und Rotons in He II	209	(12)
The Excitations (Quasiparticles) of He II	209	(2)
Thermal Properties	211	(2)
Superfluidity and the Two-Fluid Model	213	(4)
Problems	217	(4)
Real Gases, Liquids, and Solutions	221	(44)
The Ideal Molecular Gas	221	(9)
The Hamiltonian and the Partition Function	221	(2)
The Rotational Contribution	223	(3)
The Vibrational Contribution	226	(2)
The Influence of the Nuclear Spin	228	(2)
Mixtures of Ideal Molecular Gases	230	(2)
The Virial Expansion	232	(6)
Derivation	232	(2)
The Classical Approximation for the Second Virial Coefficient	234	(3)
Quantum Corrections to the Virial Coefficients	237	(1)
The Van der Waals Equation of State	238	(15)
Derivation	238	(5)
The Maxwell Construction	243	(4)
The Law of Corresponding States	247	(1)
The Vicinity of the Critical Point	247	(6)
Dilute Solutions	253	(12)
The Partition Function and the Chemical Potentials	253	(4)
Osmotic Pressure	257	(1)
Solutions of Hydrogen in Metals (Nb, Pd,...)	258	(1)
Freezing-Point Depression, Boiling-Point Elevation, and Vapor-Pressure Reduction	259	(3)
Problems	262	(3)
Magnetism	265	(62)
The Density Matrix and Thermodynamics	265	(9)
The Hamiltonian and the Canonical Density Matrix	265	(4)
Thermodynamic Relations	269	(3)
Supplementary Remarks	272	(2)
The Diamagnetism of Atoms	274	(2)
The Paramagnetism of Non-coupled Magnetic Moments	276	(4)
Pauli Spin Paramagnetism	280	(3)
Ferromagnetism	283	(20)
The Exchange Interaction	283	(2)
The Molecular Field Approximation for the Ising Model	285	(11)
Correlation Functions and Susceptibility	296	(1)
The Ornstein-Zernike Correlation Function	297	(4)
Continuum Representation	301	(2)
The Dipole Interaction, Shape Dependence, Internal and External Fields	303	(10)
The Hamiltonian	303	(1)
Thermodynamics and Magnetostatics	304	(4)
Statistical-Mechanical Justification	308	(4)
Domains	312	(1)
Applications to Related Phenomena	313	(14)
Polymers and Rubber-like Elasticity	313	(3)
Negative Temperatures	316	(3)
The Melting Curve of 3He	319	(2)
Problems	321	(6)
Phase Transitions, Renormalization Group Theory, and Percolation	327	(78)
Phase Transitions and Critical Phenomena	327	(8)
Symmetry Breaking, the Ehrenfest Classification	327	(1)
Examples of Phase Transitions and Analogies	328	(6)
Universality	334	(1)
The Static Scaling Hypothesis	335	(6)
Thermodynamic Quantities and Critical Exponents	335	(4)
The Scaling Hypothesis for the Correlation Function	339	(2)
The Renormalization Group	341	(16)
Introductory Remarks	341	(1)
The One-Dimensional Ising Model, Decimation Transformation	342	(3)
The Two-Dimensional Ising Model	345	(7)
Scaling Laws	352	(3)
General RG Transformations in Real Space	355	(2)
The Ginzburg-Landau Theory	357	(26)
Ginzburg-Landau Functionals	357	(3)
The Ginzburg-Landau Approximation	360	(2)
Fluctuations in the Gaussian Approximation	362	(7)
Continuous Symmetry and Phase Transitions of First Order	369	(7)
The Momentum-Shell Renormalization Group	376	(7)
Percolation	383	(22)
The Phenomenon of Percolation	383	(4)
Theoretical Description of Percolation	387	(1)
Percolation in One Dimension	388	(1)
The Bethe Lattice (Cayley Tree)	389	(5)
General Scaling Theory	394	(2)
Real-Space Renormalization Group Theory	396	(4)
Problems	400	(5)
Brownian Motion, Equations of Motion and the Fokker-Planck Equations	405	(28)
Langevin Equations	405	(7)
The Free Langevin Equation	405	(5)
The Langevin Equation in a Force Field	410	(2)
The Derivation of the Fokker-Planck Equation from the Langevin Equation	412	(4)
The Fokker-Planck Equation for the Langevin Equation (8.1.1)	412	(2)
Derivation of the Smoluchowski Equation for the Overdamped Langevin Equation, (8.1.23)	414	(2)
The Fokker-Planck Equation for the Langevin Equation (8.1.22b)	416	(1)
Examples and Applications	416	(17)
Integration of the Fokker-Planck Equation (8.2.6)	416	(2)
Chemical Reactions	418	(3)
Critical Dynamics	421	(4)
The Smoluchowski Equation and Supersymmetric Quantum Mechanics	425	(3)
Problems	428	(5)
The Boltzmann Equation	433	(42)
Introduction	433	(1)
Derivation of the Boltzmann Equation	434	(5)
Consequences of the Boltzmann Equation	439	(12)
The H-Theorem and Irreversibility	439	(3)
Behavior of the Boltzmann Equation under Time Reversal	442	(1)
Collision Invariants and the Local Maxwell Distribution	443	(2)
Conservation Laws	445	(2)
The Hydrodynamic Equations in Local Equilibrium	447	(4)
The Linearized Boltzmann Equation	451	(13)
Linearization	451	(2)
The Scalar Product	453	(1)
Eigenfunctions of L and the Expansion of the Solutions of the Boltzmann Equation	454	(2)
The Hydrodynamic Limit	456	(6)
Solutions of the Hydrodynamic Equations	462	(2)
Supplementary Remarks	464	(11)
Relaxation-Time Approximation	464	(1)
Calculation of W (v1, v2; v'1, v'2)	465	(7)
Problems	472	(3)
Irreversibility and the Approach to Equilibrium	475	(34)
Preliminary Remarks	475	(2)
Recurrence Time	477	(3)
The Origin of Irreversible Macroscopic Equations of Motion	480	(7)
A Microscopic Model for Brownian Motion	480	(6)
Microscopic Time-Reversible and Macroscopic Irreversible Equations of Motion, Hydrodynamics	486	(1)
The Master Equation and Irreversibility in Quantum Mechanics	487	(3)
Probability and Phase-Space Volume	490	(4)
Probabilities and the Time Interval of Large Fluctuations	490	(3)
The Ergodic Theorem	493	(1)
The Gibbs and the Boltzmann Entropies and their Time Dependences	494	(2)
The Time Derivative of Gibbs' Entropy	494	(1)
Boltzmann's Entropy	494	(2)
Irreversibility and Time Reversal	496	(7)
The Expansion of a Gas	496	(5)
Description of the Expansion Experiment in μ-Space	501	(1)
The Influence of External Perturbations on the Trajectories of the Particles	502	(1)
Entropy Death or Ordered Structures?	503	(6)
Problems	505	(4)
Appendix	509	(52)
A Nernst's Theorem (Third Law)	509	(8)
A.1 Preliminary Remarks on the Historical Development of Nernst's Theorem	509	(1)
A.2 Nernst's Theorem and its Thermodynamic Consequences	510	(2)
A.3 Residual Entropy, Metastability, etc.	512	(5)
B. The Classical Limit and Quantum Corrections	517	(15)
B.1 The Classical Limit	517	(5)
B.2 Calculation of the Quantum-Mechanical Corrections	522	(5)
B.3 Quantum Corrections to the Second Virial Coefficient B(T)	527	(5)
C. The Perturbation Expansion	532	(1)
D. The Riemann ζ-Function and the Bernoulli Numbers	533	(1)
E. Derivation of the Ginzburg-Landau Functional	534	(7)
F. The Transfer Matrix Method	541	(2)
G. Integrals Containing the Maxwell Distribution	543	(1)
H. Hydrodynamics	544	(9)
H.1 Hydrodynamic Equations, Phenomenological Discussion	545	(1)
H.2 The Kubo Relaxation Function	546	(2)
H.3 The Microscopic Derivation of the Hydrodynamic Equations	548	(5)
I. Units and Tables	553	(8)
Subject Index	561