Statistical Mechanics : An Intermediate Course

by Morandi, Giuseppe; Ercolessi, E.; Napoli, F.; Morandi, Giuseppe

• ISBN13: 9789810244774
• ISBN10: 9810244770
• Edition: 2nd
• Format: Paperback
• Copyright: 2001-08-01
• Publisher: World Scientific Pub Co Inc

Summary

The book aims to bridge the gap between standard textbooks on the subject and more advanced books, discussing in a more thorough way the foundations of both classical and quantum statistical mechanics. It is devised for a one-semester graduate course on statistical mechanics. Its prerequisite is therefore a more elementary course on the same subject. Emphasis is laid on the geometrical aspects of classical thermodynamics, on the foundational problems, and on selected applications (mainly to spin systems), the meaning of quantum statistics, phase transitions, critical phenomena, and the renormalization group.

Table of Contents

Preface	1	(1)
Thermodynamics	1	(50)
A Recollection of Basic Notions in Classical Thermodynamics	1	(14)
The Fundamental Equation of Thermodynamics	11	(4)
Thermodynamic Potentials, ...Stability Conditions	15	(13)
A Mathematical Digression: Integrating Factors and	28	(13)
An Exercise in the Use of the Gibbs-Duhem Relation: the Ideal Monoatomic Gas	41	(1)
Thermodynamics of Paramagnetic Bodies	42	(4)
Some Relations on Partial Derivatives & Jacobians	46	(2)
A Digression on: Integrability Conditions	48	(3)
Problems	49	(2)
Equilibrium Classical Statistical Mechanics	51	(112)
Foundations of Classical Statistical Mechanics	51	(31)
A resume of Hamiltonian Dynamics	52	(4)
Canonical Transformations	56	(5)
The Heisenberg and Schrodinger Pictures in Classical Dynamics	61	(4)
Integrable Dynamical Systems and Perturbations	65	(8)
The Ergodic Hypothesis and the Foundations of Classical Statistical Mechanics	73	(9)
Statistical Ensembles in CSM: Micro-canonical Ensemble	82	(12)
The Entropy Function	84	(10)
Statistical Ensembles in CSM: Canonical and Grand-Canonical Ensembles	94	(26)
The Canonical Ensemble	94	(8)
The Grand-Canonical Ensemble	102	(9)
Some Applications	111	(4)
General Remarks	115	(5)
Response, Correlations and Fluctuations: I Classical	120	(29)
Symmetry Properties of Correlation Functions	120	(12)
Fourier Transforms of Correlation Functions	132	(3)
Generating Functionals and Static Generalized Susceptibilities	135	(4)
Linear Response Theory	139	(7)
The Classical Fluctuation-Dissipation Theorem	146	(3)
Harmonic Oscillators & Ergodicity	149	(3)
The Volume Phase Space for a Perfect Gas	152	(3)
Density-Density Correlation Function of a Perfect Gas	155	(8)
Problems	158	(5)
Spin Hamiltonians I: Classical	163	(66)
Spin Hamiltonians	163	(12)
Gaussian Identities for Spin Hamiltonians	175	(6)
Mean Field Theory and Phase Transitions	181	(18)
MFA for Ising Model	181	(8)
MFA for Heisenberg Model	189	(10)
Linearized Spin Dynamics: Spin Waves, Response and Correlations	199	(5)
SSB, Goldstone and Mermin-Wagner Theorems	204	(13)
The Goldstone Theorem	209	(2)
The Mermin-Wagner Theorem	211	(6)
Poisson Description of Spin Dynamics	217	(1)
Perturbation expansions and the Classical Analogue of Wick's Theorem	218	(2)
``Conventional'' Mean Field Theory	220	(1)
Some Group-Theoretical Aspects Related to SSB	221	(8)
Problems	226	(3)
Equilibrium Quantum Statistical Mechanics	229	(54)
Resume of Quantum Mechanics	229	(22)
Basic Axioms of Quantum Mechanics	231	(7)
Mixed States and Density Matrices	238	(7)
Time Evolution	245	(6)
Foundations of Quantum Statistical Mechanics: Ensembles	251	(18)
Ensembles in Equilibrium QSM	251	(11)
The KMS conditions and the thermodynamic limit	262	(5)
Ergodicity in Quantum Mechanics	267	(2)
Response, Correlations and Fluctuations II: Quantum	269	(9)
Static Susceptibilities and Correlation Functions	272	(2)
Linear Response Theory and the Fluctuation-Dissipation Theorem	274	(4)
Two-level Systems	278	(5)
Identical Particles in Quantum Statistical Mechanics	283	(68)
Statistics and Identical Particles in QSM	283	(13)
Inequivalent Quantizations in Multiply-Connected Spaces	287	(9)
Fock Spaces & Second Quantization	296	(16)
Observables in Fock Space and Gauge Invariance	309	(3)
Quantum Gases and Beyond	312	(39)
The Free Fermi Gas	319	(8)
The Free Bose Gas and the Bose-Einstein Condensation	327	(10)
Interacting Identical Particles. The Thermal Hartree-Fock Approximation	337	(11)
Problems	348	(3)
Spin Hamiltonians II: Quantum	351	(34)
The Heisenberg Model Hamiltonian	351	(2)
Free Spins	352	(1)
Partition Function and Path Integrals	353	(21)
Correlation Functions, KMS Conditions and the Boundary Conditions on the Auxiliary Fields	358	(8)
The Static Approximation and the Classical Limit	366	(2)
The adiabatic expansion	368	(6)
Mean-Field Approximations and SSB: Ferro and Antiferro Magnetism	374	(11)
Linearized Spin Dynamics Spin Waves	380	(3)
Problems	383	(2)
Phase Transitions and Critical Phenomena	385	(28)
Introduction to Phase Transitions	385	(14)
Phase Coexistence and Phase Diagrams: Classification of Phase Transitions	385	(4)
Some Thermodynamics across the coexistence line(s)	389	(3)
Phase Transitions in Magnetic Systems	392	(2)
Fluctuations and Correlations Near a Critical Point	394	(2)
Ornstein-Zernike Type Theories	396	(3)
Some Exact Results on Phase Transitions	399	(14)
The Theorems of Lee and Yang	399	(3)
Critical Points, SSB, Order Parameters and the Goldstone Theorem	402	(6)
The Mermin-Wagner Theorem	408	(2)
Quantum Description of Phase Coexistence. Decomposition of KMS States	410	(3)
Model Systems, Scaling Laws and Mean Field Theories	413	(64)
Phase Transitions in Model Systems	413	(36)
The Gaussian Model	414	(8)
The Spherical Model	422	(2)
The 2D Ising Model	424	(16)
The XY Model and the Kosterlitz-Thouless Transition	440	(9)
Critical Exponents, Exponent Inequalities and Scaling Assumptions	449	(11)
Critical Exponents and Universality	453	(1)
Exponent Inequalities	454	(3)
Scaling Assumptions and Scaling Laws	457	(3)
The Landau Theory of Phase Transitions	460	(17)
Second-Order Transitions in the Landau Theory	464	(3)
Dimensional Analysis, Role of Fluctuations and the Ginzburg Criterion	467	(3)
Cubic Terms and First-Order Transitions. Tricritical points	470	(3)
Problems	473	(4)
Superconductivity & Superfluidity	477	(46)
Superconductivity and the Ginzburg-Landau Theory	477	(25)
Introduction	477	(4)
Two-Fluid Theories	481	(2)
The Critical Magnetic Field Hc	483	(2)
The Ginzburg-Landau Theory of Superconductivity	485	(6)
The Ginzburg-Landau Parameter	491	(4)
Magnetic Properties of Type-II Superconductors	495	(5)
Field-theoretic aspects of the Ginzburg-Landau theory. Time-dependent Ginzburg-Landau theory	500	(2)
Superfluids and Superfluidity	502	(21)
Some phenomenology of 4He. Two-fluid models	503	(4)
A microscopic approach to superfluidity	507	(11)
Vortices in HeII	518	(2)
Problems	520	(3)
The Renormalization Group and Critical Phenomena	523	(58)
Introduction: a Recollection of Basic Facts	523	(3)
Kadanoff's Block Scaling	526	(5)
Introduction to the RG. Sketch of the Method	531	(6)
Real Space Renormalization Group (RSRG). Results for Ising Spins	537	(20)
Introduction	537	(4)
Fixed Points and the Linearized RSRG	541	(7)
Results for Ising Systems	548	(9)
Momentum Space Renormalization Group. Applications to Landau-Types Models	557	(24)
Fixed Points and Exponents	561	(1)
The Gaussian Model	562	(1)
The m4 Model	563	(15)
Problems	578	(3)
Appendix A Mathematical Digression I: Differentiable Manifolds and Exterior Calculus	581	(16)
Appendix B Mathematical Digression II: Some Mathematics of Hilbert Spaces	597	(30)
Appendix C Linear Stability Theory	627	(4)
Appendix D Eigenvalue and Eigenvector Problems for Non-Symmetric Matrices	631	(4)
Bibliography	635	(8)
Index	643