Statistical physics establishes a bridge from the macroscopic world to study the microscopic world. This is a theory with the fewest assumptions and the broadest conclusions. Up to now there is no evidence to show that statistical physics itself is responsible for any mistakes. Statistical physics has become an important branch of modern theoretical physics and this course has become one of the common fundamental courses of graduate students in different majors in physics departments.Statistical physics is a branch of science engaged in studying the laws of thermal motion of macroscopic systems. The advanced statistics for graduate students mainly studies quantum statistics. The first four chapters of this book are fundamental， and should be well known. The last five chapters are recent developments， including the studies on Bose-Einstein condensation， a class of inverse problems in quantum statistics （their Chen's exact solution formulas， Dai's exact solution formulas，asymptotic behavior control theory， and concrete realizations of the inversion theories）， an introduction to the theory of Green's functions in quantum statistics， the unified diagonalization theorem for Hamiltonians of quadratic form， and an introduction to the third formulation of quantum statistics and the functional integral approach. This course was edited by revising the lecture notes of the author， from courses of quantum statistics and advanced statistics for graduate students，since 1978. At the same time， this work contains the research results of some related projects， supported by the National Natural Science Foundation of China.
Chapter1 Fundamental Principles 1.1 Introduction:The Characters of Thermodynamics and Statistical Physics and Their Relationship 1.2 Basic Thermodynamic Identities 1.3 Fundamental Principles and Conclusions of Classical Statistics 1.3.1 Microscopic and Macroscopic Descriptions，Statistical Distribution Functions 1.3.2 Liouville Theorem 1.3.3 Statistical Independence 1.3.4 Microscopical Canonical，Canonical and Grand CanonicalEnsembles 1.4 Boltzmann Gas 1.5 Density Matrix 1.5.1 DensityMatrix 1.5.2 Some General Properties of the Density Matrix 1.6 Liouville Theorem in Quantum Statistics 1.7 Canonical Ensemble 1.8 Grand Canonical Ensemble 1.8.1 Fundamental Expression of the Grand Canonical Ensemble 1.8.2 Derivation of the Fundamental Thermodynamic Identity 1.9 Probability Distribution and Slater Sum 1.9.1 Meaning of the Diagonal Elements of the Density Matrix 1.9.2 Slater Summation 1.9.3 Example:Probability of the Harmonic Ensemble 1.10 Theory of the Reduced Density Matrix Chapter2 The Perfect Gas in Quantum Statistics 2.1 Indistinguishability Principle for Identical Particles 2.2 Bose Distribution and Fermi Distribution 2.2.1 Perfect Gases in Quantum Statistics 2.2.2 Bose Distribution 2.2.3 Fermi Distribution 2.2.4 Comparison of Three Distributions;Gibbs Paradox Again 2.3 Density of States，Chemical Potential and Equation of State 2.3.1 Density of States 2.3.2 Virial Equation for Quantum Ideal Gases 2.4 Black-body Radiation 2.4.1 Thermodynamic Quantities for the Black-body Radiation Field 2.4.2 Exitance and Variety of Displacement Laws 2.4.3 Waveband Radiant Exitance and Waveband Photon Exitance 2.5 Bose-Einstein Condensation in Bulk 2.5.1 Bose Condensation，Dynamical Quantities with Temperature Lower Than theλPoint 2.5.2 Discontinuity of the Derivatives of Specific Heat and λPhenomena 2.5.3 Two-Fluid Theory 2.5.4 2-D Case 2.6 Degenerate Fermi Gases and Ferm Sphere 2.6.1 Properties of Fermi Gases at Absolute Zero 2.6.2 Specific Heat of Free Electron Gases 2.6.3 State Equation，Heat Capacity at Constant Pressure， and Heavy Fermions 2.7 Fermi Integrals and their Low Temperature Expansion 2.8 Magnetism of Fermi Gases 2.8.1 Spin Magnetism:Paramagnetism 2.8.2 Energy Spectra and Stationary States of Electrons in a Homogeneous Magnetic Field 2.8.3 Diamagnetism of Orbital Motion of Free Electrons 2.9 Peierls Perturbation Expansion of Free Energy 2.9.1 Classical Case 2.9.2 Quantum Case 2.9.3 Expansion of Free Energy of an Ideal Gas in an External Field 2.10 Appendix Chapter3 Second Quantization and Model Hamiltonians 3.1 Necessity of Second Quantization 3.2 Second Quantization for Bose System 3.3 Second Quantization·Fermi System 3.4 Some Conservation Laws 3.5 Some Model Hamiltonians 3.6 Electron Gases with Coulomb Interaction 3.6.1 Completely Ionized Gases—the High Temperature Plasma 3.6.2 The Degenerate Electron Gas with Coulomb Interaction（Metal Plasma） 3.7 Anderson Model Chapter4 Least Action Principle，F(xiàn)ield Quantization and the Electron-Phonon System 4.1 Classical Description of Lattice Vibrations 4.2 Continuous Media Model of Lattice Vibration（Classical） 4.3 The Least Action Principle，Euler-Lagrange Equation and Hamilton Equation 4.4 Lagrangian and Hamiltonian of Continuous Media 4.5 Quantization of the Lattice Vibration Field 4.6 Debye Theory of Specific Heat of Solids 4.7 The Electron-Phonon System and the Frohlich Hamiltonian Chapter5 Bose-Einste in Condensation 5.1 Spatial andMomentum Distributions of Bose-Einstein Condensation in Harmonic Traps and Bloch Summation 5.1.1 Introduction 5.1.2 Generalized Expression for Particle Density 5.1.3 Distributions for Ideal Systems 5.1.4 New Expression with Clear Physical Picture 5.1.5 Momentum Distributions 5.1.6 Results of Numerical Calculations 5.1.7 Discussion and Concluding Remarks 5.1.8 Momentum Distribution of BEC 5.2 BEC in Confined Geometry and Thermodynam icMapping 5.2.1 Introduction 5.2.2 Confinement Effects 5.2.3 Thermodynamic Mapping 5.2.4 Mapping Relation for Confined BEC 5.2.5 Determination of the Critical Temperature 5.2.6 Discussion Chapter6 Some Inverse Problems in Quantum Statistics 6.1 Introduction 6.2 Specific Heat-Phonon Spectrum Inversion 6.2.1 Technique for Eliminating Divergences 6.2.2 Unique Existence Theorem and Exact SPIE Solution 6.2.3 Summary 6.3 Concrete Realization of Inversion 6.3.1 The Specific Heat-Phonon Spectrum Inversion Problem 6.3.2 Results and Concluding Remarks 6.4 Mobius Inversion Formula 6.4.1 RiemannζFunction and Mobius Function 6.4.2 Mobius Inversion Formula 6.4.3 The Modified Mobius Inversion Formula 6.4.4 Applications in Physics 6.5 Unification of the Theories 6.5.1 Introduction 6.5.2 Deriving Chen's Formula from Dai's Exact Solution 6.5.3 Concluding Remarks 6.6 Appendix Chapter7 An Introduction to Theory of Green's Functions 7.1 Temperature-Time Green's Functions 7.1.1 Definition of Temperature-Time Green's Functions 7.1.2 The Equation of Motion of Double-Time Green's Functions 7.1.3 Time Correlation Functions 7.2 Spectral Theorem 7.2.1 Spectral Representation of Time Correlation Functions 7.2.2 Spectral Representations of Retarded and Advanced Green's Functions 7.2.3 Spectral Representation of Causal Green's Functions 7.3 Example:Ideal Quantum Gases 7.4 Theory of Superconductivity with Double-Time Green's Functions 7.5 Higher-Order Spectral Theorem，Sum Rules and Uniqueness Chapter8 A Unified Diagonalization Theorem for Quadratic Hamiltonian 8.1 A Model Hamiltonian 8.2 Diagonalization Theorem for Fermi Quadratic Forms 8.3 Conclusion:A Unified Diagonalization Theorem Chapter9 Functional Integral Approach:A Third Formulation of Quantum Statistical Mechanics 9.1 Introduction 9.1.1 Hubbard's Method 9.1.2 Difficulties 9.2 An Operator Identity 9.3 Functional Integral Formulation of Quantum Statistical Mechanics 9.4 Reality and Method of Steepest Descents 9.5 Discussion and Concluding Remarks 9.6 Some Recent Developments 9.7 Application:An Exact Solution References Index
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