Statistical Physics II

1. Brownian Motion.- 1.1 Brownian Motion as a Stochastic Process.- 1.2 Central Limit Theorem and Brownian Motion.- 1.3 Langevin Equation and Harmonic Analysis.- 1.4 Gaussian Processes.- 1.5 Brownian Motion Modeled by a Gaussian Process.- 1.6 Fluctuation-Dissipation Theorem.- 2. Physical Processes as Stochastic Processes.- 2.1 Random Frequency Modulation.- 2.2 Brownian Motion Revisited.- 2.3 Markovian Processes.- 2.4 Fokker-Planck Equation.- 2.5 Contraction of Information. Projected Processes.- 2.6 Derivation of Master Equations.- 2.7 Brownian Motion of a Quantal System.- 2.8 Boltzmann Equation.- 2.9 Generalized Langevin Equation and the Damping Theory.- 3. Relaxation and Resonance Absorption.- 3.1 Linear Irreversible Processes.- 3.1.1 Mechanical and Thermal Forces vs Displacements and Currents.- 3.1.2 Linear Relations.- 3.1.3 Response to a Pulsed Force.- 3.1.4 Relaxation Phenomena.- 3.2 Complex Admittance.- 3.2.1 Harmonic (Fourier) Analysis.- 3.2.2 Energy Dissipation.- 3.3 Debye Relaxation.- 3.3.1 Dielectric Relaxation.- 3.3.2 Response Functions with Exponential Damping.- 3.3.3 Solution of Polar Molecules.- 3.4 Resonance Absorption.- 3.4.1 Van Vleck-Weisskopf-Fröhlich Type Resonance Absorption.- 3.4.2 Nuclear Magnetic Resonance.- 3.4.3 Failure at High Frequencies.- 3.5 Wave Number-Dependent Complex Admittance.- 3.5.1 Non-Markovian Nonlocal Linear Relations.- 3.5.2 Complex Admittance for the Diffusion Phenomenon.- 3.6 Dispersion Relations.- 3.6.1 Proof of the Dispersion Relations.- 3.6.2 Dispersion Relations and Causality.- 3.6.3 Analytical Continuation into the Complex Plane.- 3.7 Sum Rules and Interpolation Formulas.- 3.7.1 Moment Sum Rules.- 3.7.2 Non-Markovian Law of Diffusion.- 4. Statistical Mechanics of Linear Response.- 4.1 Static Response to External Force.- 4.1.1 Static Admittance and the Canonical Correlation.- 4.2 Dynamic Response to External Force.- 4.2.1 The Response Function and the Poisson Bracket.- 4.2.2 Kubo Formula.- 4.2.3 Initial Values of the Response Function and Its Derivatives.- 4.3 Symmetry and the Dispersion Relations.- 4.3.1 Spectral Function and Its Symmetry.- 4.3.2 Symmetry in the Current Response.- 4.3.3 Symmetry in the Displacement Response.- 4.3.4 Proof of the Dispersion Relations.- 4.4 Fluctuation and Dissipation Theorem.- 4.4.1 Symmetrized Correlation.- 4.4.2 The Equivalence Between the Symmetrized Correlation Function and the Response or the Relaxation Function.- 4.4.3 Fluctuation-Dissipation Theorem.- 4.5 Density Response, Conduction and Diffusion.- 4.5.1 Density and Current in Response to the External Field.- 4.5.2 Relaxation of the Density Response and the Density Fluctuation.- 4.5.3 Shielding of the External Potential.- 4.5.4 Resistivity Formula.- 4.5.5 Dielectric Shielding and Electric Conductivity.- 4.5.6 Kramers-Kronig Relations and the Sum Rules.- 4.6 Response to Thermal Internal Forces.- 4.6.1 Onsager's Postulate.- 4.6.2 Fluctuation of Macrovariables as Brownian Motion.- 4.6.3 A General Formulation of Onsager's Postulate.- 4.6.4 Nonequilibrium Density Matrix.- 4.7 Some Remarks on the Linear-Response Theory.- 4.7.1 The Kinetic Method versus the Linear-Response Theory.- 4.7.2 Van Kampen's Objection.- 4.7.3 Spurious Singularities at the Zero Value of the External Field.- 4.7.4 Singularities at k = 0, ? = 0.- 5. Quantum Field Theoretical Methods in Statistical Mechanics.- 5.1 Double-Time Green's Functions.- 5.1.1 Retarded Green's Functions.- 5.1.2 Advanced Green's Functions.- 5.2 Chain of Equations of Motion and the Decoupling Approximation.- 5.2.1 Chain of Equations of Motion.- 5.2.2 Complex Dielectric Function of a Plasma in a Decoupling Approximation.- 5.3 Relation to the Kinetic Equation.- 5.3.1 Klimontovich Operator.- 5.3.2 Self-Consistent Field Approximation.- 5.3.3 Plasma Oscillation.- 5.4 Single-Particle Green's Function and the Causal Green's Function.- 5.4.1 Single-Particle Green's Functions.- 5.4.2 Single-Particle Green's Functions for Free Particles.- 5.4.3 Causal Green's Functions.- 5.5 Basic Formula for Perturbational Expansion.- 5.5.1 Perturbational Expansion of the Equilibrium Density Matrix.- 5.5.2 Perturbational Expansion of the Thermodynamic Potential.- 5.6 Temperature Green's Function.- 5.6.1 Temperature Green's Functions (Matsubara-Green's Functions).- 5.6.2 Fourier Analysis of the Temperature Green's Function.- 5.6.3 Single-Particle Temperature Green's Function for Noninteracting Particles.- 5.6.4 Abrikosov-Gor'kov-Dzyaloshinskii-Fradkin Theorem.- 5.7 Diagram Technique.- 5.7.1 Bloch-De Dominicis Theorem.- 5.7.2 Perturbational Expansion of $${\left\langle {\mathcal{S}(\beta )} \right\rangle _0}$$.- 5.7.3 Correspondence with Feynman Diagrams.- 5.7.4 Matsubara Formula.- 5.8 Dyson Equation.- 5.8.1 Single-Particle Temperature Green's Function.- 5.8.2 Graphical Summation.- 5.8.3 Feynman Rules.- 5.9 Relationship Between the Thermodynamic Potential and the Temperature Green's Function.- 5.10 Special Case of Two-Particle Green's Function.- 5.10.1 Two-Particle Green's Function of Zeroth-Order for a Plasma.- 5.10.2 Polarization Operator.- 5.10.3 Electric Charge Density Green's Function.- General Bibliography of Textbooks.- References.