This volume of Statistical Physics consititutes the second part of Statistical Physics (Springer Series in Solid-State Science, Vols. 30, 31) and is devoted to nonequilibrium theories of statistical mechanics. We start with an intro­ duction to the stochastic treatment of Brownian motion and then proceed to general problems involved in deriving a physical process from an underlying more basic process. Relaxation from nonequilibrium to equilibrium states and the response of a system to an external disturbance form the central problems of nonequilibrium statistical mechanics. These problems are treated both phenomenologically and microscopically along the lines of re­ cent developments. Emphasis is placed on fundamental concepts and methods rather than on applications which are too numerous to be treated exhaustively within the limited space of this volume. For information on the general aim of this book, the reader is referred to the Foreword. For further reading, the reader should consult the bibliographies, although these are not meant to be exhaustive.

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.