The quantum statistics of dynamic processes

1 General Aspects.- 1. The Concept of Statistical Physics.- 2. Summary of Quantum Theory.- 2.1 Observables as Operators. Commutation Relations.- 2.2 The Unitary Space $$ \mathfrak{U} $$ of States. Expectation Values.- 2.3 The Statistical Operator of a Mixed State.- 3. Quantum Theory in Liouville Space.- 3.1 The Liouville Space $$ \mathfrak{L} $$ (Without Scalar Product).- 3.1.1 The Elements in Liouville Space.- 3.1.2 Operators in $$ \mathfrak{L} $$ (Superoperators).- 3.2 The Formulations of Quantum-Theoretical Dynamics.- 3.2.1 The Fundamental Equations of Time Evolution.- 3.2.2 Dynamics in the Schrodinger Formulation.- 3.2.3 Dynamics in the Heisenberg Formulation.- 3.2.4 The Ehrenfest Theorem and Its Consequences.- 3.3 Subsystems.- 3.3.1 Combined Systems.- 3.3.2 The Product Liouville Space.- 3.3.3 Expectation Values in a Subsystem. The Reduced Statistical Operator.- 3.3.4 The Time Variation of the Reduced Statistical Operator.- 3.3.5 Transfer of Work and Heat into a Subsystem.- 3.4 Useful Operator Identities.- 3.4.1 Operator Identities for Time Evolution.- 3.4.2 Differentiation of Exponential Operators.- 4. Systems of Many Particles.- 4.1 The Mean Square Deviations of Macroscopic Observables.- 4.1.1 Microscopic Densities and Their Correlation Functions.- 4.1.2 Macroscopic Densities and Their Fluctuations.- 4.2 General Properties of the Time Evolution of Expectation Values.- 5. Information-Theoretical Construction of the Statistical Operator.- 5.1 The Uncertainty Measure of the Statistical Operator.- 5.1.1 Definition of the Uncertainty Measure n[?].- 5.1.2 Properties of the Uncertainty Measure n[?].- 5.1.3 The Relationship Between Information Theory and the Uncertainty Measure n[?].- 5.2 The Generalized Canonical Statistical Operator ?.- 5.2.1 Observation Levels.- 5.2.2 Determination of the Statistical Operator by Maximization of the Uncertainty Measure. Entropy with Respect to an Observation Level.- 5.2.3 Linear Transformations Within an Observation Level.- 5.2.4 Extension of the Observation Level.- 5.2.5 A Sufficient Observation Level. Representativity of a Generalized Canonical Statistical Operator.- 5.2.6 Stationary Generalized Canonical Statistical Operators.- 5.3 Examples of Generalized Canonical Statistical Operators.- 5.3.1 The Hamiltonian as an Observation Level.- 5.3.2 Partial Hamiltonians as Decomposable Observation Levels.- 5.3.3 Partial Hamiltonians as Nondecomposable Observation Levels.- 5.3.4 Projectors {PF} as Observation Levels.- 6. The Significance of Generalized Canonical Statistical Operators for Dynamic Processes.- 6.1 The Statistical Operator at the Beginning of a Process.- 6.2 Entropy Production in Dynamic Processes of Adiabatic Systems.- 6.3 Examples of Entropy Production in Dynamic Adiabatic Processes.- 6.3.1 The Dynamics of an Adiabatic Process in Going from One Thermal Equilibrium to Another.- 6.3.2 The Dynamics of an Adiabatic Process in Going from Thermal Equilibrium to an Inhibited Equilibrium.- 6.4 Accompanying Entropy S{G}(t) with Respect to an Observation Level {G}.- 2 Response to Time-Dependent External Fields.- 7. The Quantum-Statistical Formulation of Response Theory.- 7.1 Introduction to the Physical Problem.- 7.2 The Mathematical Formulation of the Problem.- 8. A Scalar Product in the Liouville Space for Linear Response Theory.- 8.1 Scalar Products and Projection Operators in Liouville Space.- 8.1.1 Properties of Scalar Products in L.- 8.1.2 Adjoint Operators (Superoperators) in L.- 8.1.3 Projection Operators P in L.- 8.1.4 The Generation of Orthogonal Elements in L Using Projection Operators.- 8.2 The Liouville Space with the Mori Scalar Product.- 8.2.1 Definition of the Mori Scalar Product.- 8.2.2 Properties of the Mori Scalar Product.- 8.3 The Physical Significance of the Mori Product.- 8.3.1 Interpretation of the Mori Product as a Linear Variation, Tr(d?*G).- 8.3.2 A Note on Formal Calculation with Non-Hermitian "Observables".- 8.3.3 The Isothermal Susceptibility.- 8.3.4 The Adiabatic Susceptibility.- 9. Linear Response Theory.- 9.1 The Kubo Formula.- 9.1.1 The Quantum-Statistical Formulation in the Time Domain.- 9.1.2 The Quantum-Statistical Formulation in the Frequency Domain.- 9.2 The Physical Interpretation of the Kubo Formula Using Particular Time-Dependent Fields.- 9.2.1 A Pulsed External Field.- 9.2.2 A Sudden Change in the External Field.- 9.2.3 An Harmonically Oscillating External Field.- 9.3 Properties of the Response and Relaxation Functions.- 9.3.1 The Linear Response Function.- 9.3.2 The Linear Relaxation Function.- 9.4 Properties of the Dynamic Susceptibility.- 9.4.1 Decomposition of ?MF(?) into Two Hermitian Matrices, ?'MF(?) and ?"MF(?).- 9.4.2 Relations Between ?MF(?) and ?MF(t) or ?MF(t).- 9.4.3 The Kramers-Kronig Relations.- 9.4.4 High-Frequency Behavior of ?MF(?).- 9.4.5 The Moments of the Spectral Density Function.- 9.5 The Limit of Slow Field Variation.- 9.5.1 Properties of the Isolated Susceptibility.- 9.5.2 The Physical Significance of the Isolated Susceptibility.- 9.5.3 Plateaus in the Relaxation Function.- 9.6 The Work Performed on the System..- 9.6.1 Average Power $$\bar W\left( {\omega _0 } \right)$$ in the Harmonic Steady State.- 9.6.2 The Work Performed, A(t1, t0), by a Field Acting from t0 to t1.- 9.7 Relations Between the Fourier-Transformed Time-Dependent Correlation Functions.- 9.8 The First Fluctuation-Dissipation Theorem.- 9.9 A Generalization of the Kubo Formula.- 10. Quadratic Response Theory.- 10.1 The Quadratic Response.- 10.1.1 Formulation in the Time Domain.- 10.1.2 Formulation in the Frequency Domain.- 10.1.3 Symmetrized Expressions.- 10.2 The Influence of Energy Entering the System.- 10.2.1 The Behavior of ?MMF(t1, t2) at Long Times.- 10.2.2 $$\varphi M_\alpha M_\gamma F\left( {t_1,\infty } \right) $$ as a Linear Response Function.- 10.2.3 Separation of the Response Function into ?|| and ??.- 10.3 Interpretation Using Time-Dependent Fields.- 10.3.1 The Superposition of Two Short Pulses.- 10.3.2 The Superposition of Two Harmonically Oscillating Fields.- 10.4 Concluding Remarks.- 3 Equations of Motion for Observables in the Case of Small Deviations from Equilibrium.- 11. Exact Integro-Dilferential Equations for Relaxation Processes.- 11.1 An Heuristic Introduction to the Langevin-Mori Theory.- 11.2 Mori's Integro-Differential Equations for Operators.- 11.2.1 Derivation and Interpretation.- 11.2.2 Choosing a Set of Observables G?.- 11.3 The Frequency and the Memory Matrices.- 11.3.1 The Eigenelements of the Frequency Matrix.- 11.3.2 Properties of the Memory Matrix. Dynamic Onsager-Casimir Coefficients.- 11.4 The Integro-Differential Equations for Relaxation Functions.- 11.4.1 Dynamics of the Correlation Matrix ?v?(t). Relationship to Linear Dynamic Response Theory.- 11.4.2 Integro-Differential Equations for the Expectation Values (t).- 12. Perturbation-Theoretical Treatment of the Frequency and Memory Matrix.- 12.1 The Leading Terms of a Perturbation-Theory Expansion in L1.- 12.1.1 A Set of Observables {G} as an Invariant Subspace L{G} with Respect to L0.- 12.1.2 Perturbation-Theory Expansion of the Scalar Products.- 12.1.3 The Leading Terms of a Perturbation-Theory Expansion of ?v? and ?v?(t).- 12.2 Extending the Set of Observables in a Manner Appropriate to the Perturbation.- 12.2.1 The Mori Equations for the Extended Set of Observables.- 12.2.2 Perturbation-Theoretical Approximations.- 13. The Transition to Differential Equations with Damping.- 13.1 One Slow Hermitian Observable.- 13.1.1 Separation of the Time Scales
 Simplified Argument.- 13.1.2 Validity of the Approximation.- 13.2 A Set of Slow Observables.- 13.2.1 Carrying Out the Markovian Approximation.- 13.2.2 Properties of the Markovian Approximation.- 13.3 Modification of the Approximation Due to Rapid Oscillations.- 13.3.1 Principle.- 13.3.2 Formulation Using Matrices.- 13.3.3 Discussion Based on the Damped Harmonic Oscillator.- 14. Time Derivatives as a Special Set of Observables.- 14.1 Specialization of the Mori Integro-Differential Equations.- 14.1.1 The Space L{G} Spanned by the Derivatives.- 14.1.2 The Mori Equations for Time Derivatives.- 14.1.3 OrthogonaIObservables.- 14.2 A Continued-Fraction Expression for the Correlation Function ?(?).- 14.2.1 Exact Description.- 14.2.2 Neglecting the Memory Matrix.- 14.2.3 The Markovian Approximation.- 15. Dynamic Onsager-Casimir Coefficients as Linear Response Functions for Generalized Forces.- 15.1 The Integro-Differential Equations for the Expectation Values in Externally Driven Systems..- 15.1.1 The Set {G} in the Mori Projection Operator.- 15.1.2 The Derivation of Generalized Mori Equations for the Expectation Values (t) in an Externally Driven System.- 15.1.3 Time-Dependent Lagrange Multipliers ?v.(t) for the Accompanying Generalized Canonical Statistical Operator ? as Generalized Forces.- 15.2 The Irreversible Entropy Production in Linear Dynamic Processes.- 15.2.1 The Accompanying Entropy S{G(h)}(t).- 15.2.2 Significance of the Onsager-Casimir Coefficients L'v?(?) for Entropy Production.- 15.3 The Second Fluctuation-Dissipation Theorem.- 15.3.1 The Residual Force f?(t).- 15.3.2 Equilibrium Correlation Functions of f? (?).- 16. Physical Examples.- 16.1 A Heavy Particle in an Elastic Chain: A Model Which Can Be Solved Exactly - Rubin's Model.- 16.1.1 Dynamics of the Residual Force.- 16.1.2 The Memory Function.- 16.1.3 Separation of the Time Scales.- 16.1.4 Discussion of the Exact Solution ?(t).- 16.2 Spin-Bath Relaxation.- 16.3 Magnetic Resonance.- 16.3.1 Reduction to a Single Equation for ?+ + (t).- 16.3.2 Perturbation Theory and the Markovian Approximation.- 16.3.3 Reduction to Bath Correlation Functions.- 16.4 A Local Conservation Law.- 16.4.1 Decoupling of the Fourier Components.- 16.4.2 The Wavenumber as a Slowness Parameter.- 4 Equations of Motion of the Relevant Parts of the Statistical Operator.- 17. Mappings of the Statistical Operator onto a Relevant Part.- 17.1 The Concept of the Relevant Part, ?rel(t).- 17.2 Linear Relation Between ?rel(t) and ?(t).- 17.2.1 Properties of the Operator P.- 17.2.2 Explicit Expressions for P.- 17.2.3 The Nakajima-Zwanzig Equation.- 17.2.4 Example: ?rel(t) of a Subsystem.- 17.2.5 The Explicit Time Dependence of the Operators P and L.- 17.3 Nonlinear Relation Between ?rel(t) and ?(t).- 17.3.1 Properties of the Mapping.- 17.3.2 Nonlinear Dynamical Equation for ?rel(t).- 18. The Generalized Canonical Statistical Operator ?(t) as ?rel (t).- 18.1 The Linear Case.- 18.2 The Robertson Equation.- A. Equivalence of the Nakajima-Zwanzig Equation and the Generalized-Operator Langevin Equation.- B. Symmetries.- B.1.1 Properties of D(g).- B.1.2 Selection Rules.- B.2.2 Symmetry Properties Resulting from Time-Reversal Invariance.- Solutions to the Exercises.