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Springer The Quantum Statistics of Dynamic Processes by Eugen Fick, Günter Sauermann, William D. Brewer
The methods of statistical physics have become increasingly important in recent years for the treatment of a variety of diverse physical problems. Of principal interest is the microscopic description of the dynamics of dissipative systems. Although a unified theoretical description has at present not yet been achieved, we have assumed the task of writing a textbook which summarizes those of the most important methods which are self-contained and complete in themselves. We cannot, of course, claim to have treated the field exhaustively. A microscopic description of physical phenomena must necessarily be based upon quantum theory, and we have therefore carried out the treatment of dynamic processes strictly within a quantum-theoretical framework. For this reason alone it was necessary to omit a number of extremely important theories which have up to now been formulated only in terms of classical statistics. The goal of this book is, on the one hand, to give an introduction to the general principles of the quantum statistics of dynamical processes, and, on the other, to provide readers who are interested in the treatment of particular phenomena with methods for solving specific problems. The theory is for the most part formulated within the calculational frame work of Liouville space, which, together with projector formalism, has become an expedient mathematical tool in statistical physics._x000D_ Table of contents :- _x000D_
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 Schroedinger 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 $$\varphi M_\alpha M_\gamma F\left( {t_1,\infty } \right)_x000D_
$$ 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._x000D_
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