Nonlinear Nonequilibrium Thermodynamics I

This book gives the first detailed and coherent treatment of a young and exciting branch of statistical physics. Both theoretical physicists and applied scientists will find this material of interest since the theoretical treatment is supported by numerous illustrative examples and application of the general results to a variety of electrical, thermal, mechanical and chemical systems.

1. Introduction.- 1.1 What Is Nonlinear Nonequilibrium Thermodynamics?.- 1.1.1 Foundations of Nonequilibrium Thermodynamics.- 1.1.2 What Nonequilibrium Results Are Discussed in This Book?.- 1.1.3 Distinguishing Features of Nonlinear Nonequilibrium Thermodynamics.- 1.2 Early Work on Nonlinear Nonequlibrium Thermodynamics.- 1.3 Some Particular Problems and Their Corresponding FDRs: Historical Aspects.- 1.3.1 Einstein’s Problem: Determination of the Diffusion Coefficient of a Brownian Particle.- 1.3.2 A Second Problem: Determination of the Intensity of a Random Force Acting on a Brownian Particle.- 1.3.3 The More General Linear Markov FDR.- 1.3.4 Onsager’s Reciprocal Relations.- 1.3.5 Nyquist’s Formula.- 1.3.6 The Callen-Welton FDT and Kubo’s Formula.- 1.3.7 Mori’s Relation.- 1.3.8 Thermal Noise of Nonlinear Resistance: The Markov Theory.- 1.3.9 Thermal Noise of Nonlinear Resistance: The Non-Markov Theory.- 2. Auxiliary Information Concerning Probability Theory and Equilibrium Thermodynamics.- 2.1 Moments and Correlators.- 2.1.1 Moments and the Characteristic Function.- 2.1.2 Correlators and Their Relationship with Moments.- 2.1.3 Moments and Correlators in Quantum Theory.- 2.2 Some Results of Equilibrium Statistical Thermodynamics.- 2.2.1 Entropy and Free Energy.- 2.2.2 Thermodynamic Parameters. The First Law of Thermodynamics.- 2.2.3 The Second Law of Thermodynamics.- 2.2.4 Characteristic Function of Internal Parameters and Free Energy.- 2.2.5 Thermodynamic Potential ? (a).- 2.2.6 Conditional Entropy.- 2.2.7 Formulas Determining the Equilibrium Probability Density of Internal Parameters.- 2.2.8 Conditional Thermodynamic Potentials and the First Law of Thermodynamics.- 2.2.9 The Functions S (B) and F (B) and the Second Law of Thermodynamics.- 2.2.10 The Case in which Energy Is an Argument of Conditional Entropy.- 2.2.11 Formulas of Quantum Equilibrium Statistical Thermodynamics.- 2.3 The Markov Random Process and Its Master Equation.- 2.3.1 Definition of a Markov Process.- 2.3.2 The Smoluchowski—Chapman Equation and Its Consequences.- 2.3.3 The Master Equation.- 2.3.4 The Fokker—Planck Equation and Its Invariant Form.- 2.3.5 The Stationary Markov Process.- 2.4 Infinitely Divisible Probability Densities and Markov Processes.- 2.4.1 Infinitely Divisible Probability Density.- 2.4.2 Stationary Markov Process with Independent Increments.- 2.4.3 Arbitrary Markov Processes.- 2.5 Notes on References to Chapter 2.- 3. The Generating Equation of Markov Nonlinear Nonequilibrium Thermodynamics.- 3.1 Kinetic Potential.- 3.1.1 Definition of Kinetic Potential.- 3.1.2 Relation Between the Kinetic Potential and the Free Energy: Asymptotic Formula.- 3.1.3 Example: Kinetic Potential for a System with Linear Relaxation and Quadratic Free Energy.- 3.1.4 Kinetic Potential Image.- 3.1.5 Modified Kinetic Potential.- 3.1.6 Properties of the Kinetic Potential and of Its Image.- 3.2 Consequences of Time Reversibility.- 3.2.1 Time-Reversal Symmetry of the Hamiltonian and of the One-Time Probability Density.- 3.2.2 Conditions Imposed on Transition Probabilities by Time Reversibility.- 3.2.3 Time-Reversal and the Markov Operator.- 3.2.4 Restrictions Imposed on the Kinetic Potential and on Its Image.- 3.2.5 The Modified Generating Equation.- 3.3 Examples of the Kinetic Potential and of the Validity of the Generating Equation.- 3.3.1 Consequences of the Generating Equation for a System with Linear Relaxation and Quadratic Free Energy.- 3.3.2 Diode Model of a Nonlinear Resistor: Relaxation Equation.- 3.3.3 Diode Model: Explanation of the Paradox Related to Detection of Thermal Fluctuations.- 3.3.4 Diode Model: The Kinetic Potential and Its Image.- 3.3.5 Poisson Model of Nonlinear Resistor: Construction of the Markov Operator Using Current—Voltage Characteristics.- 3.3.6 Gupta’s Formulas.- 3.4 Other Examples: Chemical Reactions and Diffusion.- 3.4.1 Chemical Reactions and Reaction Equations.- 3.4.2 Chemical Potentials.- 3.4.3 The Kinetic Potential Corresponding to (3.4.5).- 3.4.4 Extent of Reaction and the Corresponding Kinetic Potential.- 3.4.5 Chemical Reactions as Spatial or Continuum Fluctuational Processes.- 3.4.6 Diffusion of a Gaseous Admixture in a Homogeneous Gas.- 3.4.7 Chemical Reactions with Diffusion.- 3.5 Generating Equation for the Kinetic Potential Spectrum.- 3.5.1 The Kinetic Potential Spectrum.- 3.5.2 The Generating Equation.- 3.5.3 Examples of Spectra.- 3.6 Notes on References to Chapter 3.- 4. Consequences of the Markov Generating Equation.- 4.1 Markov FDRs.- 4.1.1 Relations for Images of Coefficient Functions.- 4.1.2 Basic Fluctuation—Dissipation Relations.- 4.1.3 Modified FDRs.- 4.1.4 Generalization of FDRs to the Case of an External Magnetic Field or Other Time-Odd Parameters.- 4.1.5 The Functions R+ and R? and their Relationship.- 4.1.6 Another Form of Many-Subscript Relations.- 4.2 Approximate Markov FDRs and Their Covariant Form.- 4.2.1 Approximate Relationship Between the Coefficient Function and Its Image.- 4.2.2 Markov FDRs in Zeroth and First Orders in k.- 4.2.3 The Covariant Form of One-Subscript and Two-Subscript FDRs.- 4.2.4 The Covariant Form of Quadratic FDRs.- 4.2.5 The Covariant Form of Cubic FDRs.- 4.3 Application of FDRs for Approximate Determination of the Coefficient Functions.- 4.3.1 Phenomenological Equation: Its Initial and Standard Forms.- 4.3.2 Linear Approximation.- 4.3.3 Linear—Quadratic Approximation.- 4.3.4 Linear—Quadratic—Cubic Approximation.- 4.3.5 Some Formulas of the Modified Version.- 4.3.6 Remark About the Use of Covariant FDRs for Constructing Coefficient Functions.- 4.4 Examples of the Application of Linear Nonequilibrium Thermodynamics Relations.- 4.4.1 Twofold Correlators in the Case of a System with Linear Relaxation and Quadratic Free Energy.- 4.4.2 Example: Elektrokinetic Phenomena.- 4.4.3 Thermokinetic Processes.- 4.4.4 Thermoelectric Phenomena.- 4.4.5 Anomalous Case: Circuit with Ideal Detector.- 4.4.6 The Mechanical Oscillator.- 4.4.7 Field Case: Maxwell’s Equations and Their Treatment from the Viewpoint of Nonequilibrium Thermodynamics.- 4.4.8 Simple Chemical Reactions.- 4.4.9 Thermal Fluctuations of the Velocity of a Liquid or Gas.- 4.5 Examples of the Application of the Markov FDRs of Nonlinear Nonequilibrium Thermodynamics.- 4.5.1 Simple Circuit with a Capacitor and a Nonlinear Resistor.- 4.5.2 Circuit with an Inductor and a Nonlinear Resistor.- 4.5.3 Heat Exchange of Two Bodies in the Linear—Quadratic Approximation.- 4.5.4 Chemical Reaction in the Linear—Quadratic Approximation.- 4.5.5 Linear—Cubic Approximation: Example with Capacitance.- 4.5.6 Oscillatory Circuit.- 4.5.7 Nonlinear Friction in the Case of a Spherical Body Moving in a Homogeneous Isotropic Medium.- 4.5.8 Nonlinear Electrical Conduction in an Isotropic Medium.- 4.6 H-Theorems of Markov Nonequilibrium Thermodynamics.- 4.6.1 H-Theorems of Linear Theory.- 4.6.2 The Second Law of Thermodynamics in the Linear Approximation: An Example.- 4.6.3 The H-Theorem of the Nonlinear Theory.- 4.6.4 Can Prigogine’s Theorem Be Generalized to the Nonlinear Case?.- 4.7 Notes on References to Chapter 4.- 5. Fluctuation—Dissipation Relations of Non-Markov Theory.- 5.1 Non-Markov Phenomenological Relaxation Equations and FDRs of the First Kind.- 5.1.1 Relaxation Equations with After-effects.- 5.1.2 Linear Approximation: Reciprocal Relations.- 5.1.3 Linear FDR of the First Kind.- 5.1.4 Particular Case of Equations in the Linear—Quadratic Approximation.- 5.1.5 Quadratic FDRs.- 5.1.6 General Definition of the Functions ?….- 5.1.7 Generating Functional and Generating Equation.- 5.1.8 The H-Theorem of Non-Markov Theory.- 5.1.9 Covariant Form of Non-Markov FDRs of the First Kind.- 5.2 Definition of Admittance and Auxiliary Formulas.- 5.2.1 External Forces and Admittances.- 5.2.2 Admittances in the Spectral Representation.- 5.2.3 Passing to the Quantum Case.- 5.2.4 Formulas for Functional Derivatives with Respect to External Forces.- 5.2.5 Formula for Permutation of Operators Under Averaging Sign in the Absence of External Forces.- 5.2.6 Consequences of Formula (5.2.71).- 5.2.7 Identities for Operators ?±.- 5.3 Linear and Quadratic FDRs of the Second Kind.- 5.3.1 Linear Fluctuation—Dissipation Theorem (FDT).- 5.3.2 Symmetry of Quantum Moments and Correlators Under Time-Reversal Transformation.- 5.3.3 Reciprocal Relations of Linear Theory.- 5.3.4 Quadratic FDT.- 5.3.5 Other Forms of Quadratic FDT.- 5.3.6 Three-Subscript Relation for Derivative of Correlator with Respect to External Force.- 5.3.7 Linear and Quadratic FDRs with Modified Admittances.- 5.3.8 Non-Markov FDRs of the Second Kind in the Case Where Energy Is One of the Parameters Ba.- 5.3.9 Modified Variant of FDRs of the Second Kind.- 5.4 Cubic FDRs of the Second Kind.- 5.4.1 Formulas Pertaining to Four-Subscript Correlators and Commutators.- 5.4.2 Relation for the Dissipationally Determinable Part of the Correlator.- 5.4.3 Dissipationally Determinable Part of the Triadmittance.- 5.4.4 Dissipationally Determinable Part of the Biadmittance.- 5.4.5 Two Relationships for the Dissipationally Undeterminable Part of the Biadmittance.- 5.4.6 Relations for the Dissipationally Undeterminable Part of the Triadmittance.- 5.4.7 Relations for the Dissipationally Undeterminable Part of the Quadruple Equilibrium Correlator.- 5.4.8 Using Symmetrized Triadmittance for Obtaining the Quadruple Correlator.- 5.4.9 Modified Cubic FDRs.- 5.5 Connection Between FDRs of the First and Second Kinds.- 5.5.1 Relaxation Equations with External Forces.- 5.5.2 Derivation of Reciprocal Relation of the First Kind.- 5.5.3 Linear FDR.- 5.5.4 Linear—Quadratic Approximation: The Connection Between the Function ?1,23 and the Quadratic Admittance G1,23.- 5.5.5 Stochastic Equation in the Linear—Quadratic Approximation.- 5.5.6 Derivation of the Quadratic FDR for ?12,3.- 5.5.7 Another Quadratic FDR.- 5.5.8 Necessary Conditions Imposed on the Method of External Force Inclusion in the Markov Case: Linear—Quadratic Approximation.- 5.6 Linear and Quadratic FDRs of the Third Kind.- 5.6.1 Definition of Impedances.- 5.6.2 Reciprocal Relations for Linear Impedance.- 5.6.3 Linear FDR or Nyquist’s Formula.- 5.6.4 Correlators of Random Forces in the Nonlinear Case: Their Relation to the Functions G….- 5.6.5 Formulas for L12,3 and L123.- 5.6.6 The Functions Q… and the Stochastic Representation of Random Forces.- 5.6.7 Quadratic FDRs of the Third Kind.- 5.6.8 Determination of L1 and Q1.- 5.6.9 Another Form of FDRs of the Third Kind.- 5.7 Cubic FDRs of the Third Kind.- 5.7.1 Relation Between Four-Subscript Functions L… and Functions G….- 5.7.2 Dissipationally Determinable and Dissipationally Undeterminable Parts of the Functions L….- 5.7.3 Stochastic Representation and Its Consequences.- 5.7.4 Relations for the Dissipationally Determinable Parts of the Functions Q….- 5.7.5 Relationships for the Dissipationally Undeterminable Parts of the Functions Q….- 5.7.6 Another Form of Cubic FDRs of the Third Kind.- 5.8 Notes on References to Chapter 5.- 6. Some Uses of Non-Markov FDRs.- 6.1 Calculation of Many-Time Equilibrium Correlators and Their Derivatives in the Markov Case.- 6.1.1 Linear and Quadratic Admittances in the Markov Case.- 6.1.2 Twofold and Threefold Correlators in the Markov Case.- 6.1.3 Another Approach to the Computation of Twofold and Threefold Correlators: The Impedance Method.- 6.1.4 Finding the Fourfold Correlator: Its Dissipationally Determinable Part.- 6.1.5 Dissipationally Undeterminable Functions Z12,34(2), Y12,34(2).- 6.1.6 Dissipationally Undeterminable Part of the Fourfold Correlator.- 6.1.7 Four-Subscript Derivatives of Correlators with Respect to Forces.- 6.2 Examples of Computations of Many-Fold Correlators or Spectral Densities and Their Derivatives with Respect to External Forces.- 6.2.1 A Nonlinear Resistance—Inductance Electrical Circuit.- 6.2.2 An Electrical Circuit with Capacitance and Cubic Nonlinear Resistance.- 6.2.3 The Threefold Correlator of the Internal Energy of a Body in Thermal Contact with Another Body.- 6.2.4 Velocity Correlators of a Body Travelling with Nonlinear Friction in an Isotropic Medium.- 6.2.5 A Circuit with Inductance in the Non-Markov Case.- 6.2.6 Threefold Spectral Density of the Concentration of Diffusing Gas.- 6.2.7 Impedances and Admittances of an Electromagnetic Field in a Cubically Nonlinear Medium.- 6.2.8 Dissipationally Undeterminable Functions and the Fourfold Correlator of an Electromagnetic Field.- 6.3 Other Uses of Nonlinear FDRs.- 6.3.1 Calculation of Z12,34 in the Diode Model of Nonlinear Resistance.- 6.3.2 The Dissipationally Undeterminable Function Z12,34(2) for the Example of Sect. 6.2.5.- 6.3.3 Serially Connected Nonlinear Subsystems at Different Temperatures.- 6.3.4 The Threefold Flux Correlator for Serially Connected Nonlinear Subsystems at Different Temperatures.- 6.3.5 Nonfluctuational Fluxes in Systems Containing Nonlinear Dissipative Elements at Different Temperatures.- 6.3.6 An Example of Flux Due to Temperature Difference Between Non-linear Resistances.- 6.4 Application of Cubic FDRs to Calculate Non-Gaussian Properties of Flicker Noise.- 6.4.1 Flicker Noise.- 6.4.2 How Will Flicker Noise Change if a Two-Terminal Impedor Is Connected to a Resistance with Flicker Properties?.- 6.4.3 Correlators for Flicker Noise in a Model of Fluctuating Resistance.- 6.4.4 Another Way of Obtaining Correlators.- 6.4.5 The Fourfold Correlator of Flicker Noise Applied to the Theory of the Voss and Clark Experiment.- 6.5 Notes on the References to Chapter 6.- Appendices.- A1. Relation of Conjugate Potentials in the Limit of Small Fluctuations.- A2. On the Theory of Infinitely Divisible Probability Densities.- A2.1 Justification of the Representation (2.4.9) Subject to (2.4.10).- A2.2 Example: Gaussian Distribution.- A3. Some Formulas Concerning Operator Commutation.- A5. The Contribution of Individual Terms of the Master Equation.- A6. Spectral Densities and Related Formulas.- A6.1 Definition of Many-Fold Spectral Densities.- A6.2 Space—Time Spectral Densities.- A6.3 Spectral Density of Spatial Spectra and Space—Time Spectral Density.- A6.4 Spectral Density of Nonstationary and Nonhomogeneous Random Functions.- A7. Stochastic Equations for the Markov Process.- A7.1 The Ito Stochastic Equation.- A7.2 Symmetrized Stochastic Equations.- References.