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" Nonlinear nonequilibrium thermodynamics i : "
Rouslan L Stratonovich
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BL
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753692
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b573653
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Rouslan L Stratonovich
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Nonlinear nonequilibrium thermodynamics i : : linear and nonlinear fluctuationdissipation theorems.\ Rouslan L Stratonovich
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| Publication Statement
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[Place of publication not identified] : Springer, 2012
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| ISBN
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3642773435
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: 9783642773433
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| Contents
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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.
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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.
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| LC Classification
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QC318.I7R687 2012
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| Added Entry
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Rouslan L Stratonovich
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