This book gives the first detailed coherent treatment of a relatively young branch of statistical physics - nonlinear nonequilibrium and fluctuation-dissipative thermo­ dynamics. This area of research has taken shape fairly recently: its development began in 1959. The earlier theory -linear nonequilibrium thermodynamics - is in principle a simple special case of the new theory. Despite the fact that the title of this book includes the word "nonlinear", it also covers the results of linear nonequilibrium thermodynamics. The presentation of the linear and nonlinear theories is done within a common theoretical framework that is not subject to the linearity condition. The author hopes that the reader will perceive the intrinsic unity of this discipline, and the uniformity and generality of its constituent parts. This theory has a wide variety of applications in various domains of physics and physical chemistry, enabling one to calculate thermal fluctuations in various nonlinear systems. The book is divided into two volumes. Fluctuation-dissipation theorems (or relations) of various types (linear, quadratic and cubic, classical and quantum) are considered in the first volume. Here one encounters the Markov and non-Markov fluctuation-dissipation theorems (FDTs), theorems of the first, second and third kinds. Nonlinear FDTs are less well known than their linear counterparts.

Inhalt

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 …