Extended Thermodynamics

1 Early Version of Extended Thermodynamics and Kinetic Theory of Gases.- 1 Paradoxa of Heat Conduction and Shear Diffusion.- 1.1 Heuristic Derivation of the Laws of Fourier and Navier-Stokes.- 1.2 Parabolic Laws of Heat Conduction and Shear Diffusion.- 2 Paradox Removed.- 2.1 The Cattaneo Equation.- 2.2 Extended TIP.- 2.3 Finite Pulse Speeds in Extended TIP.- 2.4 Conclusion and Criticism.- 3 Kinetic Theory of Mon-Atomic Gases.- 3.1 Boltzmann Equation and Moments.- 3.2 Equations of Balance for Moments.- 3.3 Balance of Entropy and Possible Equilibria.- 3.4 The Grad Distribution.- 3.5 Entropy and Entropy Flux in Grad’s 13-Moment Theory.- 3.6 Phenomenological Equations Derived from the Kinetic Theory.- 3.7 Pulse Speeds.- 3.8 Conclusions.- 2 Extended Thermodynamics of Mon-Atomic Gases.- 1 The Equations of Extended Thermodynamics of Mon-Atomic Gases.- 1.1 Thermodynamic Processes.- 1.2 Discussion.- 1.3 Galilean Invariance: Convective and Non-Convective Fluxes.- 1.4 Euclidean Invariance: Inertial Effects.- 2 Constitutive Theory.- 2.1 Restrictive Principles.- 2.2 Exploitation of the Principle of Material Frame Indifference.- 2.3 Exploitation of the Entropy Principle.- 2.4 Exploitation of the Requirement of Convexity and Causality.- 3 Field Equations and the Thermodynamic Limit.- 3.1 Field Equations.- 3.2 The Thermodynamic Limit.- 3.3 The Frame-Dependence of the Heat Flux.- 4 Thermal Equations of State and Specific Equation for Ideal Gases.- 4.1 The Classical Ideal Gas.- 4.2 Comparison with the Kinetic Theory.- 4.3 Comparison with Extended TIP.- 4.4 Degenerate Ideal Gases.- 3 Thermodynamics of Mixtures of Fluids.- 1 Ordinary Thermodynamics of Mixtures (TIP).- 1.1 Constitutive Equations.- 1.2 Paradox of Diffusion.- 2 Extended Thermodynamics of Non-Viscous Simple Mixtures.- 2.1 Balance Equations.- 2.2 Thermodynamic Processes.- 2.3 Constitutive Theory.- 2.4 Summary of Results.- 2.5 Wave Propagation in a Non-Reacting Binary Mixture.- 2.6 Landau Equations: First and Second Sound in He II.- 3 Ordinary and Extended Thermodynamics of Mixtures.- 3.1 The Laws of Fick and Fourier in Extended Thermodynamics.- 3.2 Onsager Relations.- 3.3 Inertial Contribution to the Laws of Diffusion.- 4 Relativistic Extended Thermodynamics of Mon-Atomic Gases.- 1 Balance Equations and Constitutive Restrictions.- 1.1 Thermodynamic Processes.- 1.2 Principles of the Constitutive Theory.- 2 Constitutive Theory.- 2.1 Scope and Structure.- 2.2 Lagrange Multipliers and the Vector Potential.- 2.3 Principle of Relativity and Linear Representations.- 2.4 Stress Deviator, Heat Flux, and Dynamic Pressure.- 2.5 Fugacity and Absolute Temperature.- 2.6 Linear Relations Between Lagrange Multipliers and n, UA, t, ?, qA, e.- 2.7 The Linear Flux Tensor.- 2.8 The Entropy Flux Vector.- 2.9 Residual Inequality.- 2.10 Causality and Convexity.- 2.11 Summary of Results.- 3 Identification of Viscosities and Heat Conductivity.- 3.1 Extended Thermodynamics and Ordinary Thermodynamics.- 3.2 Transition from Extended to Ordinary Thermodynamics.- 4 Specific Results for Relativistic and Degenerate Gases.- 4.1 Equilibrium Distribution Function.- 4.2 The Degenerate Relativistic Gas.- 4.3 Non-Degenerate Relativistic Gas.- 4.4 Degenerate Non-Relativistic Gas.- 4.5 Non-Degenerate Non-Relativistic Gas.- 4.6 Strongly Degenerate Relativistic Fermi Gas.- 4.7 A Remark on the Strongly Degenerate Relativistic Bose Gas.- 4.8 Equilibrium Properties of an Ultrarelativistic Gas.- 5 Two Applications.- 5.1 The Limit Mass of White Dwarfs.- 5.2 Thermo Acceleration Waves.- 6 The Relativistic Kinetic Theory for Non-Degenerate Gases.- 6.1 Boltzmann—Chernikov Equation.- 6.2 Equations of Transfer.- 6.3 Equations of Balance for Particle Number, Energy-Momentum, Fluxes, and Entropy.- 6.4 Maxwell-Jüttner Distribution: Equilibrium Properties.- 6.5 Possible Thermodynamic Fields in Equilibrium.- 7 The Non-Relativistic Limit of Relativistic Thermodynamics.- 7.1 The Problem.- 7.2 Variables and Constitutive Quantities.- 7.3 The Dynamic Pressure.- 5 Thermodynamics of Light and Sound.- 1 Phonons and Photons.- 1.1 Cavity Radiation and Sound in Single Crystals.- 1.2 Thermodynamic Processes.- 1.3 Motivations.- 1.4 Principles of the Constitutive Theory: The Entropy Inequality.- 1.5 Exploitation of the Entropy Principle.- 2 Energy and Momentum of Light and Sound.- 2.1 Eddington Factor, Entropy, and Entropy Flux.- 2.2 Summary of Results and Limiting Cases.- 2.3 Waves.- 3 Shock Waves.- 3.1 Rankine Hugoniot Equations and the Speed of Shock Waves.- 3.2 Entropy Criterion for the Admissibility of Shock Waves.- 6 Formal Structure of Extended Thermodynamics.- 1 The Formal Structure of Extended Thermodynamics.- 1.1 The Axioms of Extended Thermodynamics: Thermodynamic Processes.- 1.2 Universal Principles of the Constitutive Theory.- 1.3 Exploitation of the Entropy Principle.- 1.4 Symmetric Hyperbolic Field Equations.- 2 Galilean Invariance.- 2.1 Euclidean Transformations, Galilean Transformations, and Rotations.- 2.2 Explicit Dependence of Constitutive Quantities on Velocity.- 2.3 Galilean Invariance and Entropy Principle.- 3 Variables of Increasing Tensorial Rank.- 3.1 Restrictive Conditions on the Matrices X(v) and Ar.- 3.2 The Case N = 3: An Illustration.- 3.3 The Matrix X(v) for Arbitrary N.- 7 Waves in Extended Thermodynamics.- 1 Hyperbolicity and Symmetric Hyperbolic Systems.- 1.1 Hyperbolicity in the t-Direction.- 1.2 Symmetric Hyperbolic Systems.- 2 Linear Waves.- 2.1 Plane Harmonic Waves: The Dispersion Relation.- 2.2 The High Frequency Limit.- 2.3 Higher-Order Terms.- 2.4 Linear Waves in Extended Thermodynamics.- 3 Hyperbolicity in Extended Thermodynamics.- 3.1 The Characteristic Polynomial.- 3.2 Region of Hyperbolicity.- 4 Non-Linear Waves, Discontinuity Waves, or Acceleration Waves.- 4.1 Amplitude of Discontinuity Waves.- 4.2 Growth and Decay.- 4.3 Evolution of Amplitude in Extended Thermodynamics.- 5 Non-Linear Waves and Shock Waves.- 5.1 Weak Solutions.- 5.2 Rankine-Hugoniot Equations.- 5.3 Shocks in Extended Thermodynamics.- 5.4 Selection Rules for Physical Shocks: The Entropy Growth Condition.- 5.5 Selection Rules for Physical Shocks: The Lax Conditions.- 5.6 Lax Condition in Extended Thermodynamics.- 8 Second Sound in Solids at Low Temperature.- 1 Extended Thermodynamics of Phonons Generalized.- 1.1 Equations of Balance and Constitutive Equations.- 1.2 Entropy Principle.- 1.3 Adjusting v’ to Observed Wave Speeds.- 1.4 An Analytic Form of UE(T).- 2 Shock Waves in a Rigid Conductor.- 2.1 Rankine-Hugoniot Equations.- 2.2 Entropy Growth Condition.- 2.3 Lax Conditions.- 2.4 Shape Changes of Second Sound Wave.- 9 Molecular Extended Thermodynamics.- 1 Field Equations of Molecular Extended Thermodynamics of Many Moments.- 1.1 Densities, Fluxes, and Productions as Moments of the Phase Density.- 1.2 Maximizing Entropy.- 1.3 Molecular Extended Thermodynamics Implies Extended Thermodynamics.- 1.4 Galilean Invariance and Central Moments.- 1.5 Linear Relation Between Lagrange Multipliers ?? and Fields u?.- 1.6 The Case n = 20: An Illustration.- 1.7 Field Equations for n = 13, 14, 20, 21, 26, 35.- 2 More Kinetic Theory.- 2.1 Boltzmann Equation and Transfer Equations.- 2.2 Linearized Kinetic Theory.- 2.3 Eigenfunctions and Eigenvalues for Maxwellian Atoms.- 3 Equations for Mean Eigenfunctions and Closure.- 3.1 Mean Eigenfunctions.- 3.2 Field Equations for Mean Eigenfunctions and Closure.- 4 Molecular Extended Thermodynamics and Method of Eigenfunctions.- 4.1 Phase Density in Molecular Extended Thermodynamics.- 4.2 Lagrange Multipliers as Expansion Coefficients.- 10 Testing Extended Thermodynamics by Light.- 1 Basic Electrodynamics.- 1.1 Distant Field Approximation.- 1.2 Incident Plane Harmonic Wave.- 1.3 Intensity of Scattered Light.- 2 Auto-Correlation Function as the Characteristic Feature of a Fluctuating Quantity.- 2.1 The Simplest Case: One Real Fluctuating Quantity.- 2.2 Several Complex Fluctuating Quantities.- 3 The Spectral Density: A Measurable Quantity.- 3.1 Auto-Correlation Function of the Scattered Field.- 3.2 Spectral Density and Dynamic Form Factor.- 4 Onsager Hypothesis.- 4.1 Statement.- 4.2 Field Equations of a Navier-Stokes-Fourier Fluid.- 4.3 Solution of the Field Equations.- 5 Auto-Correlation Function and Its Properties.- 5.1 The Dynamic Form Factor.- 5.2 Alternative Form of the Dynamic Form Factor and the Auto-Correlation Function.- 5.3 Form Factor and Auto-Correlation Function in Forward Scattering.- 5.4 Heat Modes and Sound Modes.- 6 How to Use Light Scattering.- 6.1 Two Philosophies.- 6.2 Dimensionless Parameters and Limiting Characteristics.- 7 Light Scattering and Extended Thermodynamics.- 7.1 The Case of 13 Moments: An Illustration.- 7.2 Comparison of the Navier-Stokes-Fourier Theory and of Extended Thermodynamics of 13 and 14 Moments.- 7.3 Extended Thermodynamics with More and More Moments.- 11 Testing Extended Thermodynamics by Sound.- 1 Basic Acoustics.- 1.1 How the Acoustic Resonator Measures Phase Speeds in Principle.- 1.2 Piezoelectric Transducer and the Mechanical Impedance.- 1.3 External Mechanical Impedance and Wave Length.- 1.4 Difficulties with Many Modes and Damping.- 2 Dispersion Relations.- 2.1 Navier-Stokes-Fourier Theory.- 2.2 Extended Thermodynamics of 13 Fields.- 2.3 Extended Thermodynamics with Many Variables.- 2.4 Interpretation and Criticism.- 3 Maximum Speed.- 3.1 Modes of Least Damping.- 3.2 The Maximum Speed.