Extended Irreversible Thermodynamics (eBook)

Preface to the Fourth Edition 5
Preface to the Third Edition 7
Preface to the First Edition 9
Contents 12
Part I General Theory 18
1 Classical, Rational and Hamiltonian Formulations of Non-equilibrium Thermodynamics 19
1.1 The General Balance Laws of Continuum Physics 20
1.1.1 The One-Component System 21
1.1.2 The Multicomponent Mixture 24
1.1.3 Charged Systems 26
1.2 The Law of Balance of Entropy 29
1.3 Classical Irreversible Thermodynamics 30
1.3.1 The Local-Equilibrium Hypothesis 30
1.3.2 Entropy Production and Entropy Flux 31
1.3.3 Linear Constitutive Equations 33
1.3.4 Constraints on the Phenomenological Coefficients 36
1.3.5 The Onsager–Casimir Reciprocal Relations 36
1.3.6 Limitations 38
1.4 Rational Thermodynamics 39
1.4.1 The Basic Axioms of Rational Thermodynamics 40
1.4.1.1 The Principle of Equipresence 40
1.4.1.2 The Principle of Memory or Heredity 41
1.4.1.3 The Principle of Local Action 41
1.4.1.4 The Principle of Material Frame-Indifference 41
1.4.2 Constitutive Equations 44
1.4.3 Critical Remarks 46
1.5 A Hamiltonian Formulation: the generic Formalism 48
1.6 Problems 51
2 Extended Irreversible Thermodynamics: Evolution Equations 57
2.1 Heat Conduction in Rigid Solids 58
2.1.1 Motivations 58
2.1.2 The Generalised Gibbs Equation 64
2.3 One-Component Viscous Fluid 69
2.4 The Generalised Entropy Flux and Entropy Production 71
2.5 Linearized Evolution Equations of the Fluxes 73
2.6 Rational Extended Thermodynamics 75
2.6.1 Heat Conduction 75
2.6.2 Viscous Fluids 79
2.7 Some Comments and Perspectives 80
2.8 Problems 83
3 Extended Irreversible Thermodynamics: Non-equilibrium Equations of State 87
3.1 Physical Interpretation of the Non-equilibrium Entropy 87
3.2 Non-equilibrium Equations of State: Temperature 90
3.2.1 Zeroth Law, Second Law, and Temperature 91
3.2.2 Evaluation of in Some Special Situations 95
3.2.3 Alternative Definitions of Generalised Temperature 95
3.2.4 Experimental Hints for the Non-equilibrium Temperature 97
3.3 Non-equilibrium Equations of State: Thermodynamic Pressure 98
3.4 Concavity Requirements and Stability 101
3.4.1 Heat Conduction in a Rigid Solid 102
3.4.2 Heat Conduction in Ideal Gases 103
3.4.3 Shear Viscous Pressure in Viscous Fluids 104
3.5 Problems 105
Part II Microscopic Foundations 106
4 The Kinetic Theory of Gases 107
4.1 The Basic Concepts of Kinetic Theory 107
4.1.1 Balance Equations 109
4.1.2 The H-Theorem and the Second Law 110
4.2 Non-equilibrium Entropy and the Entropy Flux 112
4.3 Grad's Solution 113
4.4 The Relaxation-Time Approximation 118
4.5 Dilute Non-ideal Gases 120
4.5.1 Entropy and Evolution Equations 121
4.5.2 Microscopic Identification of Coefficients 123
4.6 Non-linear Transport 124
4.7 Beyond the Thirteen-Moment Approximation 128
4.8 Problems 133
5 Fluctuation Theory 136
5.1 Einstein's Formula: Second Moments of Equilibrium Fluctuations 136
5.2 Ideal Gases 141
5.3 Fluctuations and Hydrodynamic Stochastic Noise 144
5.4 The Entropy Flux 144
5.5 Application to a Radiative Gas 146
5.6 Onsager's Relations 148
5.7 Experimental Observations of Fluctuations of the Fluxes and Flux Fluctuation Theorem 150
5.8 Problems 152
6 Information Theory 156
6.1 Basic Concepts 156
6.2 Ideal Gas Under Heat Flux and Viscous Pressure 161
6.3 Ideal Gas Under Shear Flow: Non-linear Analysis 163
6.4 Ideal Gas Submitted to a Heat Flux: Non-linear Analysis 166
6.5 Relativistic Ideal Gas Under an Energy Flow 167
6.6 Heat Flow in a Linear Harmonic Chain 170
6.7 Information Theory and Non-equilibrium Fluctuations 175
6.8 Problems 179
7 Linear Response Theory 181
7.1 Projection Operator Methods 181
7.2 Evolution Equations for Simple Fluids 187
7.3 Continued-Fraction Expansions 191
7.4 Problems 193
8 Computer Simulations 195
8.1 Computer Simulations of Non-equilibrium Steady States 195
8.2 Non-equilibrium Equations of State 197
8.3 Dependence of the Free Energy on the Shear Rate: Non-linear Approach 200
8.4 Shear-Induced Heat Flux and the Zeroth Law 202
8.5 Problems 204
Part III Selected Applications 208
9 Hyperbolic Heat Transport in Rigid Conductors 209
9.1 Linear Wave Propagation: Second Soundand Telegrapher Equation 210
9.2 Transient Heat Conduction: Parabolic VersusHyperbolic Regimes 212
9.2.1 Formulation of the Problem in Absence of Internal Source 212
9.2.2 Results for Diamond Films 214
9.2.3 Heating in Presence of Internal Energy Source: Application to Thermal Ignition 215
9.2.3.1 The Model 216
9.2.3.2 Results for Solids Propellants 217
9.3 Beyond the Cattaneo Equation 218
9.3.1 Guyer–Krumhansl's Model 219
9.3.2 A Generalised Guyer–Krumhansl's Model 220
9.3.2.1 Heat Propagation Velocity: Non-linear Acceleration Waves 225
9.3.2.2 Application to Dielectric Crystals at Low Temperature (< 20 K)
9.3.3 Other Examples of Flux Limiters 229
9.4 Phonon Hydrodynamics 232
9.5 Two-Temperature Models 233
9.6 Other Applications 234
9.7 Problems 235
10 Heat Transport in Micro- and Nano-systems 242
10.1 EIT Description of Heat Conductionat Micro- and Nano-scales 243
10.1.1 Effective Heat Conductivity 244
10.1.2 Transient Temperature Distribution in a Micro Film 248
10.2 The Equation of Phonon Radiative Transfer 250
10.3 The Ballistic Diffusion Equation 254
10.3.1 The Model Equations 255
10.3.2 Illustration: Transient Heat Transport in Thin Films 258
10.4 Problems 260
11 Waves in Fluids: Sound, Ultrasound, and Shock Waves 262
11.1 Sound Propagation in Fluids: Linear Waves 262
11.1.1 The Classical Theory 263
11.1.2 Extended Thermodynamic Theory 265
11.1.3 Particular Results for Monatomic Gases 267
11.2 Non-linear Acceleration Waves in Monatomic Ideal Gases 1
11.3 Shock Waves 273
11.3.1 The Classical Navier–Fourier–Stokes Approach 273
11.3.2 The Extended Irreversible Thermodynamics Approach 275
11.3.3 Shock Structure 278
11.4 Problems 280
12 Generalised Hydrodynamics 283
12.1 Density and Current Correlation Functions 283
12.2 Spectral Density Correlation 285
12.2.1 The Classical Hydrodynamical Approximation 285
12.2.2 The EIT Description 288
12.3 The Transverse Velocity Correlation Function:The EIT Description 290
12.4 The Longitudinal Velocity Correlation Function: The EIT Description 293
12.5 Influence of Higher-Order Fluxes 296
12.6 Problems 297
13 Non-classical Diffusion, Thermo-diffusion and Suspensions 299
13.1 Molecular Diffusion in Perfect Fluid Mixtures 300
13.2 Telegrapher's Equation and Stochastic Processes 305
13.2.1 Correlated Random Walk 306
13.2.2 H–Theorem for Telegrapher Type Equation 307
13.3 Non-Fickian Diffusion in Polymers 309
13.4 Hyperbolic Reaction–Diffusion Systems 314
13.5 Thermo-diffusion in Binary Mixtures 316
13.6 Suspensions of Solid Particles in Fluids 319
13.6.1 Suspensions Versus Molecular Diffusion 320
13.6.2 EIT Description of Suspensions 321
13.6.3 Comparison with Other Models 324
13.6.3.1 Internal Variables Theory 324
13.6.3.2 The Two-Fluid Model 325
13.7 Microstructure in Rapid Solidification of Binary Alloys 326
13.8 Problems 329
14 Electrical Systems and Micro-devices Modelization 334
14.1 Electrical Systems: Evolution Equations 334
14.2 Cross Terms in Constitutive Equations: Onsager's Relations 338
14.3 Hydrodynamical Models of Transport in Semiconductors and Plasmas 341
14.3.1 Transport in Semiconductors 341
14.3.2 Transport in Plasmas 345
14.4 Dielectric Relaxation of Polar Liquids 347
14.5 Problems 350
15 From Thermoelastic Solids to Rheological Materials 354
15.1 Thermoelasticity 355
15.2 Viscoelasticity 358
15.3 Plasticity 361
15.4 Relation of EIT to Kinetic Polymer Models 366
15.4.1 The Rouse and Zimm Models 366
15.4.2 EIT Description of the Rouse and Zimm Models 367
15.4.3 Kinetic Justification of the EIT Results 368
15.5 Non-Newtonian Fluids 372
15.5.1 General Considerations 372
15.5.2 EIT Description of Second-Order Non-Newtonian Fluids 376
15.5.2.1 A Three-Parameter Model 376
15.5.2.2 Giesekus Four-Parameter Model 382
15.6 Problems 383
16 Thermodynamics of Polymer Solutions Under Shear Flow 389
16.1 The Chemical Potential Under Shear 390
16.2 Explicit Solution for the Rouse–Zimm Model 394
16.2 Chemical Reactions Under Flow 399
16.2.1 Shear-Induced Effect on the Affinity 399
16.2.2 Illustration: Polymer Degradation Under Flow 400
16.4 Dynamical Approach 403
16.5 Shear-Induced Migration of Polymers 405
16.6 Problems 409
17 Relativistic Formulation 413
17.1 The Macroscopic Theory 414
17.2 Characteristic Speeds 417
17.3 Relativistic Kinetic Theory 420
17.4 Nuclear Matter and Relativistic Ion Collisions 423
17.5 Problems 425
18 Viscous Cosmological Models and Cosmological Horizons 428
18.1 Viscous Cosmological Models and Accelerated Expansion 429
18.2 Particle Production and Effective Bulk Viscosity 435
18.3 Extended Thermodynamics and Cosmological Horizons 437
18.4 Astrophysical Problems: Gravitational Collapse 440
18.5 Problems 441
A Summary of Vector and Tensor Notation 444
A.1 Symmetric and Antisymmetric Tensors 444
A.2 Decomposition of a Tensor 444
A.3 Scalar (or Dot) and Tensorial (Inner) Products 445
A.4 (Inner) Tensorial Product (also Named Dyadic Product) 446
A.5 Cross Multiplication Between Two Vectors and Between a Tensor and a Vector 446
A.6 Differentiation 446
A.7 Tensor Invariants 447
B Useful Integrals in the Kinetic Theory of Gases 448
C Some Physical Constants 449
References 450
Author Index 471
Subject Index 478