RATIONAL EXTENDED
THERMODYNAMICS
Ingo Müller
Thermodynamik -
Technische Universität Berlin.
&
Tommaso Ruggeri
Dipartimento di Matematica e CIRAM -
Università di Bologna.
Preface
Ordinary thermodynamics provides reliable results when
the thermodynamic fields are smooth, in the sense that there are no steep
gradients and no rapid changes. In fluids and gases this is the domain of the
equations of Navier-Stokes and Fourier. Extended thermodynamics becomes
relevant for rapidly varying and strongly inhomogeneous processes. Thus the
propagation of high-frequency-waves, and the shape of shock waves, and the
regression of small-scale-fluctuation are governed by extended thermodynamics.
The field equations of ordinary thermodynamics are
parabolic while extended
thermodynamics is governed by hyperbolic systems. The
main ingredients of
extended thermodynamics are
·
field equations of balance type,
·
constitutive quantities depending on the present local
state and
·
entropy as a concave function of the state variables
This set of assumptions leads to first order
quasi-linear symmetric hyperbolic systems of field equations; it guarantees the
well-posedness of initial value problems and finite speeds of propagation.
Several tenets of irreversible thermodynamics had to
be changed in subtle ways to make extended thermodynamics work. Thus, the
entropy is allowed to depend on non-equilibrium variables, the entropy flux is
a general constitutive quantity, and the equations for stress and heat flux
contain inertial terms. New insight is therefore provided into the principle of
material frame indifference.
With these modifications an elegant formal structure
can be set up in which, just as in classical thermostatic, all restrictive
conditions – derived from the entropy principle -- take the form of
integrability conditions.
Also the modifications made by extended thermodynamics
render the theory fully consistent with the kinetic theory of gases, in
particular Grad's 13-moment version of the kinetic theory of gases. In fact,
extended thermodynamics is most restrictive for gases or, more generally, for
bodies whose constituent particles have large mean free path. Most of this
book, therefore, deals with gases: classical ideal gases, degenerate gases,
relativistic gases, and mixtures of gases. It puts into perspective the various
phenomena called second sound, viz. heat propagation, propagation of
shear stress, and the second sound in superfluid helium.
Phonons and photons may have large mean free paths as well,
and therefore they are amenable to a treatment by extended thermodynamics. Two
chapters describe the present status of the systematic theory in this field,
which is still progressing.
A certain disappointment with extended thermodynamics
of 13 or 14 fields is created by the observation that it describes resonance
experiments and light scattering data only slightly better than the
conventional theory. These data require further extensions to many moments.
Also the shock wave structure calculated in extended thermodynamics of 13
fields is worse than the shock wave structure in ordinary thermodynamics; and
again: many moments are needed to put things right. All this will be
demonstrated in the second half of this book.
When enough moments are used to describe the state,
extended thermodynamics leads to perfect agreement of theory and experiment.
Actually -- even without reference to experiments -- extended thermodynamics
carries its own evaluation of the range of validity: As soon as more moments do
not change the predictions, the extant number of moments provides a proper
description of the state.
The present
book is a new edition; at least half of the material is new and the rest is
revised and streamlined to a considerable degree. Also the title is changed: Rational
Extended Thermodynamics. The literature is full of papers referring to extended
thermodynamics which, however, are devoid of rational methodology and
mathematical cohesion. The epithet rational in the present title is
chosen so as to emphasize the
systematic procedure which the book espouses, -- a procedure typical for a
deductive science.
Three chapters, Chapters 12 through 14, carry the
names of Drs. Struchtrup and Weiss, because the material presented there is the
work of these authors. However, the chapters are entirely embedded - in
contents, format, and style -- into the scope of this book. We are grateful to
Drs. Struchtrup and Weiss for their contribution.
We wish to thank Mrs. Marlies Hentschel who spent long
hours on the word processor with unfailing enthusiasm for this work.
Contents
·
1 Tour d'Horizon 1
·
2 Early Version of Extended Thermodynamics 9
o 1
Paradoxa of Heat Conduction and Shear Diffusion 10
§
1.1 Heuristic Derivation of the Laws of Fourier
and Navier-Stokes 10
§
1.2 Parabolic Laws of Heat Conduction and Shear
Diffusion 11
o 2
Paradox Removed 12
§
2.1 The Cattaneo Equation 12
§
2.2 Extended TIP 14
§
2.3 Finite Pulse Speeds in Extended TIP 16
§
2.4 Conclusion and Criticism 18
o 3
Kinetic Theory of Mon-atomic Gases
19
§
3.1 Boltzmann-Equation and Moments 19
§
3.2 Equations of Balance for Moments 20
§
3.3 Balance of Entropy and Possible
Equilibria 22
§
3.4 The Grad Distribution 23
§
3.5 Entropy and Entropy Flux in Grad's 13-Moment
Theory. 24
§
3.6 Phenomenological Equations derived from the
Kinetic Theory 24
§
3.7 Pulse Speeds 26
§
3.8 Conclusions 26
·
3 Formal Structure of Extended
Thermodynamics 27
o 1
Field Equations 28
§
1.1 Thermodynamic Processes and Principles of
the Constitutive Theory 28
§
1.2 Universal Principles of the Constitutive
Theory 28
o 2
Entropy Inequality, Symmetric Hyperbolicity 29
§
2.1 Exploitation of the Entropy Inequality 29
§
2.2 Symmetric Hyperbolic Field Equations 31
§
2.3 Discussion
31
§
2.4 Characteristic Speeds 32
o 3
Main Subsystems 33
§
3.1 Constraints on the Main Field 33
§
3.2 A Main Subsystem implies an Entropy
Inequality 33
§
3.3 A Main Subsystem is Symmetric
Hyperbolic 34
§
3.4 Characteristic Speeds of the Subsystems 34
§
3.5 Other Subsystems 35
o 4
Galilean Invariance 35
§
4.1 Tensors, Galilean Tensors and Euclidean
Tensors 35
§
4.2 Principle of Relativity 36
§
4.3 Exploitation of the Principle of Relativity
for the Entropy Balance 37
§
4.4 Exploitation of the Principle of Relativity
for the Field Equations 37
§
4.5 Field Equations for Internal Quantities 38
§
4.6 Galilei Invariance for Subsystems 39
§
4.7 Galilean Invariance and Entropy
Principle 40
§
4.8 Explicit Velocity Dependence of Constitutive
Quantities. The Determination of Ar. 41
o 5
Thermodynamics of an Euler Fluid
43
§
5.1 The Euler Fluid 43
§
5.2 Lagrange Multipliers 44
§
5.3 Internal Lagrange Multipliers 45
§
5.4 Absolute Temperature 46
§
5.5 Vector Potential 46
§
5.6 Convexity
47
§
5.7 Characteristic Speed 47
§
5.8 Subsystems
48
§
5.9 Discussion
49
·
4 Extended Thermodynamics of Mon-Atomic
Gases 51
o 1
The Equations of Extended Thermodynamics of Mon-Atomic Gases 52
§
1.1 Thermodynamic Processes 52
§
1.2 Discussion
53
§
1.3 Galilean Invariance. Convective and Non-convective
Fluxes. 53
§
1.4 Euclidean Invariance. Inertial Effects. 55
o 2
Constitutive Theory. 56
§
2.1 Restrictive Principles. 56
§
2.2 Exploitation of the Principle of Material
Frame Indifference 58
§
2.3 Exploitation of the Entropy Principle 59
1. Six Steps
59
2. Lagrange Multipliers (Step i.) 59
3. Removal of
Velocity-Dependent Terms (Step ii)
60
4. Lagrange Multipliers
as Variables. The Scalar Potential and the Vector Potential. (Step
iii) 61
5. Equilibrium. (Step
iv) 62
6. The Potentials
near Equilibrium (Step v.) 63
7. Residual
Inequality (Step vi) 65
8. Summary of Results
of the Entropy Principle (Step vii)
65
§
2.4 Exploitation of the Requirement of Convexity
and Causality 67
o 3
Field Equations and the Thermodynamic Limit 67
§
3.1 Field Equations. 67
§
3.2 The Thermodynamic Limit. 69
§
3.3 The Frame-Dependence of the Heat Flux. 71
§
3.4 Material Frame-Indifference in Ordinary and
Extended Thermodynamics 73
o 4
Thermal Equations of State and Ideal Gases. 73
§
4.1 The Classical Ideal Gas 73
§
4.2 Comparison with the Kinetic Theory. 74
§
4.3 Comparison with Extended TIP. 75
§
4.4 Degenerate Ideal Gases. 75
·
5 Thermodynamics of Mixtures of Euler
Fluids 79
o 1
Ordinary Thermodynamics of Mixtures (TIP) 80
§
1.1 Constitutive Equations 80
§
1.2 Paradox of Diffusion 83
o 2
Extended Thermodynamics of Mixtures of Euler Fluids 83
§
2.1 Balance Equations 83
§
2.2 Thermodynamic Processes 85
§
2.3 Constitutive Theory 86
§
2.4 Summary of Results 90
§
2.5 Wave Propagation in a Non-Reacting Binary
Mixture 92
§
2.6 Landau Equations. First and Second Sound in
He II. 96
o 3
Ordinary and Extended Thermodynamics of Mixtures. 99
§
3.1 The Laws of Fick and Fourier in Extended
Thermodynamics. 99
§
3.2 Onsager Relations 101
§
3.3 Inertial Contribution to the Laws of
Diffusion. 102
·
6 Relativistic Thermodynamics 105
o 1
Balance Equations and Constitutive Restrictions. 106
§
1.1 Thermodynamic Processes 106
§
1.2 Principles of the Constitutive Theory. 107
o 2
Constitutive Theory. 108
§
2.1 Scope and Structure. 108
§
2.2 Lagrange Multipliers and the Vector
Potential. Step i. 108
§
2.3 Principle of Relativity and Linear
Representations. Step ii. 110
§
2.4 Stress Deviator, Heat Flux and Dynamic
Pressure. Step iii. 112
§
2.5 Fugacity and Absolute Temperature. Step
iv. 113
§
2.6 Linear Relations between Lagrange
Multipliers and n, UA, tAB
, π, qA, e 114
§
2.7 The Linear Flux Tensor. Step vi. 116
§
2.8 The Entropy Flux Vector. Step vii. 117
§
2.9 Residual Inequality. Step viii. 118
§
2.10 Causality and Convexity. Step ix. 118
§
2.11 Summary of Results. Step x. 120
o 3
Identification of Viscosities and Heat-Conductivity 123
§
3.1 Extended Thermodynamics and Ordinary
Thermodynamics. 123
§
3.2 Transition from Extended to Ordinary
Thermodynamics 124
o 4
Specific Results for Relativistic and Degenerate Gases. 126
§
4.1 Equilibrium Distribution Function. 126
§
4.2 The Degenerate Relativistic Gas. 127
§
4.3
Non-Degenerate Relativistic Gas.
131
§
4.4
Degenerate Non-relativistic Gas.
133
§
4.5
Non-degenerate Non-relativistic Gas.
135
§
4.6 Strongly Degenerate Relativistic Fermi
Gas 136
§
4.7 A Remark on the Strongly Degenerate
Relativistic Bose Gas. 139
§
4.8 Equilibrium Properties of an
Ultrarelativistic Gas. 139
o 5 An
Application: The Mass-limit of a White Dwarf.
140
o 6
The Relativistic Kinetic Theory for Non-Degenerate Gases. 145
§
6.1 Boltzmann-Chernikov Equation 145
§
6.2 Equations of Transfer. 145
§
6.3 Equations of Balance for Particle Number,
Energy-Momentum, Fluxes and Entropy
146
§
6.4 Maxwell-Jüttner Distribution, Equilibrium
Properties 147
§
6.5 Possible Thermodynamic Fields in
Equilibrium. 148
o 7
The Non-Relativistic Limit of Relativistic Thermodynamics 149
§
7.1 The Problem 149
§
7.2 Variables and Constitutive Quantities 149
§
7.3 The Dynamic Pressure 151
§
7.4 Order of Magnitude of the Dynamic
Pressure 152
·
7 Extended Thermodynamics of Reacting
Mixtures 155
o 1
Motivation, Results and Discussion
156
§
1.1 Motivation
156
§
1.2 Results
157
§
1.3 Discussion
158
o 2
Fields 159
§
2.1 A Conventional Choice 159
§
2.2 Absolute Temperature, Fugacities and
Chemical Affinity 159
§
2.3 Summary of Fields 162
o 3
Field Equation 162
§
3.1 Balance Laws 162
§
3.2 Constitutive Theory 162
§
3.3 Principle of Relativity 163
o 4
Entropy Inequality 164
§
4.1 Lagrange Multipliers 164
§
4.2 Exploitation 164
o
5
Non-Relativistic Limit 164
§
5.1
Discussion 164
§
5.2 Dynamic Pressure and Bulk Viscosity 166
§
5.3 Thermal Conductivity and Viscosity 167
·
8 Waves in Extended Thermodynamics 169
o 1
Hyperbolicity and Symmetric Hyperbolic Systems. 170
§
1.1 Hyperbolicity in the t-direction. 170
§
1.2 Symmetric Hyperbolic Systems 170
o 2
Linear Waves 171
§
2.1 Plane Harmonic Waves, the Dispersion
Relation. 171
§
2.2 The High Frequency Limit 172
§
2.3 Higher Order Terms. 173
§
2.4 Linear Waves in Extended Thermodynamics 173
o 3
Hyperbolicity and Non-linear Waves.
175
§
3.1 The Characteristic Polynomial. 175
§
3.2 Region of Hyperbolicity. 176
o 4
Acceleration Waves. 178
§
4.1 Amplitude of Discontinuity Waves. 178
§
4.2 Growth and Decay. 180
§
4.3 Evolution of Amplitude in Extended
Thermodynamics. 180
§
4.4 Acceleration Waves in Relativistic Extended
Thermodynamics 183
o 5
Weak Solutions and Shock Waves.
184
§
5.1 Weak Solutions. 184
§
5.2 Rankine-Hugoniot Equations 184
§
5.3 Shocks in Extended Thermodynamics. 186
§
5.4 Selection Rules for Physical Shocks, the
Entropy Growth Condition. 190
§
5.5 Selection Rules for Physical Shocks. The Lax
Conditions. 190
§
5.6 Lax Condition in Extended
Thermodynamics. 191
·
9 Extended Thermodynamics of Moments 193
o 1
Field Equations for Moments
194
§
1.1 Densities, Fluxes and Productions as Moments
of the Phase Density 194
§
1.2 Extended Thermodynamics of Moments 195
§
1.3 Specific Phase Densities 196
§
1.4 Field Equations for Λα
and Equations for uα near Equilibrium 197
§
1.5 The Case N=3: An Illustration 198
§
1.6 Field Equations for n=13,14,20,21,26,35 200
o 2
Characteristic Speeds 203
§
2.1 Field Equations near Equilibrium 203
§
2.2 Pulse Speed 204
§
2.3 Discussion
206
§
2.4 The Relativistic Case; Speeds Smaller than
c. 206
o 3
Mean Eigenfunctions 207
§
3.1 Eigenfunctions and Eigenvalues 207
§
3.2 Mean Eigenfunctions as the Main Field 210
§
3.3 Linear Field Equations for the
Mean-Eigenfunctions 211
o 4
Maximization of Entropy
213
§
4.1 Maximizing Entropy 213
§
4.2 Maximizing Entropy is Equivalent to Extended
Thermodynamics of Moments 214
·
10 Extended Thermodynamics and Light
Scattering 215
o 1
Basic Electrodynamics 216
§
1.1 Distant Field Approximation 216
§
1.2 Incident Plane Harmonic Wave 217
o 2 A
Modicum of Fluctuation Theory 219
§
2.1 Expectation Values 219
1. Fluctuation
Theory 219
2. Expectation
Values 221
3. Length Scales 223
§
2.2 Temporal Evolution of a Fluctuation 224
1. Mean regression
and auto-correlation 224
§
2.3 Auto-correlation of Es (R ,t) 225
1. Auto-correlation 225
2. Spectral Density
and Dynamic Form Factor 226
3. Onsager
Hypothesis 227
o 3
Measuring the Spectral Density
227
§
3.1 Signal and Spectral Density 227
§
3.2 Measured Data and their Dependence on
Pressure 230
o 4
Navier-Stokes-Fourier Fluid
231
§
4.1 Dynamic Form Factor 231
§
4.2 An Alternative Form of the Dynamic Form
Factor. Also: An Approximate Form for Forward Scattering. 233
§
4.3 Graphical Representation of the Dynamic Form
Factor for a Monatomic Ideal Gas
234
§
4.4 Comparison with Experimental Data 236
§
4.5 Auto-Correlation 237
§
4.6 Heat and Sound Modes 237
o 5
Extended Thermodynamics
239
§
5.1 Introducing Extended Thermodynamics. The
Case of 13 Moments. 239
§
5.2 Dynamic Form Factors for n=20,35,84 241
§
5.3 Heat and Sound Modes in Extended
Thermodynamics 244
§
5.4 Higher Moments by Method of
Eigenfunctions 244
§
5.5 Dynamic Form Factors for Many Moments 248
§
5.6 Evaluation of Moment Theories 248
§
5.7 Characteristic speeds 251
§
5.8 More Experimental and Theoretical
Evidence 252
o 6
Extrapolation of S 252
§
6.1 The Problem 252
§
6.2 The Boltzmann Equation in the Krook
Approximation 255
§
6.3 The Dynamic Form Factor S(q ,ω);
General Formula 255
§
6.4 Fluctuations in Phase Space 255
§
6.5 The Dynamic Form Factor S(q ,ω);
Specific Form 257
§
6.6 Discussion
257
·
11 Testing Extended Thermodynamics by Sound 259
o 1
Basic Acoustics 260
§
1.1 How the Acoustic Resonator measures Phase
Speeds in Principle. 260
§
1.2 Piezoelectric Transducer and the Mechanical
Impedance 261
§
1.3 External Mechanical Impedance and
Wave-Length 263
§
1.4 Difficulties with Many Modes and
Damping 263
o 2
Dispersion Relations 264
§
2.1 Navier-Stokes-Fourier Theory 264
§
2.2 Extended Thermodynamics of 13 Fields 265
§
2.3 Extended Thermodynamics with Many
Variables 267
§
2.4 Conclusion and Estimate 267
o 3
Maximum Speed 268
§
3.1 Modes of Least Damping 268
§
3.2 The Maximum Speed 269
·
12 Structure of Shock-Waves 271
o 1
Experimental Evidence 272
o 2
Review of Previous Work
273
§
2.1 Rankine-Hugoniot Relations 273
§
2.2 Becker's Solutions 275
§
2.3 Singular Perturbation Analysis 277
§
2.4 Numerical Solution by Gilbarg and
Paolucci 278
§
2.5 The 13-Moment Theory by Grad 278
§
2.6 The 13-Moment Theory by Anile &
Majorana 279
§
2.7 Criticism of Moment Methods for Shock
Structure 279
§
2.8 Alternative Methods for Shock Structure
Calculations 280
o 3
Preliminaries on Singular Points and Characteristic Speeds 280
§
3.1 Field Equations and Boundary Values 280
§
3.2 Singular Points and Stationary Points 281
§
3.3 The Singularities D=0 282
§
3.4 Regular and Irregular Singularities 283
o 4
Numerical Calculation of the Shock Structure 284
§
4.1 Initial and Boundary Value Problems 284
§
4.2 Algorithm for the Initial Value Problem 285
§
4.3 Algorithm for the Boundary Value
Problem 288
§
4.4 The 13-Moment Case 288
§
4.5 The 14-Moment Case 291
§
4.6 The 21-Moment Case 296
o 5
Conclusion 299
o 6
Addendum on Initial Value Problem for 13 Moments 299
o 7
Quantitative Results and Conclusions
301
·
13 Extended Thermodynamics of Radiation 303
o 1
Structure of Extended Thermodynamics of Photons 304
§
1.1 Energy and Momentum of Individual
Photons 304
§
1.2 Radiative Transfer Equation 304
§
1.3 Moments and Moment Equations. The Closure
Problem 305
§
1.4 Entropy and Maximization of Entropy 305
§
1.5 Closure
306
o 2
Equilibrium 307
§
2.1 The First Few Moments 307
§
2.2 Equilibrium of Radiation with Matter 307
1. Matter in Motion 307
2. Matter at Rest 309
3. Momentum of
Radiation in a Moving Body 309
o 3 Near
Equilibrium 310
§
3.1 Phase Density in Near-Equilibrium 310
§
3.2 Approximate Lagrange Multipliers 311
o 4
Field Equations 311
§
4.1 Closure for Moments 311
§
4.2 Closure for Productions 312
§
4.3 The Hierarchies of Field Equations 312
§
4.4 Absorption and Emission of Bremsstrahlung.
Thomson Scattering 313
o 5
Local Radiative Equilibrium
313
§
5.1 The Rosseland Mean Value of the Absorption
Coefficient 313
§
5.2 Maxwell Iteration 314
§
5.3 Conclusion
315
o 6
Compression of Radiation
315
§
6.1 A Thought Experiment 315
§
6.2 Solution of the Radiative Transfer
Equation 317
§
6.3 Solution of Moment Equations 318
§
6.4 Conclusion
318
o 7
Penetration of a Beam of Radiation into Matter 318
§
7.1 Field Equations 318
§
7.2 Characteristic Speeds and Amplitudes of the
Propagating Beam 319
§
7.3 Plane Harmonic Waves and Dispersion Relation
(General) 320
§
7.4 Intense Absorption. The Damped Wave
Limit 322
§
7.5 Intense Scattering. The Diffusion
Limit. 323
§
7.6 General Case and a Simple Example 324
o 8
Radiative Entropy in Gray Bodies
325
§
8.1 Photon Gas and an Eulerian Fluid 325
§
8.2 Equilibrium of Radiation with Matter at
Rest 326
§
8.3 Entropy Production due to Matter-Photon
Interaction 326
§
8.4 Thermodynamic Fields of Radiation in the
Neighbourhood of a Spherical Source 327
§
8.5 Absorption of Radiation from a Spherical
Source in an Eulerian Fluid at Temperature T
329
§
8.6 Entropic Production for Incident Rays 331
§
8.7 Pseudo-Temperature 332
§
8.8 Entropy Flux and Entropy 334
·
14 Extended Thermodynamics of Phonons 337
o 1
Phonon Transfer Equation
338
§
1.1 Energy and Momentum of Phonons 338
§
1.2 Phonon Transfer Equation, Energy and
Momentum 338
§
1.3 The Phase Density of Production 338
o 2
Moments and Moment Equations 340
§
2.1 Moments and their Equilibrium Values 340
§
2.2 Moment Equations and Conservation Laws 340
§
2.3 Closure Problem 341
o 3
The Heat Pulse Experiment
342
§
3.1 Experimental Results and One-dimensional
Equations. 342
§
3.2 Ballistic Phonons 343
§
3.3 Second Sound in its Purest Form 344
§
3.4 Damped Second Sound and Pure Diffusion 345
§
3.5 The 9-Field Theory of Extended
Thermodynamics 345
§
3.6 Heat Pulses. Numerical Solutions 347
·
15 Thermodynamics of Metal Electrons 351
o 1
Equations of Balance 352
§
1.1 Kinetic Theory of Metal-Electrons 352
§
1.2 Equations of Balance of Mass, Momentum,
Energy and Energy Flux 353
§
1.3 Entropy Principle and Phase Density Close to
Equilibrium 354
o 2
Extended Thermodynamics and Kinetic Theory 355
§
2.1 Toward Extended Thermodynamics of Electrons
in Metals 355
§
2.2 A Convenient Shortcut via the Kinetic Theory
of Electrons 356
§
2.3 Characteristic Speeds 357
§
2.4 The Laws of Ohm and Fourier 357
§
2.5 Hall and Coriolis Effects 358
§
2.6 Discussion
359
·
16 Viscoelastic Fluids 361
o 1
Viscoelastic Fluids of Second Grade
362
§
1.1 The Stress of a Second Grade Fluid 362
§
1.2 Ordinary Thermodynamics of Second Grade
Fluids 362
§
1.3 Discussion
364
o 2
Rate Type versus Differential Type Constitutive Equations 365
§
2.1 Cattaneo and Stability 365
§
2.2 Viscoelasticity and Stability 365
§
2.3 Conclusion
366
o 3
Toward Extended Thermodynamics of Viscoelasticity 366
§
3.1 Fields and Field Equations 366
§
3.2 Incompressible Adiabatic Fluid 367
§
3.3 Entropy Inequality 368
§
3.4 Partial Exploitation of the Entropy
Inequality 369
§
3.5 Evaluation
370
§
3.6 Criticism and Outlook 371
·
Bibliography 373