Variational and Extremum Principles in Macroscopic Systems (eBook)

Front Cover 1
VARIATIONAL AND EXTREMUM PRINCIPLES IN MACROSCOPIC SYSTEMS 4
Copyright Page 5
CONTENTS 22
Foreword 6
Volume Editors’ Preface 8
Preface 10
List of Contributors 16
Part I: THEORY 40
Chapter 1. Progress in variational formulations for macroscopic processes 42
1. Theoretical aspects 42
2. Applications 44
3. Discussion 46
Acknowledgements 61
References 61
Chapter 2. Lagrange formalism and irreversible thermodynamics: The Second Law of Thermodynamics and the Principle of Least Entropy Production 64
1. The traditional structure of thermodynamics of irreversible processes versus universal structures of Lagrange formalism (LF)—a challenge 64
2. Preparatories: a few formal structures of Lagrange formalism 70
3. Lagrange formalism as applied to thermodynamics of irreversible processes 82
References 94
Chapter 3. Fundamental problems of variational principles: objectivity, symmetries and construction 96
1. Introduction 96
2. Objectivity in nonrelativistic mechanics 98
3. Lagrange formalism 101
4. Lagrange formalism in field theory 103
5. Construction of variational principles 105
6. Spacetime symmetries of dissipative systems 108
7. Discussion 110
Acknowledgements 112
References 112
Chapter 4. Semi-inverse method for establishment of variational theory for incremental thermoelasticity with voids 114
1. A brief introduction to inverse method of calculus of variations 114
2. The semi-inverse method 116
3. Mathematical formulation of thermoelasticity with voids 123
4. Variational formulas for incremental thermoelasticity with voids 126
5. Conclusion 133
References 133
Chapter 5. Variational formulations of relativistic elasticity and thermoelasticity 136
1. Introduction 136
2. Unconstrained variational formulation of relativistic elasticity 137
3. Constrained variational principles 144
4. Thermodynamics of isentropic flows 148
5. Hamiltonian version of the unconstrained formulation 150
6. Concluding remarks 151
References 152
Chapter 6. The geometric variational framework for entropy in General Relativity 154
1. Introduction 154
2. The geometry of entropy in General Relativity 156
3. A variational setting for conserved currents and entropy in relativistic field theories 158
4. Boundary conditions and reference backgrounds in General Relativity 163
5. Conclusion and perspectives 166
References 168
Chapter 7. Translational and rotational motion of a uniaxial liquid crystal as derived using Hamilton’s principle of least action 170
1. Introduction 170
2. Lagrangian and Hamiltonian dynamics of discrete rigid particles 173
3. Rotational motion in terms of true vectors 181
4. The material description of continuous fluids of rod-like particles 187
5. The spatial description of the Leslie–Ericksen fluid 190
6. Conclusion 194
References 195
Chapter 8. An introduction to variational derivation of the pseudomomentum conservation in thermohydrodynamics 196
1. Introduction 196
2. Notion of pseudomomentum vector and pseudomomentum flux tensor 198
3. The Thomson–Tait variational principle 199
4. Gibbs’ principle extended formulation (1877) 200
5. Thermokinetic variational principle of Natanson (1899) 204
6. The Eckart variational principle 208
7. Eulerian representation of pseudomomentum 213
8. Capillarity extended Gibbs principle 216
9. The acoustic pseudomomentum 217
10. Generalized Lagrangian Mean approach 219
11. Conclusions 221
References 221
Chapter 9. Towards a variational mechanics of dissipative continua? 226
1. Introduction: the drive to a variational formulation 226
2. Special attention paid to canonical balance laws: purely elastic case 227
3. The case of thermoelastic conductors 232
4. The notion of thermal material force 233
5. The case of dissipative continua with nondissipative heat conduction 235
6. Canonical four-dimensional space–time formulation 239
7. Comparison with some nonlinear wave processes 240
8. Conclusion 242
Acknowledgement 242
References 242
Chapter 10. On the principle of least action and its role in the alternative theory of nonequilibrium processes 246
1. Space, time, and generic physical quantities 246
2. Alternative theory of nonequilibrium processes 248
3. Callen’s principle 249
4. Matter and forces 250
5. Hamiltonian theory and the principle of least action 252
6. The least, the best, and other variations 254
7. The principle of least action and the natural trend 260
8. Summary 263
Acknowledgement 264
References 264
Chapter 11. Variational principles for the linearly damped flow of barotropic and Madelung-type fluids 266
1. Introduction 266
2. Dissipative quantum theory 267
3. Linear damping in classical hydrodynamics 275
4. Summary 282
References 282
Chapter 12. Least action principle for dissipative processes 284
1. Introduction 284
2. Basic concepts 285
3. Hamilton–Lagrange formalism for linear parabolic dissipative processes 287
4. Invariance properties, symmetries 291
5. Stochastic properties of parabolic processes 296
6. Hyperbolic dissipative processes 301
Acknowledgement 304
References 304
Chapter 13. Hamiltonian formulation as a basis of quantized thermal processes 306
1. Introduction 306
2. Hamiltonian of heat conduction 307
3. Energy and number operator of heat conduction 309
4. Description of energy fluctuation 311
5. A Bose system 314
6. Infinitesimally deformed Bose distribution 316
7. q-Boson approximation 319
8. Bohmian quantum dynamics of particles 320
9. Hamilton–Jacobi equation, the action, the kernel, and a wave function 321
10. On the thermodynamical potentials: classical and quantum-thermodynamical 324
11. Fisher, bound, and extreme physical information 326
12. Another choice of probability 327
References 329
Chapter 14. Conservation laws and variational conditions for wave propagation in planarly stratified media 332
1. Introduction 332
2. Wave propagation in planarly stratified media 334
3. Rays and turning points 337
4. First integrals 339
5. Wave propagation in the presence of turning points 341
6. Variational conditions and jump relations 344
7. Application to reflection–transmission processes 347
8. Remarks on the first integral and the WKB solution 349
9. Conclusions 351
Acknowledgements 351
References 352
Chapter 15. Master equations and path-integral formulation of variational principles for reactions 354
1. Reaction–diffusion systems 354
2. Master equation and Fokker–Planck equation 355
3. Hamilton–Jacobi theories 357
4. Construction, uniqueness and critical points of ø 361
5. Path integrals for the master equation and Fokker–Planck equation 364
6. A single chemical species: exact and approximate solutions 366
7. Exit times and rate constants 368
8. Progress variables 369
9. Dissipation 374
Acknowledgement 375
References 375
Chapter 16. Variational principles for the speed of traveling fronts of reaction–diffusion equations 378
1. Variational principles for the speed of propagation of traveling fronts 380
2. Variational principle for the asymptotic speed of fronts of the density-dependent reaction–diffusion equation 384
3. Variational calculations for thermal combustion waves 388
4. Minimal speed of fronts for reaction–convection–diffusion equations 390
Acknowledgements 391
References 391
Chapter 17. The Fermat principle and chemical waves 394
1. Introduction 395
2. The Fermat principle of least time 398
3. Aplanatic surfaces 403
Acknowledgements 412
References 412
Part II: APPLICATIONS 414
STATISTICAL PHYSICS AND THERMODYNAMICS 416
Chapter 1. Fisher variational principle and thermodynamics 418
1. Introduction 418
2. Fisher’s measure and translation families 419
3. Variational techniques and Fisher’s measure 420
4. The excited solutions’ role 423
5. Application: viscosity 425
6. Application: electrical conductivity 427
7. Conclusions 430
Appendix A. BTE and the method of moments 430
Appendix B. The BTE-Grad treatment of viscosity 431
Appendix C. The Grad treatment of electrical conductivity 432
References 433
Chapter 2. Generalized entropy and the Hamiltonian structure of statistical mechanics 434
1. Introduction 434
2. Microcanonical and canonical ensembles 435
3. Temperature 439
4. Thermodynamic entropy 439
5. Boltzmann–Gibbs canonical ensemble 443
6. Tsallis ensemble and the additivity of E 443
7. Additivity of k ln p0 444
8. Numerical example 446
9. Conclusions 448
References 449
HYDRODYNAMICS AND CONTINUUM MECHANICS 450
Chapter 3. Some observations of entropy extrema in physical processes 452
1. Introduction 452
2. Kinetic systems—incompressible flow 452
3. Kinetic systems—compressible flow 460
4. Equilibrium 466
5. Combined—Chapman–Jouget detonation 470
6. Entropy extremization and the available energy of Gibbs 470
7. Conclusion 472
References 476
Chapter 4. On a variational principle for the drag in linear hydrodynamics 478
1. Introduction 478
2. Review of the variational principle 479
3. The Oseen drag on a disk broadside to the stream 482
4. A Hellmann–Feynman theorem for the drag 489
5. Conclusion and discussion 491
References 492
Chapter 5. A variational principle for the impinging-streams problem 494
1. Introduction 494
2. The fundamental problem of the calculus of variations and its extension to consider variable endpoints and coupled domains 497
3. The further extension to higher dimensions 501
4. Boundary-value problem for impinging streams 505
5. Variational principle 506
6. Potential resolution of indeterminacy 509
7. Conclusions 509
References 510
Chapter 6. Variational principles in stability analysis of composite structures 512
1. Introduction 512
2. Literature review 513
3. Variational principles: theoretical background 516
4. Application I: local buckling of FRP composite structures 517
5. Application II: global buckling of FRP composite structures 524
6. Conclusions 532
Acknowledgements 532
References 532
TRANSPORT PHENOMENA AND ENERGY CONVERSION 534
Chapter 7. Field variational principles for irreversible energy and mass transfer 536
1. Introduction 536
2. Lagrange multipliers as adjoints and potentials of variational formulation 538
3. Basic equations for damped-wave heat transfer 539
4. Action and extremum conditions in entropy representation (variables q and pe) 541
5. Source terms in internal energy equation 543
6. Inhomogeneous waves for variational adjoints 544
7. Telegraphers equations 545
8. Special case of a reversible process 546
9. Action and extremum conditions in energy representation (variables js and ps) 547
10. Waves for potentials and physical variables in the energy representation 549
11. Energy-momentum tensor and conservation laws in the heat theory 550
12. Entropy production and Second Law of Thermodynamics in the heat theory 552
13. Matter tensor and balance laws in the energy representation (variables js and rs) 553
14. Energy representation with no entropy generation 555
15. Potential representations of vector equations of change 557
16. Conclusions 559
Acknowledgement 560
References 560
Chapter 8. Variational principles for irreversible hyperbolic transport 562
1. Introduction 562
2. Restricted variational principles and EIT 564
3. Hyperbolic transport within the variational potential approach 567
4. Path-integral formulation of hyperbolic transport 574
5. Final comments and remarks 577
References 578
Chapter 9. A variational principle for transport processes in continuous systems: derivation and application 582
1. Introduction 582
2. Statement of the problem 583
3. Particular case 1: fluxes determined by the space derivatives of intensive variables 585
4. Case II: fluxes determined by the time derivatives of extensive variables 592
5. Thermodynamic analysis of eutectic solidification 592
6. Conclusion 596
Acknowledgements 597
References 597
Chapter 10. Do the Navier–Stokes equations admit of a variational formulation? 600
1. The problem of the variational formulation for the equations of fluid motion 600
2. The exergy content of a fluid in motion 608
3. Variational derivation of the flow field 609
4. Conclusions 610
Appendix A. Explicit derivation of the final equations from the exergy functional 611
References 614
Chapter 11. Entropy-generation minimization in steady-state heat conduction 616
1. Heat conduction in isotropic solids 616
2. Heat conduction in anisotropic solids 629
3. Conclusion 640
Acknowledgements 640
References 641
Chapter 12. The nonequilibrium thermodynamics of radiation interaction 642
1. Introduction 642
2. Entropy with and without equilibrium 646
3. Entropy generation and variational principles 653
4. Free radiation and equilibrium matter 656
5. Radiation and matter in equilibrium 660
6. Summary and conclusion 663
Acknowledgements 664
References 665
Chapter 13. Optimal finite-time endoreversible processes— general theory and applications 666
1. Introduction 666
2. General problem 668
3. Constant entropy-production rate 670
4. The boundary conditions 672
5. Solved examples 672
6. Conclusions 677
References 677
Chapter 14. Evolutionary Energy Method (EEM): an aerothermoservoelectroelastic application 680
1. Introduction 680
2. Evolutionary energy method 682
3. Algebraic evolutionary energy equations of motion for dynamic systems 688
4. Initial-value problems in the evolutionary energy method 690
5. AEM for direct optimal control of a thermal-structural dynamic system 691
6. The LEE (ÐE equations) for the thermoelectric and elastoelectric system 694
7. Structural skin-temperature control at Mach 10 hypersonic flight 699
8. Concluding remarks 704
Appendix A. A constructive and demonstrative proof of the law of evolutionary energy for Newtonian dynamics 705
Acknowledgements 709
References 709
ECOLOGY 712
Chapter 15. Maximization of ecoexergy in ecosystems 714
1. Introduction 714
2. What is ecoexergy? 715
3. Ecoexergy and information 718
4. How to calculate ecoexergy of organic matter and organisms? 718
5. Why have living systems such a high level of ecoexergy? 722
6. Formulation of a thermodynamic hypothesis (maximization of ecoexergy) for ecosystems 723
7. Support to the hypothesis 724
8. Growth and development of ecosystems 727
References 730
SELFORGANIZATION AND ECONOPHYSICS 732
Chapter 16. Self-organized criticality within the framework of the variational principle 734
1. Introduction 734
2. Field approach and optimal trajectories 736
3. Dynamics of the system exhibiting self-organized criticality 739
4. Conclusions 752
References 753
Chapter 17. Extremum criteria for nonequilibrium states of dissipative macroeconomic systems 756
1. Introduction 756
2. The maximum principle of entropy and the distribution of merchandizing prices 758
3. The distributions of incomes and wealth 760
4. A kinetic insight. The Boltzmann kinetic equation 763
5. Uniting the Boltzmann and Onsager pictures and the minimum production of entropy 764
6. Economic systems in far-from-equilibrium steady states. Economic cycles 767
7. Noise-induced transitions between nonequilibrium steady states 769
8. Some final remarks 771
References 773
Chapter 18. Extremal principles and limiting possibilities of open thermodynamic and economic systems 774
1. Introduction 774
2. Thermodynamic system including an active subsystem 775
3. Open microeconomic system 785
4. Conclusion 790
Acknowledgements 790
References 791
Glossary of principal symbols 792
Subject Index 796