Alternative Mathematical Theory of Non-equilibrium Phenomena (eBook)

Dieter Straub (Autor)

eBook Download: PDF

1996 | 1. Auflage
377 Seiten
Elsevier Science (Verlag)
978-0-08-052707-9 (ISBN)

Infos

Alternative Mathematical Theory of Non-equilibrium Phenomena presents an entirely new theoretical approach to complex non-equilibrium phenomena, especially Gibbs/Falk thermodynamics and fluid mechanics. This innovative new theory allows for inclusion of all state variables and introduces a new vector-dissipation velocity-which leads to useful restatements of momentum, the Second Law, and tensors for the laws of motion, friction, and heat conduction. This application-oriented text is relatively self-contained and is an excellent guide-book for engineers with a strong interest in fundamentals, or for professionals using applied mathematics and physics in engineering applications.
This book emphasizes macroscopic phenomena, focusing specifically on gaseous states, though relations to liquid and crystalline states are also considered. The author presents a new Alternative Continuum Theory of Compressible Fluids (AT) which providesa qualitative description of the subject in predominantly physical terms, minimizing the mathematical premises. The methodology discussed has applications in a wide range of fields outside of physics in areas including General System Theory, TheoreticalEconomics, and Biophysics and Medicine.

Key Features
* Presents the first theory capable of handling non-equilibria phenomena
* Offers a unified theory of all branches of macroscopic physics
* Considers a consistent and uniform view of reality, supported by modern mathematics, leading to results different than those produced by classical theories
* Results in a change of paradigms in physics, engineering, and natural philosophy

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Alternative Mathematical Theory of Non-equilibrium Phenomena presents an entirely new theoretical approach to complex non-equilibrium phenomena, especially Gibbs/Falk thermodynamics and fluid mechanics. This innovative new theory allows for inclusion of all state variables and introduces a new vector-dissipation velocity-which leads to useful restatements of momentum, the Second Law, and tensors for the laws of motion, friction, and heat conduction. This application-oriented text is relatively self-contained and is an excellent guide-book for engineers with a strong interest in fundamentals, or for professionals using applied mathematics and physics in engineering applications. This book emphasizes macroscopic phenomena, focusing specifically on gaseous states, though relations to liquid and crystalline states are also considered. The author presents a new Alternative Continuum Theory of Compressible Fluids (AT) which providesa qualitative description of the subject in predominantly physical terms, minimizing the mathematical premises. The methodology discussed has applications in a wide range of fields outside of physics in areas including General System Theory, TheoreticalEconomics, and Biophysics and Medicine. - Presents the first theory capable of handling non-equilibria phenomena- Offers a unified theory of all branches of macroscopic physics- Considers a consistent and uniform view of reality, supported by modern mathematics, leading to results different than those produced by classical theories- Results in a change of paradigms in physics, engineering, and natural philosophy

Inhalt

Cover 1
Contents 8
Preface 12
Acknowledgments 15
Chapter 1. Physics Today: Perspectives 16
1.1 Motivation 16
1.2 Origin and Importance of Non-equilibrium Phenomena 17
1.3 Today's Mechanical Worldview: A Short Historical Outline 20
1.4 Continuum Theories of Mass-Point Fluids 27
1.5 Gibbsian Thermostatics? 34
Chapter 2. Falkian Dynamics: An Introduction 40
2.1 Falk's Principle 40
2.2 Gibbs' Fundamental Equation and System Modeling 46
2.3 Equilibria and Criteria of Stability 54
2.4 Mathematical Foundation of Falk's Dynamics I: Mappings 63
2.5 Mathematical Foundation of Falk's Dynamics II: Systems 72
Chapter 3. Motion and Matter 84
3.1 Basic Questions 84
3.2 Callen's Principle 85
3.3 Energy–Momentum Transport and Matter Model 93
3.4 Realistic Concept of Real Matter 102
Chapter 4. Systems and Symmetries 111
4.1 An Approach to Implant Space and Time in Physics 111
4.2 Falk's Dynamics of Hamiltonian Systems 112
4.3 Review of the Noether Theorem 118
4.4 Phases, Heating, and Power as Interacting Phenomena 129
Chapters 5. Barriers and Balances 142
5.1 Body-Field Systems 142
5.2 Multicomponent Single-Phase and Multiphase Properties 146
5.3 Time Parameters in Thermodynamics of Fluid Systems 153
5.4 Balance Equations 165
Chapter 6. Non-equilibrium Processes 176
6.1 Dissipation Velocity 176
6.2 Kinetic Equilibrium in Fields 181
6.3 Three Additional Theorems Concerning Non-equilibrium 183
6.4 Hypothetical State at Rest 198
6.5 Constitutive Properties of Matter 203
Chapter 7. General Equation of Motion and Its Approximations 210
7.1 Elementary Picture of Dissipation 210
7.2 General Equation of Motion 212
7.3 Some Remarks on Turbulent Flows 220
7.4 Conservation of Angular Momentum 225
7.5 Navier–Stokes–Fourier Fluids 229
7.6 Simplified Models of Dissipative Flows 235
Chapter 8. Paradigmata Are the Winners' Dogmata 241
8.1 Selection of Paragons 241
8.2 Vorticity-influenced Flows 244
8.3 Basic Applications of Gasdynamics 261
8.4 Complex Flow Phenomena 281
Chapter 9. Gibbs–Falkian Electromagnetism 295
9.1 A Quandary Concerning Electromagnetic Field Variables 295
9.2 Perspectives and Electromagnetic Units 298
9.3 Falk's Dynamics of Electromagnetic Phenomena 301
9.4 Maxwell's Equations 308
9.5 Non-equilibrium Flows in Polarized Fluids 312
9.6 Sundry Remarks on an Electromagnetic Dilemma 320
Appendix 1. Atomism or The Art of Honest Dissimulation 329
A1.1 Motivation 329
A1.2 Pre-Socratic Atomism and its Tradition in the Ancient World 330
A1.3 Atomism in the Dark Middle Ages 334
A1.4 The Galilei Affair 340
A1.5 On the Early History of the Eulerian Mass-Point 348
Appendix 2. Mathematical Supplements 354
A2.1 Supplements to the Noether Theorem 354
A2.2 Some Useful Tensor Rules 358
A2.3 On Vorticity in Navier–Stokes Flows 359
Appendix 3. Computation Scheme for a One-Component Single-Phase Body-Field System 362
List of Relevant Symbols 365
References 368
Name Index 378
Subject Index 382

Textauszug

Chapter 1

Physics Today: Perspectives

Dieter Straub

“If the Universe is the Answer what is the Question?”—Leon Lederman

1.1 Motivation

That part of classical flow mechanics belonging to the basics of engineering science and education is in competition today with theories relating to some branches of thermodynamics. These theories, specified as linear or extended irreversible thermodynamics or as rational thermodynamics, are ordinarily placed on the periphery of modern physics. Aside from this constellation, the range of solvable problems remains predominantly confined to the conventional Navier–Stokes theory, particularly over the last decade and especially in worldwide industrial practice.

Due to the rapid progress in computer design and efficiency, combined with sophisticated numerical methods, the solving of actual flow problems has led to a modern discipline, well-known as Computational Fluid Dynamics. Therefore an increasing number of young students, scientists, and engineers are interested in computer applications. When using computers for increasingly complicated CFD-problems, they must frequently concentrate their work on programming languages or software packages to cultivate their talents.

At present, the scientific literature substantiates the obvious tendency that the standards of computing techniques and applied mathematics are steadily growing. Unfortunately, this trend may mean that professional competence in the fundamentals of physics, combined with thorough knowledge of the empirical background, is increasingly diminishing. This would be a hazardous development, delicate and expensive for the prospective technologies.

Therefore, this book deals with those fundamentals in an essentially new way to provide genuine qualification for any professional work in science and engineering. One who is familiar with standard mathematics will have no difficulty in following the discussion. Nevertheless, some knowledge of recent progress in applied mathematics is required. Thus, one would benefit greatly from such complementary work as the mathematical construction and analysis of the full invariance group related to a given set of differential equations and appropriate boundary conditions. Once the group is known, one can study the action of the group on the complete set.

The invariance requirement, related to the respective symmetry group with characteristic properties and consisting of a set of specific transformation rules, may be utilized to generate new solutions from known ones (Rogers and Ames, 1989). Furthermore, the admissible group offers algebraic structures that often allow relevant classifications of solutions. Such information is always helpful for any computer-aided integration procedure needed, for example for numerical solutions of partial differential equations (Ames, 1992).

The main subjects of this book are briefly stated in the preface where the essential ideas are related to key words such as instability, non-equilibrium, motion, and irreversibility. Only recently, scientists have begun to deal more comprehensively with these phenomena, which act as a basic part of the atomistic world.

In this context it is remarkable that in theoretical physics, nonlinear non-equilibrium thermodynamics has only taken shape in the last 50 years or so. But this refers mainly to quite different extensions of statistical physics in connection with fluctuation-dissipation theorems and quantum generalizations of Markov processes. Model systems are preferred to extract exact results based on the premise that random thermal fluctuations play a decisive role in governing the evolution of non-equilibrium thermodynamic processes in space-time (see, e.g., Kreuzer, 1883, Lavenda, 1985, and Stratonovich, 1994). Applied sciences, such as chemistry or process and aerospace engineering, unfortunately have little interest in these topics.

In the following sections of this chapter, the key words mentioned above will be treated in depth since they are essential to a serious understanding of the physical essence and evolution in both the micro- and the macrolevels of the universe. Furthermore, in a cursory historical comment, we will see why it is so hard for the scientific community to adequately consider and accept these basic notions that are encountered in daily life.

1.2 Origin and Importance of Non-equilibrium Phenomena

In recent years, some radical changes have occurred in the natural sciences that are evidently inconsistent with the prevalent paradigms. For the first time, it appears possible to overcome the gap between the basic micro- and macrophysical ideas of matter and motion.

The extensive research of Ilya Prigogine and his co-workers showed in a unique way how the contemporary dualistic theory of the physical world could be superseded by a uniform description (Petrosky and Prigogine, 1988). This has, indeed, far-reaching consequences: The dogma universally suggested by physics textbooks can no longer be justified by an irreversible phenomenological macroworld on one hand and, separately, by an atomistic microworld determined by linear and reversible quantum laws on the other. The former is quite obviously controlled by phenomenological laws with a broken symmetry in time, whereas the latter can be represented by particle trajectories and wave functions, assumed to be independent of reversals in time and velocity.

Physical systems, distinguished by permanently stable and reversible behavior, are rare. They manifest special cases or refer to certain borderlines in nature that can be elegantly formulated with the help of flexible mathematics, but do not provide reliable information about intricate scenarios in space and time. Therefore, when treating real problems with quantum theory, the so-called measuring problem arises, because the broken symmetry in time is needed paradoxically as a precondition of experimentally detecting the quantum world, for example by means of irreversible photon scattering (Petrosky and Prigogine, 1993).

In the last few years some ideas of unstable microsystems have been proven compatible with that of atomistic interaction and correlation events, provided that a break of symmetry between past and future occurs. One conclusion follows immediately: Quantum events need irreversibility as an elementary mechanism inherent in each physical system consisting of many-particle collectives. Surely, this is a crucial point, because instability is predominant in most of the physical systems for which, in Prigogine’s view, the term dynamical chaos is an intrinsic one (Prigogine, 1993).

As a complementary notion, Prigogine introduces the tautological term dissipative chaos as an intermediate concept between any pure accident and redundant order (Prigogine and Stengers, 1993, p. 123).

Following Prigogine and Stengers (1993, p. 319), all resulting unstable phenomena occurring macroscopically in the physical system as an event or process are caused primarily by inherent fluctuations of the respective state quantities. Close to equilibrium the fluctuations are attenuated. As a consequence of their wave structure, the trend toward equilibrium is distinguished by asymptotically vanishing dissipative contributions. Given enough time, the system does in the end approach its equilibrium values. Only for such a condition can each fluid always be controlled.

In contrast, non-equilibrium states can amplify fluctuations, locally and spontaneously. For this reason, a systematic control is hardly feasible without impacting on the system in question. Due to these non-equilibrium effects, any local disturbance can move even a whole system into an unstable state. They are also substantial preconditions of phase transition. Another preeminent case refers to the turbulence phenomenon in gaseous and liquid flows.

Naturally, such macroscopic states and processes have their own atomistic equivalents at the microscopic level. Indeed, the actual behavior of every particle collective is determined by the local correlations, the action range and strength of which regulate how the principal properties of the whole system are influenced by local particle events.

Correlations play an important role in statistical mechanics. In general, they relate interactive connections measured in space and time to measurable properties of two or more particles. Expressed by the so-called product-moment correlation coefficient with values ranging from −1 and + 1, correlation functions are referred in a complicated way to the main tool of particle mechanics: The joint probability depends in principle on many random variables. By definition it can be reduced to a product of corresponding elementary probabilities, each one assigned to the respective random variable. This leads to a relatively primitive method of simplifying the model chosen for the statistics of complex physical events. However, if there are correlation effects, then the joint probability cannot be split into a simple product.

Equilibrium states, for instance, can be classified by correlations having an averaged range of action with typical values of about 10−10 m (or 1 Ångström). Correlations existing far from equilibrium can extend to macroscopic distances in an order of one centimeter or more. This immense difference is a striking indication of qualitative disparity...

Produkt-Details

• Erscheint lt. Verlag: 9.10.1996
• Mitarbeit: Herausgeber (Serie): William F. Ames
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Thermodynamik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-052707-8 / 0080527078
• ISBN-13: 978-0-08-052707-9 / 9780080527079