Variational and Extremum Principles in Macroscopic Systems

Recent years have seen a growing trend to derive models of macroscopic phenomena encountered in the fields of engineering, physics, chemistry, ecology, self-organisation theory and econophysics from various variational or extremum principles. Through the link between the integral extremum of a functional and the local extremum of a function (explicit, for example, in the Pontryagin’s maximum principle variational and extremum principles are mutually related. Thus it makes sense to consider them within a common context.

The main goal of Variational and Extremum Principles in Macroscopic Systems is to collect various mathematical formulations and examples of physical reasoning that involve both basic theoretical aspects and applications of variational and extremum approaches to systems of the macroscopic world.

The first part of the book is focused on the theory, whereas the second focuses on applications. The unifying variational approach is used to derive the balance or conservation equations, phenomenological equations linking fluxes and forces, equations of change for processes with coupled transfer of energy and substance, and optimal conditions for energy management.

• A unique multidisciplinary synthesis of variational and extremum principles in theory and application
• A comprehensive review of current and past achievements in variational formulations for macroscopic processes
• Uses Lagrangian and Hamiltonian formalisms as a basis for the exposition of novel approaches to transfer and conversion of thermal, solar and chemical energy

Readership is extremely broad and includes applied mathematicians, mathematical physicists, applied physicists, chemists, geologists, ecologists, mechanical engineers, chemical engineers, economists and system theorists, undergraduates, graduates and instructors, both from academia and industry.

List of contributors

Preface

Part I: Theory

I.1. Progress in Variational Formulations for Macroscopic Processes I.2. Lagrange-Formalism and Thermodynamics of Irreversible Processes: The 2nd Law of Thermodynamics and the Principle of Least Entropy Production as Straightforward Structures in Lagrange-Formalism I.3. Fundamental Problems of Variational Principles: Objectivity, Symmetries and Construction I.4. Semi-Inverse Method for Establishment of Variational Principles for Incremental Thermoelasticity with Voids I.5. Variational Formulations of Relativistic Elasticity and Thermo-Elasticity I.6. The Geometric Variational Framework for Entropy in General Relativity I.7. Translational and Rotational Motion of a Unaxial Liquid Crystal as Derived Using Hamilton’s Principle of Least Action I.8. An Introduction to Variational Derivation of the Pseudo-Momentum Conservation in Thermo-Hydrodynamics I.9. Towards a Variational Mechanics of Dissipative Continua? I.10. On the Principle of Least Action and its Role in the Alternative Theory of Non-Equilibrium Processes I.11. Variational Principles for the Linearly Damped Flow of Barotropic and Madelung-Type Fluids I.12. Least Action Principle for Dissipative Processes I.13. Hamiltonian Formulation as a Basis of Quantized Thermal Processes I.14. Conservation Laws and Variational Conditions for Wave Propagation in Planarly-Stratified Media I.15. Master Equations and Path-Integral Formulation of Variational Principles for Reactions I.16. Variational Principles for the Speed of Traveling Fronts of Reaction-Diffusion Equations I.17. The Fermat Principle and Chemical Waves

Part II: Applications

Statistical Physics and Thermodynamics

II.1. Fisher Variational Principle and Thermodynamics II.2. Generalized Entropy and the Hamiltonian Structure of Statistical Mechanics

Hydrodynamics and Continuum Mechanics

II.3. Some Observations of Entropy Extrema in Physical Processes II.4. A Variational Principle for the Drag in Linear Hydrodynamics II.5. A Variational Principle for the Impinging Streams Problem II.6. Variational Principles in Stability Analysis of Composite Structures

Transport Phenomena and Energy Conversion

II.7. Field Variational Principles for Irreversible Energy and Mass Transfer II.8. Variational Principles for Irreversible Hyperbolic Transport II.9. A Variational Principle for Transport Processes in Continuous Systems: Derivation and Application II.10. Do the Navier-Stokes Equations Admit a Variational Formulation? II.11. Entropy Generation Minimization in Steady State Heat Conduction II.12. The Nonequilibrium Thermodynamics of Radiation Interaction II.13. Optimal Finite-Time Endoreversible Processes- General Theory and Applications II.14. Evolutionary Energy Method (EEM) – An Aerothermoservoelectrostatic Application

Ecology

II.15. Maximization of Eco-Exergy in Ecosystems

Selforganization and Econophysics

II.16. Self-Organized Criticality within the Framework of Variational Principle II.17. Extremum Criteria for Nonequilibrium States of Dissipative Macroeconomic systems II.18. Extremal Principles and Limiting Possibilities of Open Thermodynamic and Economic Systems

Glossary of principal symbols

Index