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    Thermodynamics: A Dynamical Systems Approach
    Thermodynamics: A Dynamical Systems Approach
    Thermodynamics: A Dynamical Systems Approach
    Ebook204 pages2 hours

    Thermodynamics: A Dynamical Systems Approach

    By Wassim M. Haddad, VijaySekhar Chellaboina and Sergey G. Nersesov

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    This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics.


    The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth.


    This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.

    LanguageEnglish
    PublisherPrinceton University Press
    Release dateJan 10, 2009
    ISBN9781400826971
    Thermodynamics: A Dynamical Systems Approach

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    Book preview

    Thermodynamics - Wassim M. Haddad

    Thermodynamics

    PRINCETON SERIES IN APPLIED MATHEMATICS

    Edited by

    Ingrid Daubechies, Princeton University

    Weinan E, Princeton University

    Jan Karel Lenstra, Eindhoven University

    Endre Süli, University of Oxford

    The Princeton Series in Applied Mathematics publishes high quality advanced texts and monographs in all areas of applied mathematics. Books include those of a theoretical and general nature as well as those dealing with the mathematics of specific applications areas and real-world situations.

    Thermodynamics

    A Dynamical Systems Approach

    Wassim M. Haddad

    VijaySekhar Chellaboina

    Sergey G. Nersesov

    PRINCETON UNIVERSITY PRESS

    PRINCETON AND OXFORD

    Copyright © 2005 by Princeton University Press

    Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

    In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY

    All Rights Reserved

    Library of Congress Cataloging-in-Publication Data

    Haddad, Wassim M., 1961–

    Thermodynamics : a dynamical systems approach / Wassim M. Haddad, VijaySekhar Chellaboina, and Sergey G. Nersesov.

    p.cm. — (Princeton series in applied mathematics)

    Includes bibliographical references and index.

    eISBN: 978-1-40082-697-1

    1. Thermodynamics—Mathematics. 2. Differentiable dynamical systems. I.

    >Chellaboina, VijaySekhar, 1970– II. Nersesov, Sergey G., 1976– III. Title. IV. Series.

    QC311.2.H33 2005

    536´.7—dc222004066029 536

    British Library Cataloging-in-Publication Data is available

    The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed.

    Printed on acid-free paper. ∞

    pup.princeton.edu

    Printed in the United States of America

    10987654321

    To my mother Sofia who made it possible for me to pursue my passion for science, and my wife Lydia who provides theequipoise between this passion and the other joys of life.

    W. M. H.

    To my children SriHarsha and Saankhya, the entropy agents of my life.

    V. C.

    To my parents Garry and Ekatherina and my brother Artyom.

    S. G. N.

    —Herakleitos

    [Thermodynamics] is the only physical theory of a universal nature of which I am convinced that it will never be overthrown.

    —Albert Einstein

    The law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of Nature.

    —Sir Arthur Eddington

    The future belongs to those who can manipulate entropy; those who understand but energy will be only accountants.

    —Frederic Keffer

    The energy of the Universe is constant. The entropy of the Universe tends to a maximum. The total state of the Universe will inevitably approach a limiting state.

    —Rudolf Clausius

    Time flows on, never comes back. When the physicist is confronted with this fact he is greatly disturbed.

    —Leon Brillouin

    The world has signed a pact with the devil; it had to. It is a covenant to which everything, even every hydrogen atom, is bound. The terms are clear: if you want to live, you have to die. The world came into being with the signing of this contract. A scientist calls it the Second Law of Thermodynamics.

    —Annie Dillard

    Contents

    Preface

    Chapter 1. Introduction

    1.1 An Overview of Thermodynamics

    1.2 System Thermodynamics

    1.3 A Brief Outline of the Monograph

    Chapter 2. Dynamical System Theory

    2.1 Notation, Definitions, and Mathematical Preliminaries

    2.2 Stability Theory for Nonnegative Dynamical Systems

    2.3 Reversibility, Irreversibility, Recoverability, and Irrecoverability

    2.4 Reversible Dynamical Systems, Volume-Preserving Flows, and Poincaré Recurrence

    Chapter 3. A Systems Foundation for Thermodynamics

    3.1 Introduction

    3.2 Conservation of Energy and the First Law of Thermodynamics

    3.3 Entropy and the Second Law of Thermodynamics

    3.4 Ectropy

    3.5 Semistability, Energy Equipartition, Irreversibility, and the Arrow of Time

    3.6 Entropy Increase and the Second Law of Thermodynamics

    3.7 Interconnections of Thermodynamic Systems

    3.8 Monotonicity of System Energies in Thermodynamic Processes

    Chapter 4. Temperature Equipartition and the Kinetic Theory of Gases

    4.1 Semistability and Temperature Equipartition

    4.2 Boltzmann Thermodynamics

    Chapter 5. Work, Heat, and the Carnot Cycle

    5.1 On the Equivalence of Work and Heat: The First Law Revisited

    5.2 The Carnot Cycle and the Second Law of Thermodynamics

    Chapter 6. Thermodynamic Systems with Linear Energy Exchange

    6.1 Linear Thermodynamic System Models

    6.2 Semistability and Energy Equipartition in Linear Thermodynamic Models

    Chapter 7. Continuum Thermodynamics

    7.1 Conservation Laws in Continuum Thermodynamics

    7.2 Entropy and Ectropy for Continuum Thermodynamics

    7.3 Semistability and Energy Equipartition in Continuum Thermodynamics

    Chapter 8. Conclusion

    Bibliography

    Preface

    Thermodynamics is a physical branch of science that governs the thermal behavior of dynamical systems from those as simple as refrigerators to those as complex as our expanding universe. The laws of thermodynamics involving conservation of energy and nonconservation of entropy are, without a doubt, two of the most useful and general laws in all sciences. The first law of thermodynamics, according to which energy cannot be created or destroyed, merely transformed from one form to another, and the second law of thermodynamics, according to which the usable energy in an adiabatically isolated dynamical system is always diminishing in spite of the fact that energy is conserved, have had an impact far beyond science and engineering. The second law of thermodynamics is intimately connected to the irreversibility of dynamical processes. In particular, the second law asserts that a dynamical system undergoing a transformation from one state to another cannot be restored to its original state and at the same time restore its environment to its original condition. That is, the status quo cannot be restored everywhere. This gives rise to an increasing quantity known as entropy.

    Entropy permeates the whole of nature, and unlike energy, which describes the state of a dynamical system, entropy is a measure of change in the status quo of a dynamical system. Hence, the law that entropy always increases, the second law of thermodynamics, defines the direction of time flow and shows that a dynamical system state will continually change in that direction and thus inevitably approach a limiting state corresponding to a state of maximum entropy. It is precisely this irreversibility of all dynamical processes connoting the running down and eventual demise of the universe that has led writers, historians, philosophers, and theologians to ask profound questions such as: How is it possible for life to come into being in a universe governed by a supreme law that impedes the very existence of life?

    Even though thermodynamics has provided the foundation for speculation about some of science’s most puzzling questions concerning the beginning and the end of the universe, the development of thermodynamics grew out of steam tables and the desire to design and build efficient heat engines, with many scientists and mathematicians expressing concerns about the completeness and clarity of its mathematical foundation over its long and tortuous history. Indeed, many formulations of classical thermodynamics, especially most textbook presentations, poorly amalgamate physics with rigorous mathematics and have had a hard time in finding a balance between nineteenth century steam and heat engine engineering, and twenty first century science and mathematics. In fact, no other discipline in mathematical science is riddled with so many logical and mathematical inconsistencies, differences in definitions, and ill-defined notation as classical thermodynamics. With a notable few exceptions, more than a century of mathematicians have turned away in disquietude from classical thermodynamics, often overlooking its grandiose unsubstantiated claims and allowing it to slip into an abyss of ambiguity.

    The development of the theory of thermodynamics followed two conceptually rather different lines of thought. The first (historically), known as classical thermodynamics, is based on fundamental laws that are assumed as axioms, which in turn are based on experimental evidence. Conclusions are subsequently drawn from them using the notion of a thermodynamic state of a system, which includes temperature, volume, and pressure, among others. The second, known as statistical thermodynamics, has its foundation in classical mechanics. However, since the state of a dynamical system in mechanics is completely specified point-wise in time by each point-mass position and velocity and since thermodynamic systems contain large numbers of particles (atoms or molecules, typically on the order of 10²³), an ensemble average of different configurations of molecular motion is considered as the state of the system. In this case, the equivalence between heat and dynamical energy is based on a kinetic theory interpretation reducing all thermal behavior to the statistical motions of atoms and molecules. In addition, the second law of thermodynamics has only statistical certainty wherein entropy is directly related to the relative probability of various states of a collection of molecules.

    In this monograph, we utilize the language of modern mathematics within a theorem-proof format to develop a general dynamical systems theory for reversible and irreversible (non)equilibrium thermodynamics. The monograph is written from a system-theoretic point of view and can be viewed as a contribution to the fields of thermodynamics and mathematical system theory. In particular, we develop a novel formulation of thermodynamics using a middle-ground theory involving deterministic large-scale dynamical system models that bridges the gap between classical and statistical thermodynamics. The benefits of such a theory include the advantage of being independent of the simplifying assumptions that are often made in statistical mechanics and at the same time providing a thermodynamic framework with enough detail of how the system really evolves without ever needing to resort to statistical (subjective or informational) probabilities. In particular, we develop a system-theoretic foundation for thermodynamics using a large-scale dynamical systems perspective. Specifically, using compartmental dynamical system energy flow models, we place the universal energy conservation, energy equipartition, temperature equipartition, and entropy nonconservation laws of thermodynamics on a system-theoretic foundation.

    Next, we establish the existence of a new and dual notion to entropy, namely, ectropy, as a measure of the tendency of a dynamical system to do useful work and grow more organized, and we show that conservation of energy in an adiabatically isolated thermodynamic system necessarily leads to nonconservation of ectropy and entropy. In addition, using the system ectropy as a Lyapunov function candidate, we show that our large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the large-scale system initial subsystem energies. Furthermore, using the system entropy and ectropy functions, we establish a clear connection between irreversibility, the second law of thermodynamics, and the arrow of time. Finally, these results are generalized to continuum thermodynamics involving infinite-dimensional energy flow conservation models. Since in this case the resulting dynamical system is defined on an infinite-dimensional Banach space that is not locally compact, stability, convergence, and energy equipartition are shown using Sobolev

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