Thermodynamics: A dynamical systems approach

200 Pages·2005·0.99 MB·English

by  Wassim M. Haddad

Thermodynamics
PRINCETON SERIES IN APPLIED MATHEMATICS
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Thermodynamics
A Dynamical Systems Approach
Wassim M. Haddad
VijaySekhar Chellaboina
Sergey G. Nersesov
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright (cid:1)c 2005 by Princeton University Press
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey
08540
In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock,
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All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Haddad, Wassim M., 1961–
Thermodynamics : a dynamical systems approach / Wassim M. Haddad, Vi-
jaySekhar Chellaboina, and Sergey G. Nersesov.
p. cm. — (Princeton series in applied mathematics)
Includes bibliographical references and index.
ISBN: 0-691-12327-6 (acid-free paper)
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(cid:1).7—dc22 2004066029
British Library Cataloging-in-Publication Data is available
This book has been composed in Times Roman in LATEX
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
10 9 8 7 6 5 4 3 2 1
To my mother Sofia who made it possible for me to pursue
my passion for science, and my wife Lydia who provides the
equipoise 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.
vi
Tα πα´ντα ρε´ι. Πoταµε´ις τo´ις αυτo´ις εµβα´ινoµεν τε κα´ι oυκ
εµβα´ινoµεν, ε´ιµεν τε κα´ι oυκ ε´ιµεν.
—Herakleitos
[Thermodynamics] is the only physical theory of a universal na-
ture 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 Uni-
verse 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 con-
fronted with this fact he is greatly disturbed.
—Leon Brillouin
The world has signed a pact with the devil; it had to. It is a
covenanttowhicheverything,eveneveryhydrogenatom,isbound.
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 scien-
tist calls it the Second Law of Thermodynamics.
—Annie Dillard
Contents
Preface ix
Chapter 1. Introduction 1
1.1 An Overview of Thermodynamics 1
1.2 System Thermodynamics 11
1.3 A Brief Outline of the Monograph 14
Chapter 2. Dynamical System Theory 17
2.1 Notation, Definitions, and Mathematical Preliminaries 17
2.2 Stability Theory for Nonnegative Dynamical Systems 20
2.3 Reversibility, Irreversibility, Recoverability, and Irrecoverability 27
2.4 Reversible Dynamical Systems, Volume-Preserving Flows, and
Poincar´e Recurrence 34
Chapter 3. A Systems Foundation for Thermodynamics 45
3.1 Introduction 45
3.2 Conservation of Energy and the First Law of Thermodynamics 46
3.3 Entropy and the Second Law of Thermodynamics 55
3.4 Ectropy 72
3.5 Semistability,EnergyEquipartition,Irreversibility,andtheArrow
of Time 81
3.6 Entropy Increase and the Second Law of Thermodynamics 89
3.7 Interconnections of Thermodynamic Systems 91
3.8 Monotonicity of System Energies in Thermodynamic Processes 98
Chapter 4. Temperature Equipartition and the Kinetic Theory of Gases 103
4.1 Semistability and Temperature Equipartition 103
4.2 Boltzmann Thermodynamics 110
Chapter 5. Work, Heat, and the Carnot Cycle 115
5.1 On the Equivalence of Work and Heat: The First Law Revisited 115
5.2 The Carnot Cycle and the Second Law of Thermodynamics 126
Chapter 6. Thermodynamic Systems with Linear Energy Exchange 131
6.1 Linear Thermodynamic System Models 131
6.2 SemistabilityandEnergyEquipartitioninLinearThermodynamic
Models 136
viii CONTENTS
Chapter 7. Continuum Thermodynamics 141
7.1 Conservation Laws in Continuum Thermodynamics 141
7.2 Entropy and Ectropy for Continuum Thermodynamics 148
7.3 Semistability and Energy Equipartition in Continuum Thermo-
dynamics 160
Chapter 8. Conclusion 169
Bibliography 175
Index 185
Preface
Thermodynamicsisaphysicalbranchofsciencethatgovernsthether-
mal behavior of dynamical systems from those as simple as refrigera-
tors to those as complex as our expanding universe. The laws of ther-
modynamics 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 sys-
temisalwaysdiminishinginspiteofthefactthatenergyisconserved,
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 an-
other 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
runningdownandeventualdemiseoftheuniversethathasledwriters,
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?
Eventhoughthermodynamicshasprovidedthefoundationforspec-
ulation about some of science’s most puzzling questions concerning
the beginning and the end of the universe, the development of ther-
modynamics grew out of steam tables and the desire to design and