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Michael C. Mackey Time's Arrow: The Origins of Thermodynamic Behavior With 24 Figures Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

Michael C. Mackey
Center for Nonlinear Dynamics
    in Physiology and Medicine
McGill University
Montreal PQ, Canada H3G 1Y6

Library of Congress Cataloging-in-Publication Data

Mackey, Michael C, 1942-

Time's arrow : The origins of thermodynamic behavior/ Michael C. Mackey. - 1st ed.

p. cm.

Includes bibliographical references and index.

ISBN 0-387-94093-6 (New York : alk. paper : pbk.). - ISBN 3-540-94093-6 (Berlin : alk. paper : pbk.)

I. Thermodynamnics. 2. Entropy. I. Title.

QC311.M176 1993

536'.7--dc20                    93-30004

Printed on acid-free paper.

© 1992,by Springer-Verlag New York, Inc.

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To Fraser
who loves to discuss all these questions

"It is not very difficult to show that the combination of the reversible laws of mechanics with Gibbsian statistics does not lead to irreversibility but that the notion of irreversibility must be added as a special ingredient...

...the explanation of irreversibility in nature is to my mind still open.”

Bergmann (1967)

PREFACE

In 1935, Eddington wrote “The law that entropy always increases—the second law of thermodynamics — holds, I think, the supreme position among the laws of Nature.” Much has changed in science in the intervening half century, but I believe that Eddington's pronouncement still carries a great deal of truth.

The central question this book addresses is the dynamic origin of the Second Law of thermodynamics. Specifically, the goal is to define the dynamical foundation of the evolution of entropy to maximal states. This is accomplished through an application of recent results in ergodic theory to so-called “chaotic” dynamical systems (Lasota and Mackey, 1985; M.C. Mackey, 1989).

The Second Law of thermodynamics comes in so many forms that it is often confusing to understand precisely what a given author understands by the use of this term. This is unfortunate since the first statement of the Second Law was so clear. It was first enunciated by Clausius (1879), in his remarkable work building on the considerations of Carnot, when he wrote “Die Energie der Welt ist Konstant. Die Entropie der Welt strebt einem Maximum zu” (The energy of the world is constant. The entropy of the world tends to a maximum.). Though this simple declaration has been rephrased so many times that it is often unrecognizable, I find that Clausius' formulation is the most transparent. However, I will distinguish two variations on the original theme. Let S_TD(t) denote the thermodynamic entropy at time t.

Weak Form of the Second Law.

–∞ < S_TD(t₀) < S_TD(t) ≤ 0 for all times t₀ < t and there exists a set (finite or not) of equilibrium entropies dependent on the initial preparation f of the system such that

Thus the entropy difference satisfies ΔS(t) ≤ 0 and

In this case system entropy converges to a steady state value which may not be unique. If it is not unique it characterizes a metastable state. The second form of the Second Law of thermodynamics is more interesting.

Strong Form of the Second Law.

–∞ < S_TD(t₀) < S_TD(t) ≤ 0 for all times t₀ < t and there is a unique limit (i.e. independent of the initial system preparation f) such that

for all initial system preparations f. Under these circumstances,

In this case the system entropy evolves to a unique maximum irrespective of the way in which the system was prepared.

In my investigations of the connection between dynamics and entropy evolution, I have been heavily influenced by the work of Khinchin (1949), Dugas (1959), Kurth (I960), Truesdell (1960), Farquhar (1964), O. Penrose (1970, 1979), Lebowitz (1973), Lebowitz and Penrose (1973), G.W. Mackey (1974), Wehrl (1978), and Prigogine (1980). Because of the approach taken here and the nature of the material presented, a brief outline of the main points may be helpful.

Chapter 1 defines a thermodynamic system in terms of measure spaces, draws a one to one correspondence between a density and a thermodynamic state, and introduces the Boltzmann-Gibbs entropy of a density.

In Chapter 2, using a Maximal Entropy Postulate, it is a simple demonstration that the entropy of a density will assume a maximal value if and only if this density is (in the terminology of Gibbs) either the density of the microcanonical or a generalized canonical ensemble. Then it is shown that the Boltzmann-Gibbs entropy of a density can be plausibly argued to coincide with the thermodynamic entropy S_TD of a system characterized by that density.

Chapter 3 introduces Markov operators. These are linear integral operators that describe the evolution of densities by dynamical or semi-dynamical systems. Fixed points of Markov operators, known as stationary densities, define states of relative or absolute thermodynamic equilibrium depending on whether there are multiple or unique stationary densities. Thus, a central question that must be answered in any treatment of thermodynamics is under what circumstance will the entropy change from its original value (determined by the way in which the system was prepared) to a final state corresponding to one of these states of relative or absolute equilibrium. Following this the conditional entropy, a generalization of the Boltzmann-Gibbs entropy, is introduced and identified with ΔS. Under particular conditions, the conditional entropy is shown to have its maximal value of zero if the stationary density of the state of thermodynamic equilibrium is that of the canonical ensemble. Then the distinction between invertible and noninvertible systems is made. This is used to provide the not too surprising proof that entropy is constant for invertible systems. It is only in noninvertible systems that the entropy may increase. Thus, irreversibility is necessary but not sufficient for the entropy to increase. Following this, a variety of sufficient conditions are derived for the existence of at least one state of thermodynamic equilibrium based on convergence properties of the system state density average.

Chapter 4 introduces a special type of Markov operator, the Frobenius-Perron operator. Following illustrative material demonstrating its utility  {xi}  in studying the evolution of densities by a variety of dynamical and semi-dynamical systems, we turn to a consideration of the conditions that guarantee the existence of a unique state of thermodynamic equilibrium. The necessary and sufficient condition for this existence is the property of ergodicity, which may be shared by both invertible and noninvertible systems.

Chapter 5 presents the concept of mixing, introduced in a qualitative sense by Gibbs, which is a stronger property than ergodicity though it still may be shared by noninvertible and invertible systems. However, it is not sufficient to permit the entropy of a system to change from its initial value.

Chapter 6 introduces a particular form of dynamical behavior, called asymptotic periodicity, that is sufficient for the evolution of the entropy to a metastable state of relative equilibrium (weak form of the Second Law).

Chapter 7 is, in a sense, the core of this work. There it is shown that for there to be a global evolution of the entropy to its maximal value of zero (strong form of the Second Law) it is necessary and sufficient that the system have a property known as exactness.

In a very real way, the results of Chapter 7 raise as many questions as they answer. Though providing totally clear criteria for the global evolution of system entropy, at the same time these criteria suggest that all currently formulated physical laws may not be at the foundation of the thermodynamic behavior we observe every day of our lives. This is simply because these laws are formulated as (invertible) dynamical systems, and exactness is a property that only noninvertible systems may display.

One possibility is that the current invertible, dynamical system statements of physical laws are incorrect and that more appropriate formulations in terms of noninvertible semidynamical systems await discovery. Alternately, other phenomena may mask the operation of these invertible systems so they appear to be noninvertible to the observer. Chapters 8 through 11 explore this latter possibility.

In Chapter 8, we examine the effects of coarse graining of phase space, due either to measurement error or to an inherent graininess of phase space that is imposed by Nature. It is easy to show that if we observe a system with mixing dynamics, but operating in a coarse grained phase space, then the entropy of the coarse grained density will evolve to a maximum as time goes either forward (t → +∞) or backward (t → –∞). Thus, though coarse graining induces entropy increase to a maximum of zero it fails to single out any unique direction of time for this to occur. This illustrates that the origin of noninvertible behavior is not a consequence of invertible dynamics operating in a coarse grained phase space.

Chapter 9 explores the consequence of taking a trace in which we observe only some of the important dynamical variables of a dynamics operating in a higher dimensional space (hidden variables). In this case the complete dynamics may be invertible and, consequently, have a constant entropy while the entropy of the trace system may smoothly evolve to a maximum (weak or strong form of the Second Law).  {xii} 

Chapters 10 and 11, respectively, examine the effects of external perturbations on discrete and continuous time dynamics. This situation is usually called interaction with a heat bath. Interactions with a heat bath, depending on how they occur, can be shown to lead to either local (metastable) or global states of thermodynamic equilibrium.

In Chapter 10 we show that under very mild assumptions concerning the nature of the perturbation, discrete time systems with the most uninteresting dynamics in the unperturbed situation will become either asymptotically periodic or exact in the presence of perturbations. Thus they will display evolution of entropy toward states of thermodynamic equilibrium (either form of the Second Law).

Chapter 11 continues this theme by examining the effects of white noise perturbations of continuous time systems whose dynamics are described by systems of ordinary differential equations. Again these perturbations induce exactness and the consequent increase of the conditional entropy to its maximum value of zero (strong form of the Second Law).

As should be evident from this survey of the contents, it is not my intent to develop statistical mechanics as a subject. This is done rather nicely from several points of view in a variety of texts. Kestin and Dorfman (1971), Reichl (1980), Ma (1985), Pathria (1985), and Grandy (1988) are representative of some of the more thought provoking of these.

Throughout, I have tried to include as much material as necessary so this book can be read as a unit. Proofs of almost all of the theorems are given, though they need not be read to grasp the thread of the argument. Examples are offered to try to illustrate the physical significance of the results discussed. To more clearly delineate material, the end of proofs are marked with a “□” and the end of examples by a “•”.

This work was started at the Universities of Oxford and Bremen, 1986-1987, and I thank Profs. J.D. Murray (Oxford) and H. Schwegler (Bremen) for their hospitality during this period. Several colleagues have helped me clarify various points, and I hope that they will not be offended by my lack of explicit acknowledgment of their interest. They know who they are. I am especially indebted to Helmut Schwegler for his continued interest and support in reading and commenting on almost every aspect of this work. He has given of his time and energy as only a true friend can.

My wife, Nancy, and my children — Fraser, David, Alastair, Linda, and Christopher — have all contributed a great deal through their love, interest, and encouragement.

And Fraser, to whom I dedicate this book, has always asked “Hi Mike, how's it going?”

CONTENTS

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INDEX

A

Abstract ergodic theorem, 35

Additivity, 6, 14

Adjoint operators, 43, 145

Anosov

        diffeomorphism, 59

        flow, 60

Asymptotic periodicity, 72

        conditions for, 73, 75, 84

        and correlations, 86

        and entropy, 80

        and ergodicity, 74

        and exactness, 96

        and Keener map, 126

        in loosely coupled systems, 118

        period of, 72, 125

        and perturbations, 118, 122, 126

        and quadratic map, 78

        and smoothing, 71

        and stationary densities, 72

        in strongly coupled systems, 122, 123

        and tent map, 75, 82

        and Weak Form of Second Law, 80

Average, 67

B

Baker transformation, 60

        and additive perturbations, 123

        and dyadic transformation, 112

        and entropy, 64, 66, 106

        Frobenius-Perron operator for, 64, 113

        and invertibility, 60, 113

        and K-systems, 67, 112

        and Lebesgue measure, 60

        and Liouville equation, 114

        and microcanonical ensemble, 65

        and mixing, 60, 63

        quadratic, 112

        stationary density for, 65

        and trace, 112, 123

Birkhoff ergodic theorem, 44

        extended to ergodic dynamics, 53

Blackbody radiation, 18

Boltzmann, 6, 31-33

        constant, 16

        equation, 114

        μ-space, 5

        and Loschmidt, 32, 45

        statistics, 16

        and Zermelo, 45

Boltzmann-Gibbs entropy, 5

        additivity, 6, 14

        and baker transformation, 64, 106

        and canonical ensemble, 11

        coarse grained, 104

        and conditional entropy, 28

        and correlations, 86

        and exactness, 100

        and grand canonical ensemble, 19

        maximal, 9, 19

        and microcanonical ensemble, 9

        and mixing, 64

        and quadratic map, 28

        sampled, 107

        and tent map, 100

        and thermodynamic entropy, 13

        uniqueness of, 6

C

Canonical ensemble, 14

        and entropy, 11, 101

        and exactness, 92, 94, 98, 101

        and mixing, 63

        and stationary density, 64

Cauchy-Holder inequality, 93

Cesaro convergence, 56  {168} 

Chebyshev inequality, 118

Clausius, 33

Coarse graining, 103

        and baker transformation, 106

        of densities, 104

        and dynamics, 106

        of entropy, 104, 105, 112

        and errors, 105

        and Gibbs, 103, 106, 109

        and invertibility, 107

        of mixing systems, 106

        of phase space, 103

        and Strong Form of Second Law, 107

        temporal, 107

        and trace, 113

Conditional entropy, 27

        and asymptotic periodicity, 80

        and baker transformation, 66

        and Boltzmann-Gibbs entropy, 28, 29

        coarse grained, 105

        and correlations, 87

        dependence on initial density, 80, 126

        and exactness, 98

        and invertible dynamics, 31, 66

        and Keener map, 126

        and Liouville equation, 32

        and mixing, 64, 66

        and ordinary differential equations, 32

        and perturbations, 119, 126, 131, 149, 150

        properties of, 29

        and quadratic map, 78, 100

        sampled, 107

        and tent map, 82, 95

        and thermodynamic entropy difference, 29

Convergence

        Cesaro, 56

        strong, 35, 92

        weak, 35, 63

Correlation, 67

        and asymptotic periodicity, 86

        and entropy, 86

        and Frobenius-Perron operator, 68

        and Koopman operator, 68

        and mixing, 68

Covariance, 67

        and mixing, 68

        normalized, 67

D

Delay differential equation

        and exactness, 91

        and laws of physics, 101

Density, 4

        and thermodynamic states, 4

        coarse grained, 104

        evolution, 43

        Gaussian, 141, 151

        illustrated, 4

        of measure, 4

        stationary, 23, 38

Diffusion term, 145, 148

Drift term, 145

Dyadic transformation, 94

        and baker transformation, 112

        and exactness, 94

        as a factor, 112, 114, 123

        Frobenius-Perron operator for, 94

        and Liouville equation, 114

        and microcanonical ensemble, 94

        and noninvertibility, 114

        and perturbations, 123

        stationary density for, 95

        and trace, 112

Dynamical system, 3

Dynamics

        asymptotically periodic, 72

        and coarse grained entropy, 106

        constructing exact, 95

        ergodic, 48

        exact, 89

        and Frobenius-Perron operator, 39

        Hamiltonian, 33, 45, 48, 101

        invertible, 2

        irreversible, 3

        and Koopman operator, 43

        loosely coupled to perturbations, 117

        measurable, 39  {169} 

        measure preserving, 41

        mixing, 58

        noninvertible, 3

        nonsingular, 39

        reversible, 2

        and strong perturbations, 120

        time reversal invariant, 2

        trajectory of, 2, 110

        versus perturbations, 120

E

Energy, 10, 12

        continuous spectrum, 17

        discrete spectrum, 18

        Helmholtz, 16

        and Maxwell-Boltzmann distribution, 17

Ensemble, 5

Entropy

        for baker transformation, 64

        Boltzmann-Gibbs, 5

        coarse grained, 104, 105, 107, 114

        conditional, 28

        and correlations, 86

        dependence on preparation, 80, 125

        and dynamics, 30

        and invertibility, 31

        for K-systems, 67

        and mixing, 66

        for quadratic map, 28, 78, 83, 100

        for tent map, 82, 95

        and trace, 111

Ergodic theorem

        abstract, 35

        Birkhoff, 44

        for space and trajectory averages, 53

Ergodicity, 48

        and asymptotic periodicity, 74

        and Birkhoff theorem, 53

        and canonical ensemble, 54

        and Cesaro convergence, 56

        and Hamiltonian dynamics, 51

        and harmonic oscillator, 49

        and Frobenius-Perron operator, 52, 56

        illustrated, 49

        and Koopman operator, 51

        and Markov operator, 57

        and microcanonical ensemble, 54

        and mixing, 63

        and perturbations, 121

        and sampled systems, 109

        and space and time averages, 54

        and stationary density, 52

        and tent map, 75

        and thermodynamic equilibrium, 52, 54

Errors, see Coarse graining; Sampling

Euler-Bernstein equations, 142

Exactness, 89

        and asymptotic periodicity, 96

        and canonical ensemble, 92, 94, 98, 101

        constructing, 95

        convergence rate, 96

        and delay differential equations, 91

        and dyadic transformation, 94

        and entropy, 98, 100

        and factors, 113

        illustrated, 90

        and invertibility, 90

        and Keener map, 131

        and K-systems, 113

        and lower bound function, 97

        and Markov operator, 93

        and microcanonical ensemble, 94, 100

        and partial differential equations, 91

        and perturbations, 119,122, 131, 135, 149, 152, 158

        and quadratic map, 95, 100

        and sampled sequence, 108

        and strong convergence, 92

        and Strong Form of Second Law, 98, 100, 150, 152, 158

        and tent map, 91, 95, 96, 100

Extension in phase, 10

F

Factor, 112  {170} 

        and dyadic transformation, 112, 114, 123

        and exactness, 113

        and K-system, 113

        and projection operators, 113

        and quadratic baker transformation, 112

        and Strong Form of Second Law, 113

        and trace, 112

Fixed point, 23

Fokker-Planck equation, 22, 144

        derivation, 142

        generalized solution, 146

        and Liouville equation, 22, 148

        and Markov operator, 22, 145, 147

        and noninvertibility, 24

        and stationary density, 23, 150

        and thermodynamic equilibrium, 150

Frobenius-Perron operator, 24, 39

        approximating Markov operators, 136

        for baker transformation, 64, 113

        and correlations, 68

        for dyadic map, 94

        and ergodicity, 52, 56

        and evolution of densities, 44

        for Hamiltonian systems, 48

        for invertible dynamics, 41

        and Koopman operator, 44

        and Liouville equation, 47, 114, 148

        and mixing, 63

        and ordinary differential equations, 23, 47, 114, 148

        properties of, 40

        for quadratic baker transformation, 113

        for quadratic map, 28, 78

        stationary density of, 23, 25

        for tent map, 25, 75

G

γ-space, 5

Gas dynamics, 61

Gaussian density, 141, 151

Generalized solution, 146

Geodesic flows, 62

Gibbs

        and coarse graining, 103, 106, 109

        and mixing, 64

        extension in phase, 10

        γ-space, 5

        index of probability, 5

        microcanonical ensemble, 10

Gibbs function, 16

Gibbs inequality, 7

        integrated, 8

God theorem, 111

Grand canonical ensemble, 19

H

Hamiltonian dynamics, 33, 45, 48, 101

        and ergodicity, 51

        and Frobenius-Perron operator, 48

        and Lebesgue measure, 48, 101

        and maximal entropy, 101

Harmonic oscillator, 49

Hat map, see Tent map

Helmholtz free energy, 16

I

Index of probability, 5

Initial density

        dependence of entropy on, 80, 126

Irreversible, see Noninvertible

Invariant measure, 41

        illustrated, 42

        and stationary density, 42

        and thermodynamic equilibrium, 43

Invariant set, 48

Invertibility, 2

        and baker transformation, 60, 113

        and coarse graining, 107

        and entropy, 31, 66

        and exactness, 90  {171} 

        and laws of physics, 101

        and Liouville equation, 24, 114

        and Markov operator, 23

        and mixing, 60

Ito calculus, 141

J

Jensen inequality, 30

K

Keener map, 126

        asymptotic periodicity in, 126

        and entropy, 126

        exactness in, 131

        and perturbations, 126

Kolmogorov equation, 144, 145

Koopman operator, 43

        and correlations, 68

        and ergodicity, 51

        and evolution of densities, 44

        and Probenius-Perron operator, 44

        and ordinary differential equations, 46

        and time averages, 43

Krylov, 64

K-systems, 66

        and baker transformation, 67, 112

        and entropy, 67

        and exactness, 113

        and factors, 113

        and mixing, 67

        and traces, 112

L

Landau equation, 153

Laws of physics, 101

Lebesgue measure, 4, 42

        and baker transformation, 60

        and Hamiltonian systems, 48, 101

        and microcanonical ensemble, 9, 42, 54, 63, 94, 100

        and mixing, 60

        and quadratic map, 42

        and Radon-Nikodym theorem, 26

        and tent map, 91

Liapunov function, 122, 148

Liouville equation, 22, 47

        and baker transformation, 114

        and dyadic transformation, 114

        and entropy, 32

        and Fokker-Planck equation, 22, 148

        and Probenius-Perron operator, 47, 114, 148

        and invertibility, 24, 114

        and perturbations, 148

Liouville theorem, 47

Loschmidt, 32, 45

Lower bound function, 97

M

Markov operator, 22

        approximated by Probenius-Perron operators, 136

        asymptotically periodic, 72

        ergodic, 57

        exact, 93

        fixed point of, 23

        from perturbations, 117, 119, 120, 124, 131

        and Fokker-Planck equation, 22, 145, 147

        invertible, 23

        mixing, 64

        noninvertible, 24

        properties, 23

        smoothing, 70

        stationary density of, 23, 38

        and stochastic differential equations, 22, 147

Maximal entropy

        and blackbody radiation, 18

        and canonical ensemble, 11, 101

        and grand canonical ensemble, 19

        and Hamiltonian dynamics, 101

        and Maxwell-Boltzmann distribution, 17

        and microcanonical ensemble, 9

        and thermodynamic equilibrium, 12  {172} 

Maxwell-Boltzmann distribution

        continuous, 17

        discrete, 18

Measurable transformation, 39

Measure, 4

        and density, 4

        illustrated, 4, 42

        invariant, 41

        Lebesgue, 4, 42

Measure preserving transformation, 41

Measure space, 1

        σ-finite, 2

        and thermodynamic systems, 2

Metric indecomposability, see Ergodic

Metric transitivity, see Ergodic

Microcanonical ensemble, 9

        and baker transformation, 65

        and Boltzmann-Gibbs entropy, 9

        density of, 10

        and dyadic transformation, 94

        and ergodicity, 54

        and exactness, 94, 100

        and Gibbs, 10

        and Lebesgue measure, 9, 42, 54, 63, 94, 100

        and maximal entropy, 9

        and stationary density, 54, 64

Mixing, 59

        and baker transformation, 60, 63

        and canonical ensemble, 63

        and coarse graining, 106

        and correlations, 68

        and covariance, 67

        and entropy, 64, 66

        and ergodicity, 63

        and Frobenius-Perron operator, 63

        gas model, 61

        and geodesic flows, 62

        and Gibbs, 64

        illustrated, 59

        and invertibility, 60

        and K-systerns, 67

        and Lebesgue measure, 60

        Markov operator, 64

        and perturbations, 121

        and sampling, 108

        and weak convergence, 63

        μ-space, 5

N

Noise, see Perturbations

Noninvertibility, 3

        and Boltzmann equation, 114

        and conditional entropy, 33

        and dyadic transformation, 114

        and exactness, 90

        and Fokker-Planck equation, 24

        and laws of physics, 101

        and Markov operators, 24

Nonsingular transformation, 39

Normalized covariance, 67

O

Observable, 5

        and canonical ensemble, 11

        and grand canonical ensemble, 19

        and maximal entropy, 12

        and Maxwell-Boltzmann distribution, 16

Ordinary differential equations

        and conditional entropy, 32, 45

        and density evolution, 22

        and Probenius-Perron operator, 23, 47, 114, 148

        Hamiltonian system, 33, 45, 48, 101

        and Koopman operator, 46

        and Liouville equation, 22, 48

        and perturbations, 140

P

Partial differential equations and exactness, 91

Partition, 103

Partition function, 12

Perturbations

        additive, 121, 123, 153, 155

        and asymptotic periodicity, 118, 122, 126

        and baker transformation, 123

        and dyadic transformation, 123  {173} 

        and dynamics, 115

        and entropy, 119, 126, 131, 149, 150

        and ergodicity, 121

        and exactness, 119, 122, 131, 149, 152, 158

        of Keener map, 126

        and Liouville equation, 148

        and mixing, 121

        of ordinary differential equations, 139

        by other systems 119, 121

        parametric, 131, 156

        and phase transitions, 151

        and stationary density, 119, 125, 150, 152, 153

        and strong form of Second law, 119, 122, 131, 135, 149, 150

        and thermodynamic equilibrium, 119, 125

        and traces, 119, 121

        and weak form of Second law, 118, 122, 123, 124, 126, 133

        versus dynamics, 120, 121

Phase space, 1

        coarse grained, 103

Phase transitions

        first order, 155, 157

        second order, 154, 158

Planck blackbody radiatiori, 18

Poincare recurrence theorem, 45

        extension, 55

Probability current, 152

Projection operator, 113

Q

Quadratic baker transformation, 112

        and factors, 112

        Frobenius-Perron operator for, 113

        stationary density for, 113

        and trace, 112

Quadratic map, 27

        and asymptotic periodicity, 78, 83

        and entropy, 28, 78, 83, 100

        exactness of, 95, 100

        as factor, 113

        Frobenius-Perron operator for, 28, 78

        invariant measure for, 42

        stationary density for, 28, 42

        and tent map, 27

Quantized behavior, 72

R

Radon-Nikodym theorem, 26

Rayliegh-Jeans radiation law, 19

Recurrence, 45

        theorem, 45, 54

recurrent point, 45

Reversibility, see Invertibility

S

Sampling, 107

        and entropy, 109

        and ergodicity, 109

        and exactness, 108

        illustrated, 108

        and mixing, 108

Scalar product, 35

Second Law of Thermodynamics, see Strong Form of Second Law of Thermodynamics; Weak Form of Second Law of Thermodynamics

Semidynamical system, 3

        σ-algebra, 1

        σ-finite measure space, 2

Smoothing operator, 70

Spatial averages

        convergence, 96

        in ergodic systems, 54

Spectral decomposition theorem, 71

Stationary density, 23, 38

        and asymptotic periodicity, 72

        for baker transformation, 65

        and canonical ensemble, 64

        for dyadic map, 95

        and ergodicity, 52

        for Fokker-Planck equation, 23, 150

        for Frobenius-Perron operator, 23, 25

        and invariant measures, 42  {174} 

        and microcanonical ensemble, 54, 64

        and perturbations, 119, 125, 150, 152, 153

        for quadratic baker transformation, 113

        for quadratic map, 28, 42

        for tent map, 25, 76

        and thermodynamic equilibrium, 4, 23, 34, 42, 52

Stochastic differential equations, 22, 141

        Fokker-Planck equation for, 22, 144

        and Markov operators, 22, 147

        and phase transitions, 151

Stochastic kernel, 117

Stratonovich calculus, 141

Strong convergence, 35

        and exactness, 92

        and thermodynamic equilibrium, 36

Strong Form of Second Law of Thermodynamics, ix

        and coarse graining, 107

        and exactness, 98, 100, 150, 152, 158

        and factors, 113

        and global stability, 13

        and laws of physics, 101

        and perturbations, 119, 121,131, 135, 149, 150

System state density averages, 34

T

Temperature, 13

Temporal coarse graining, see Sampling

Tent map, 24

        and asymptotic periodicity, 75, 82

        and entropy, 82, 95, 100

        ergodicity of, 75

        and exactness, 91, 95, 96, 100

        Frobenius-Perron operator for, 25, 75

        and Lebesgue measure, 91

        and quadratic map, 27

        and stationary density, 25, 76

        and Weak Form of Second Law

        of Thermodynamics, 83

        stationary density for, 25, 76

Thermodynamic entropy difference, ix

        and conditional entropy, 29

Thermodynamic entropy, ix, 12, 13

        and Boltzmann-Gibbs entropy, 13

Thermodynamic equilibria, ix, 13

        and ergodicity, 52

        and Fokker-Planck equation, 150

        and invariant measures, 43

        and maximal entropy, 12

        and perturbations, 119, 125

        and stationary density, 4, 23, 34, 42, 52

        and strong convergence, 36

        and upper bound function, 36

        and weak convergence, 35

        and weak precompactness, 36

        conditions for existence, 34

        global, 13

        metastable, 13

Thermodynamic limit, 5

Thermodynamic state, 4

Thermodynamic system, 4

        and density, 4

        and measure space, 2

        state of, 4

Time reversal invariance, see Invertibility

Trace, 110

        and baker transformation, 112, 123

        and coarse grained entropy, 113

        and dyadic transformation, 112

        and entropy, 111

        and factor, 112

        and God theorem, 111

        and K-systems, 112

        and perturbations, 119, 121

        and quadratic baker transformation, 112

        and trajectories, 110

Trajectories, 2, 110

        average along, 43, 54  {175} 

        convergence of, 96

        and trace, 110

Transient operator, 71

Trivial invariant set, 48

U

Uniform parabolicity condition, 143

V

Voigts theorem, 30

W

Weak convergence, 35

        and existence of equilibria, 35

        and mixing, 63

Weak Form of Second Law of Thermodynamics, ix

        and asymptotic periodicity, 80

        dependence on initial preparation, 80, 83

        and metastable states, 13

        and perturbations, 118, 122, 123, 124, 126, 133

        and quadratic map, 83

        and tent map, 83

Weak precompactness, 36

White noise, 141

Wien radiation law, 19

Wiener process, 141

Z

Zermelo, 45