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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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David Ruelle
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Thermodynamic Formalism
The Mathematical Structures of Equilibrium Statistical Mechanics
Second Edition
Reissued in the Cambridge Mathematical Library this classic book outlines the theory
of thermodynamic formalism which was developed to describe the properties of
certain physical systems consisting of a large number of subunits. It is aimed at
mathematicians interested in ergodic theory, topological dynamics, constructive
quantum field theory, and the study of certain differentiable dynamical systems,
notably Anosov diffeomorphisms and flows. It is also of interest to theoretical
physicists concerned with the computational basis of equilibrium statistical mechanics.
The level of the presentation is generally advanced, the objective being to provide an
efficient research tool and a text for use in graduate teaching. Background material on
physics has been collected in appendices to help the reader. Extra material is given in
the form of updates of problems that were open at the original time of writing and as a
new preface specially written for this edition by the author.
David Ruelle is a Professor Emeritus at the Institut des Hautes Etudes
Scientifiques, Bures-sur-Yvette, Paris.
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David Ruelle
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
Statistical Mechanics, Second Edition
David Ruelle
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Thermodynamic Formalism
The Mathematical Structures of Equilibrium
Statistical Mechanics
Second Edition
DAVID RUELLE
Institut des Hautes Etudes Scientifiques
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David Ruelle
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published by the press syndicate of the university of cambridge
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C Addison-Wesley Publishing Company, Inc. 1978
First edition C Cambridge University Press 2004
Second edition This book is in copyright. Subject to statutory exception
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First published in Encyclopedia of Mathematics and Its Applications 1978
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or the last, either.
You know us.
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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David Ruelle
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Contents
Foreword to the first edition
Preface to the first edition
Preface to the second edition
0.1
0.2
0.3
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
page xv
xvii
xix
Introduction
Generalities
Description of the thermodynamic formalism
Summary of contents
Theory of Gibbs states
Configuration space
Interactions
Gibbs ensembles and thermodynamic limit
Proposition
Gibbs states
Thermodynamic limit of Gibbs ensembles
Boundary terms
Theorem
Theorem
Algebra at infinity
Theorem (characterization of pure Gibbs states)
The operators M
Theorem (characterization of unique Gibbs states)
Remark
Notes
Exercises
1
1
3
9
11
11
12
13
14
14
15
16
18
18
19
20
20
21
22
23
23
vii
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viii
Contents
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Gibbs states: complements
Morphisms of lattice systems
Example
The interaction F ∗ Lemma
Proposition
Remarks
Systems of conditional probabilities
Properties of Gibbs states
Remark
Notes
Exercises
24
24
25
25
26
26
27
28
29
30
30
31
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
Translation invariance. Theory of equilibrium states
Translation invariance
The function A
Partition functions
Theorem
Invariant states
Proposition
Theorem
Entropy
Infinite limit in the sense of van Hove
Theorem
Lemma
Theorem
Corollary
Corollary
Physical interpretation
Theorem
Corollary
Approximation of invariant states by equilibrium states
Lemma
Theorem
Coexistence of phases
Notes
Exercises
33
33
34
35
36
39
39
40
42
43
43
45
45
47
48
48
49
49
50
50
52
53
54
54
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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David Ruelle
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Contents
ix
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
Connection between Gibbs states and Equilibrium states
Generalities
Theorem
Physical interpretation
Proposition
Remark
Strict convexity of the pressure
Proposition
Zν -lattice systems and Zν -morphisms
Proposition
Corollary
Remark
Proposition
Restriction of Zν to a subgroup G
Proposition
Undecidability and non-periodicity
Notes
Exercises
57
57
58
59
59
60
61
61
62
62
63
63
64
64
65
65
66
66
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
One-dimensional systems
Lemma
Theorem
Theorem
Lemma
Proof of theorems 5.2 and 5.3
Corollaries to theorems 5.2 and 5.3
Theorem
Mixing Z-lattice systems
Lemma
Theorem
The transfer matrix and the operator L
The function ψ>
Proposition
The operator S
Lemma
Proposition
Remark
Exponentially decreasing interactions
69
70
70
71
72
73
75
76
78
78
79
80
81
81
82
82
82
83
83
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x
Contents
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
5.27
5.28
5.29
5.30
The space F θ and related spaces
Proposition
Theorem
Remarks
Lemma
Proposition
Remark
Theorem
Corollary
Zeta functions
Theorem
Remark
Notes
Exercises
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
Extension of the thermodynamic formalism
Generalities
Expansiveness
Covers
Entropy
Proposition
Pressure
Other definitions of the pressure
Properties of the pressure
The action τ a
Lemma
Lemma
Theorem (variational principle)
Equilibrium states
Theorem
Remark
Commuting continuous maps
Extension to a Zν -action
ν
Results for Z
-actions
Remark
Topological entropy
Relative pressure
Theorem
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84
85
85
86
86
87
88
88
89
89
90
93
93
94
101
101
101
102
103
103
104
105
106
107
107
107
108
110
111
111
112
112
113
115
115
115
116
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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David Ruelle
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Contents
xi
6.23 Corollary
Notes
Exercises
117
117
118
7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
7.19
7.20
7.21
7.22
7.23
7.24
7.25
7.26
7.27
7.28
7.29
7.30
7.31
121
121
123
123
124
124
125
126
126
128
128
128
129
129
130
131
132
132
133
133
134
135
135
135
137
137
138
139
140
140
141
141
143
144
Statistical mechanics on Smale spaces
Smale spaces
Example
Properties of Smale spaces
Smale’s “spectral decomposition”
Markov partitions and symbolic dynamics
Theorem
Hölder continuous functions
Pressure and equilibrium states
Theorem
Corollary
Remark
Corollary
Corollary
Equilibrium states for A not Hölder continuous
Conjugate points and conjugating homeomorphisms
Proposition
Theorem
Gibbs states
Periodic points
Theorem
Study of periodic points by symbolic dynamics
Proposition
Zeta functions
Theorem
Corollary
Expanding maps
Remarks
Results for expanding maps
Markov partitions
Theorem
Applications
Notes
Exercises
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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David Ruelle
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xii
Contents
Appendix A.1 Miscellaneous definitions and results
A.1.1 Order
A.1.2 Residual sets
A.1.3 Upper semi-continuity
A.1.4 Subadditivity
146
146
146
147
147
Appendix A.2
148
Topological dynamics
Appendix A.3 Convexity
A.3.1 Generalities
A.3.2 Hahn–Banach theorem
A.3.3 Separation theorems
A.3.4 Convex compact sets
A.3.5 Extremal points
A.3.6 Tangent functionals to convex functions
A.3.7 Multiplicity of tangent functionals
150
150
150
151
151
151
152
152
Appendix A.4 Measures and abstract dynamical systems
A.4.1 Measures on compact sets
A.4.2 Abstract measure theory
A.4.3 Abstract dynamical systems
A.4.4 Bernoulli shifts
A.4.5 Partitions
A.4.6 Isomorphism theorems
153
153
154
154
155
155
156
Appendix A.5 Integral representations on convex
compact sets
A.5.1 Resultant of a measure
A.5.2 Maximal measures
A.5.3 Uniqueness problem
A.5.4 Maximal measures and extremal points
A.5.5 Simplexes of measures
A.5.6 Zν -invariant measures
157
157
158
158
158
159
159
Appendix B Open problems
B.1
Systems of conditional probabilities (Chapter 2)
B.2
Theory of phase transitions (Chapter 3)
B.3
Abstract measure-theory viewpoint (Chapter 4)
B.4
A theorem of Dobrushin (Chapter 5)
B.5
Definition of the pressure (Chapter 6)
160
160
160
160
160
161
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Contents
B.6
B.7
B.8
B.9
B.10
Shub’s entropy conjecture (Chapter 6)
The condition (SS3) (Chapter 7)
Gibbs states on Smale spaces (Chapter 7)
Cohomological interpretation (Chapter 7)
Smale flows (Chapter 7 and Appendix C)
xiii
161
161
161
161
161
Appendix C Flows
C.1 Thermodynamic formalism on a metrizable compact set
C.2 Special flows
C.3 Special flow over a Smale space
C.4 Problems
162
162
163
163
164
Appendix D
165
Update of open problems
References
Index
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167
172
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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David Ruelle
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Foreword to the first edition
Thermodynamics is still, as it always was, at the center of physics, the standardbearer of successful science. As happens with many a theory, rich in applications, its foundations have been murky from the start and have provided a
traditional challenge on which physicists and mathematicians alike have tested
their latest skills.
Ruelle’s book is perhaps the first entirely rigorous account of the foundations
of thermodynamics. It makes heavier demands on the reader’s mathematical
background than any volume published so far. It is hoped that ancillary volumes
in time will be published which will ease the ascent onto this beautiful and
deep theory; at present, much of the background material can be gleaned from
standard texts in mathematical analysis. In any case, the timeliness of the content
shall be ample reward for the austerity of the text.
Giovanni Gallavotti
General Editor, Section on Statistical Mechanics
and
Gian-Carlo Rota
Editor, Encyclopedia of Mathematics and its Applications.
xv
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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David Ruelle
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Preface to the first edition
The present monograph is based on lectures given in the mathematics departments of Berkeley (1973) and of Orsay (1974–1975). My aim has been to
describe the mathematical structures underlying the thermodynamic formalism
of equilibrium statistical mechanics, in the simplest case of classical lattice spin
systems.
The thermodynamic formalism has its origins in physics, but it has now
invaded topological dynamics and differentiable dynamical systems, with applications to questions as diverse as the study of invariant measures for an
Anosov diffeomorphism (Sinai [3]), or the meromorphy of Selberg’s zeta function (Ruelle [7]). The present text is an introduction to such questions, as well
as to more traditional problems of statistical mechanics, like that of phase transitions. I have developed the general theory – which has considerable unity –
in some detail. I have, however, left aside particular techniques (like that of
correlation inequalities) which are important in discussing examples of phase
transitions, but should be the object of a special study.
Statistical mechanics extends to systems vastly more general than the classical lattice spin systems discussed here (in particular to quantum systems). One
can therefore predict that the theory discussed in this monograph will extend
to vastly more general mathematical setups (in particular to non-commutative
situations). I hope that the present text may contribute some inspiration to the
construction of the more general theories, as well as clarifying the conceptual
structure of the existing formalism.
xvii
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0521546494 - Thermodynamic Formalism: The Mathematical Structures of Equilibrium
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Preface to the second edition
Twenty-five years have elapsed since the first printing of Thermodynamic Formalism, and in the meantime a number of significant developments have taken
place in the area indicated by the subtitle The Mathematical Structures of Equilibrium Statistical Mechanics. Fortunately, our monograph was concerned with
basics, which have remained relatively unchanged, so that Thermodynamic Formalism remains frequently quoted. In the present re-issue, some misprints have
been corrected, and an update on the open problems of Appendix B has been
added. We shall now outline briefly some new developments and indicate unsystematically some source material for these developments. The mathematical
aspects of the statistical mechanics of lattice systems, including phase transitions, are covered in the monographs of Sinai [a], and Simon [b]. It may be
mentioned that research in this important domain has become less active than
it was in the 1960s, ’70s, and ’80s (but a really good idea might reverse this
evolution again). The relation between Gibbs and equilibrium states has been
extended to more general topological situations (see Haydn and Ruelle [c]). For
a connection of Gibbs states with non-commutative algebras and K-theory, see
for instance [d] and the references given there, in particular to the work of Putnam. Particularly fruitful developments have taken place which use the concepts
of transfer operators and dynamical zeta functions. In the present monograph
these concepts are introduced (in Chapters 5 and 7) in a situation corresponding
to uniformly hyperbolic smooth dynamics (Anosov and Axiom A systems, here
presented in the topological setting of Smale spaces). The hyperbolic orientation has led to very interesting results concerning the distribution of periods of
periodic orbits for hyperbolic flows (in particular the lengths of geodesics on a
manifold of negative curvature). These results have been beautifully presented
in the monograph of Parry and Pollicott [e]. More recent results of Dolgopyat on
exponential decay of correlation for hyperbolic flows [f,g] may be mentioned at
this point. It was realized by Baladi and Keller that the ideas of transfer operators
xix
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xx
Preface to the second edition
and dynamical zeta functions work well also for piecewise monotone maps of
the interval (which are not uniformly hyperbolic dynamical systems). This new
development can in particular be related to the kneading theory of Thurston and
Milnor. We have thus now a much more general theory of transfer operators,
very usefully presented in the monograph of Baladi [h], which has in particular
an extensive bibliography of the subject. For a general presentation of dynamical zeta functions see also Ruelle [i]. As we have seen, the ideas of statistical
mechanics, of a rather algebraic nature, have found geometric applications in
smooth dynamics, and particularly the study of hyperbolic systems. Extensions
to nonuniformly hyperbolic dynamical systems are currently an active domain
of research, with SRB states playing an important role for Sinai, Ruelle, Bowen,
Strelcyn, Ledrappier, Young, Viana, . . . ). This, however, is another story.
References
[a] Y. G. Sinai. Phase Transitions: Rigorous Results. Pergamon Press, Oxford, 1982.
[b] B. Simon. The Statistical Mechanics of Lattice Gases I. Princeton University Press,
Princeton, 1993.
[c] N. T. A. Haydn and D. Ruelle. “Equivalence of Gibbs and equilibrium states
for homeomorphisms satisfying expansiveness and specification,” Commun. Math.
Phys. 148, 155–167 (1992).
[d] A. Kumjan and D. Pask. “Actions of Zk associated to higher rank graphs,” Ergod.
Th. and Dynam. Syst. 23, 1153–1172 (2003).
[e] W. Parry and M. Pollicott. Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Astérisque 187–188, Soc. Math. de France, Paris, 1990.
[f] D. Dolgopyat. “On decay of correlations in Anosov flows,” Ann. of Math. 147,
357–390 (1998).
[g] D. Dolgopyat. “Prevalence of rapid mixing for hyperbolic flows,” Ergod. Th. and
Dynam. Syst. 18, 1097–1114 (1998).
[h] V. Baladi. Positive Transfer Operators and Decay of Correlations. World Scientific,
Singapore, 2000.
[i] D. Ruelle. “Dynamical zeta functions and transfer operators,” Notices Amer. Math.
Soc. 49, 887–895 (2002).
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