Canonical Gibbs Measures Some Extensions of de Finetti's Representat
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760 H. 0. Georgii
Canonical Gibbs Measures Some Extensions of de Finetti's Representation Theorem for Interacting Particle Systems
Springer-Verlag Berlin Heidelberg New York 1979
Author H. 0. Georgii Fakultat fOr Mathematik Universitat Bielefeld Postfach 8640 D-4800 Bielefeld
AMS Subject Classifications (1970): 60 K35, 82A60
ISBN 3-540-09712-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09712-0 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data Georgii, Hans-Otto. Canonical Gibbs measures. (Lecture notes in mathematics; v. 760) Bibliography: p. Includes index. 1. Probabilities. 2. Representations of groups. 3. Measure theory. I. Title. II. Title: Gibbs measures. Ill. Title: De Finetti's representation theorem. IV. Series: Lecture notes in mathematics (Berlin); v. 760. OA3.L28 no. 760 [OC20.7.P7]510'.8s [519.2]79-23184 ISBN 0-387-09712-0
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Contents
Introduction
§ 1 Basic concepts 1.1 The discrete model 1.2 The continuous model § 2 Equilibrium states for systems of moving particles
2.1 2.2
Particle motions in the discrete model Particle motions in the continuous model
§ 3 Spatially homogeneous models
3.1 The variational principle 3.2 The free energy as a function of time
§ 4 Independent models 4.1 The discrete case 4.2 The continuous case § 5 Discrete models with interaction
§ 6
2
15 29 29
39 64 64
78 91 91 104 110
5.1 Formulation of results 5.2 Conditional probabilities and the activity function 5.3 Estimating the activity function
110 118 132
Continuous models with interaction
139
6.1 6.2 6.3
139 146 159
Formulation of results Conditional densities and the a~tivity functions Estimating the activity functions
§ 7 Some further results on homogeneous models
170
7.1 Properties of the activity function 7.2 The equivalence of ensembles
170 174
References
182 188
Index
Introduction
Symmetric probability measures on the infinite product
F
~
of a set F (equiped with theory:
a
X
F
X
a - algebra) are a well-known concept in probability
A probability measure on
is said to be symmetric if it is invariant under
~
the group of those transformations of
~
which are defined by a permutation of finite-
ly many coordinates. Apparently the first place in which this notion appeared was a contribution of J. Haag in
1924 .
to the International Congress of Mathematicians at Toronto
B. de Finetti
(1931)
probability measures. Fo
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