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This book provides a detailed introduction to the ergodic theory of
equilibrium states giving equal weight to two of its most important
applications, namely to equilibrium statistical mechanics on
lattices and to (time discrete) dynamical systems. It starts with a
chapter on equilibrium states on finite probability spaces which
introduces the main examples for the theory on an elementary level.
After two chapters on abstract ergodic theory and entropy,
equilibrium states and variational principles on compact metric
spaces are introduced emphasizing their convex geometric
interpretation. Stationary Gibbs measures, large deviations, the
Ising model with external field, Markov measures,
Sinai-Bowen-Ruelle measures for interval maps and dimension maximal
measures for iterated function systems are the topics to which the
general theory is applied in the last part of the book. The text is
self contained except for some measure theoretic prerequisites
which are listed (with references to the literature) in an
appendix.
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Entropy (Hardcover, New)
Andreas Greven, Gerhard Keller, Gerald Warnecke
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R3,104
R2,718
Discovery Miles 27 180
Save R386 (12%)
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Ships in 12 - 17 working days
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The concept of entropy arose in the physical sciences during the
nineteenth century, particularly in thermodynamics and statistical
physics, as a measure of the equilibria and evolution of
thermodynamic systems. Two main views developed: the macroscopic
view formulated originally by Carnot, Clausius, Gibbs, Planck, and
Caratheodory and the microscopic approach associated with Boltzmann
and Maxwell. Since then both approaches have made possible deep
insights into the nature and behavior of thermodynamic and other
microscopically unpredictable processes. However, the mathematical
tools used have later developed independently of their original
physical background and have led to a plethora of methods and
differing conventions.
The aim of this book is to identify the unifying threads by
providing surveys of the uses and concepts of entropy in diverse
areas of mathematics and the physical sciences. Two major threads,
emphasized throughout the book, are variational principles and
Ljapunov functionals. The book starts by providing basic concepts
and terminology, illustrated by examples from both the macroscopic
and microscopic lines of thought. In-depth surveys covering the
macroscopic, microscopic and probabilistic approaches follow. Part
I gives a basic introduction from the views of thermodynamics and
probability theory. Part II collects surveys that look at the
macroscopic approach of continuum mechanics and physics. Part III
deals with the microscopic approach exposing the role of entropy as
a concept in probability theory, namely in the analysis of the
large time behavior of stochastic processes and in the study of
qualitative properties of models in statistical physics. Finally in
Part IV applications in dynamical systems, ergodic and information
theory are presented.
The chapters were written to provide as cohesive an account as
possible, making the book accessible to a wide range of graduate
students and researchers. Any scientist dealing with systems that
exhibit entropy will find the book an invaluable aid to their
understanding.
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