Mathematical Theory of Nonequilibrium Steady States On the Frontier
This volume provides a systematic mathematical exposition of the conceptual problems of nonequilibrium statistical physics, such as entropy production, irreversibility, and ordered phenomena. Markov chains, diffusion processes, and hyperbolic dynamic
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Da-Quan Jiang Min Qian Min-Ping Qian
Mathematical Theory of Nonequilibrium Steady States On the Frontier of Probability and Dynamical Systems
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Authors Da-Quan JIANG Min QIAN Min-Ping QIAN LMAM School of Mathematical Sciences Peking University Beijing 100871 People’s Republic of China e-mail: [email protected] [email protected]
Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 37D20, 37D25, 37D35, 37D45, 37H15, 58J65, 60F10, 60G10, 60H10, 60J10, 60J27, 60J35, 60J60, 82C05, 82C31, 82C35 ISSN 0075-8434 ISBN 3-540-20611-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science + Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany
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Preface
The title of this book already says something about its contents and historical origin, but since it is meant in a rigorous mathematical context, a few words of explanation may be added. Boltzmann [36] introduced the concept of detailed balance as a way of maintaining equilibrium. On the other hand, Kolmogorov defined the reversibility of a Markov chain; what he meant is actually the reversibility in a statistical sense. It is not a mere accident that these two concepts are mathematically identical. When Prigogine’s work [188, 344] became known to the public, how to define nonequilibrium steady states seemed to be the first question to ask. Reversibility had already been an accepted notion in mathematics, so a small group of Chinese mathematicians just took “irreversibility” as the equivalent of “nonequilibrium” and tried to find out what mathematics could be derived from this definition. A small meeting of about eight people was then held in Beijing in 1978, and as a consequence, a pamphlet [55] in Chine
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