Scaling and Self-Similarity in Physics: Renormalization in Statistical Mechanics and Dynamics PDF
Preview Scaling and Self-Similarity in Physics: Renormalization in Statistical Mechanics and Dynamics
Progress in Physics Vol. 7 Editedby A. Jaffe, G. Parisi, and D. Ruelle Springer Science+Business Media, LLC Scaling and Self-Similarity in Physics Renormalization in Statistical Mechanics and Dynamics Jiirg Frohlich, editor 1983 Springer Science+Business Media, LLC Editor: JürgFröhlich DepartmentofTheoreticalPhysics ETH-Höriggerberg 8093Zürich,Switzerland LibraryofCongressCatalogingin PublicationData Scalingand self-similarity inphysics. (Progressinphysics ;v.7) 1.Renormalizationgroup. 2.Statisticalmechanics. 3.Fieldtheory. 4.Dynamics. 5.Phasetransformations (Statisticalphysics) 6.CriticalPhenomena (Physics) I.Fröhlich,Jürg. II.TitIe:Self-similarity, Ill.Series:Progressinphysics (Boston, Mass.) ;v.7. Oc20.7.R43S33 1983 530.1 83-15788 Cll--KurztitelaufuahmederDeutschenBibliothek Scalingand self-similarityin physics: renormalizationinstatist. mechanicsand dynamicsled.byJürg Fröhlich.-Boston; Basel;Stuttgart:Birkhäuser, 1983. (Progressinphysics;Vol.7) NE:Fröhlich,Jürg [Hrsg.];GT Allrightsreserved.No partofthispublicationmay bereproduced,storedina retrievalsystem,or transmitted,inanyform or byany means,electronic, mechanical,photocopying,recordingor otherwise,withoutpriorpermissionof the copyrightowner. eSpringerScience+BusinessMediaNewYork 1983 Ursprünglicherschienenbei BirkhäuserBoston, Inc. 1983. Softcoverreprintofthe hardcover 1st edition 1983 ISBN978-1-4899-6764-0 ISBN978-1-4899-6762-6 (eBook) DOI 10.1007/978-1-4899-6762-6 987654321 TABLE OF CONTENTS FOREWORD 1 PART I : Equilibrium statistical mechanics and field theory Large Fluctuations of Random Fields and Renormalization Group: Some Perspeutives, G. Jona-Lasinio 11 The Berezinskii-Kosterlitz-Thouless Transition (Energy - Entropy Arguments and Renormalization in Defect Gases), J.FrBhlich and T.Spencer 29 Interface and Surface Tension in Ising Model, C.-E.Pfister 139 Iterated Mayer Exnansions and their Application to Coulomb Gases, J.Z.Imbrie 163 Rigorous Results on the Critical Behavior in Statistical Mechanics, M.Aizenman 181 Non-Perturbativ6 Methods for the Study of Massless Models, J.-R.Fontaine 203 Rigorous Renormalization Group and Asymptotic Freedom, K.Gaw~dzki and A.Kupiainen 227 On Infrared Superrenormalization, J.Magnen and R.Seneor 263 4 Ultraviolet Stability in Field Theory The ~3 Model, T.Balaban 297 PART II : Dynamical Systems Renormalization Group Analysis for Dynamical Systems, P.Collet and H.Koch 323 v vi Bowen's Formula for the Hausdorff Dimension of Self-Similar Sets, D.Ruelle 351 Perturbation Theory for Classical Hamiltonian Systems, G.Gallavotti 359 FOREWORD The .articles collected in this book have grown out of a series of seminar talks held at the Institut des Hautes Etudes Scientifiques. Bures-sur-Yvette. between spring 1981 and summer 1982. These talks were presented by people who. during that period. happened to be visitors at the Institut or worked in the Paris area. Most of them were (and still are) thinking about problems in theoretical and mathemati cal physics related. in a general way. to ideas of scaling and self-similarity; problems which nowadays are often studied with the help of what has become known under the name of "renormalization group methods". Their ideas and their results illustrate in which way mathematical physics has interacted with one of the big themes of present-day theoretical physics. To present a brief status report about where these interactions have led us. so far. is a guiding idea behind this volume. During the past ten to twenty years. various concepts of scaling. self-similarity and renormalization have as sumed a well defined shape and proven to provide a very productive framework to think about a surprisingly large class of physical problems. Powerful computational and analytical tools have emerged from them. Most readers will be familiar with one or another among many excellent text books and review articles devoted to scaling and renormal ization in the theory of critical phenomena in statistical mechanics and of differentiable dynamical systems. [Ref erences to some of these texts may be found in several articles in this book.] Numerous theoretical physicists 1 2 have contributed very important insights and results to these fields. Thanks to their original and persistent ef forts we now do have some considerable qualitative and quantitative understanding of "continuous transitions and critical phenomena", not only in equilibrium statistical mechanics and quantum field theory, but also in dynamics and in the theory of disordered systems. Precisely because very many people have contributed a large numer of substantial result s to the subject addressed in this book, it unfortunately turned out to be necessary to apply rather arbitrary, but strict criteria to choose authors of articles for the present book. Such criteria are usually unsatisfactory. For I still believe in a basic unity of theoretical and mathematical physics. The criteria which we finally adopted are not intended to express any value judgments. [This is no attempt towards "eine Umwer tung aller Werte".] Rather , they were imposed upon us by convenience and by consideration of what was within our modest possibilities. They are: 1. Every author was supposed to have spent time or given a seminar talk at the Institut des Hautes Etudes Scien tifiques during the period between spring 1981 and summer 1982. 2. Only contributions were accepted which describe mathematically rigorous results related to the main theme of this book. 3. Every article was supposed to be of relevance to statistical physics and/or dynamical systems theory. 4. Contributions concerning "scaling and self-similar ity in mathematics" were not invited. These criteria may explain why many ideas, results and calculations of enormous theoretical and practical import ance to critical phenomena in equilibrium statistical mech anics and dynamics cannot be found in any of the articles published in this volume. Moreover, they make it clear why 3 most of the original creators of many important ideas and methods are not among the contributors. Crit erion 3 explains why "scaling and renormalization in relativistic quantum field theory", for example, is great ly underemphasized in this book. This is most regrettable, for several reasons: a) Many of the important ideas on "scaling and renorm alization" actually originated in relativistic quantum field theory. b) The discovery of many relations and deep connections, heuristic and mathematical, between the theory of continuous transitions and critical phenomena on the one hand and quan tum field theory on the other hand, and the many uses of these connections in actual calculations belong to the great achievements of theoretical physics in recent times and has had a profound impact on the development of statis tical physics and quantum field theory. [Various connec tions and analogies between quantum field theory and fluid dynamics may eventually turn out to be of comparable signi ficance.] c) Quant um field theory is one of a few basic theories in modern theoretical physics, and it is the most plausible and most efficient theoretical basis for particle physics. As such it is of eminent import ance , and it still attracts many among the most brilliant theoreticians. The theory of disordered systems is not represented in this book, at all. This, too, is really quite regret table; especially since, during the past few years, nice mathematically rigorous results about disordered systems, related to the themes of this book, have started to appear. Moreover, the theory of disordered systems bears quite a lot of promise for the near - and not so near - future. Criterion 4 has excluded all developments related to scaling, self-similarity and renormalization in mathematics from this book. That there are many and important such de velopments is presumably clear to many or most readers. In
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