Rigidity Theory and Applications PDF
Preview Rigidity Theory and Applications
Rigidity Theory and Applications FUNDAMENTAL MATERIALS RESEARCH Series Editor: M. F. Thorpe, Michigan State University East Lansing, Michigan ACCESS IN NANOPOROUS MATERIALS Edited by Thomas J. Pinnavaia and M. F. Thorpe DYNAMICS OF CRYSTAL SURFACES AND INTERFACES Edited by P. M. Duxbury and T. J. Pence ELECTRONIC PROPERTIES OF SOLIDS USING CLUSTER METHODS Edited by T. A. Kaplan and S. D. Mahanti LOCAL STRUCTURE FROM DIFFRACTION Edited by S. J. L. Billinge and M. F. Thorpe RIGIDITY THEORY AND APPLICATIONS Edited by M. F. Thorpe and P. M. Duxbury A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further informationplease contact the publisher. Rigidity Theory and Applications Edited by M. F. Thorpe and P. M. Duxbury Michigan State University East Lansing, Michigan KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 0-306-47089-6 Print ISBN: 0-306-46115-3 ©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://www.kluweronline.com and Kluwer's eBookstore at: http://www.ebooks.kluweronline.com SERIES PREFACE This series of books, which is published at the rate of about one per year, addresses fundamental problems in materials science. The contents cover a broad range of topics from small clusters of atoms to engineering materials and involve chemistry, physics, materials science, and engineering, with length scales ranging from Ångstroms up to millimeters. The emphasis is on basic science rather than on applications. Each book focuses on a single area of current interest and brings together leading experts to give an up-to-date discussion of their work and the work of others. Each article contains enough references that the interested reader can access the relevant literature. Thanks are given to the Center for Fundamental Materials Research at Michigan State University for supporting this series. M.F. Thorpe, Series Editor E-mail: [email protected] v PREFACE Although rigidity has been studied since the time of Lagrange (1788) and Maxwell (1864), it is only in the last twenty-five years that it has begun to find applications in the basic sciences. The modern era starts with the important theorem of Laman (1970), which made the subject rigorous in two dimensions, followed by the development of computer algorithms that can test over a million sites in seconds and find the rigid regions, and the associated pivots, leading to many applications. This workshop on Rigidity Theory and Applications was organized to bring together leading researchers studying the underlying theory, and to explore the various areas of science where applications of these ideas are being implemented. This is certainly the first workshop of this type, and it was interesting to see how researchers from different fields (mathematics, computer science, statistical physics, experimental physics, biochemistry, and ceramics) struggled with the language and concepts of the other disciplines. Technical language is always a barrier, but with a little effort and patience, this was largely overcome. This focused workshop was held at the picturesque and historic Park Place Hotel in Traverse City, Michigan, USA from June 1998. All participants were by invitation only and 23 gave presentations, most of which resulted in contributions which form this book. These proceedings provide a unique tutorial snapshot of the state of the study of rigidity in the basic sciences in 1998. One perhaps unexpected outcome of this activity is to show that the insights of Maxwell were quite accurate, and that the constraint counting approach associated with his name, which ignores the possibility of redundant elements, is a remarkably good starting point in many cases. A number of papers at this workshop used the Maxwell approach to gain insight into glasses and ceramic materials. The concept of a floppy mode, a distortion of the system with no associated cost in energy, proved to be useful in all the applications covered at the workshop. The ideas of Laman have filtered through applied mathematics and computer science to produce powerful methods to enumerate the rigid clusters in large systems and the recent applications of this work in glasses and proteins are covered in this book. Many questions remain. The theory of rigidity in three dimensions is not as robust as we would like, and this is important as most systems of interest in science are three- rather than two- dimensional in nature. The number of people working on problems associated with rigidity remains quite small, and the language rather specialized, so that an effort is needed to integrate these approaches into the mainstream of science. This is beginning to happen as words like rigid, flexible, constraintand floppy mode are being used increasingly in science and engineering, and we hope that this book will help. vii We would like to thank Michigan State University for financing this workshop and the Center for Fundamental Materials Research at MSU for contributing to the cost of producing these proceedings. The efforts of Lorie Neuman and Janet King, who organized the workshop and proceedings, are greatly appreciated as was the advice and help of the Advisory Committee members: Punit Boolchand, Leslie Kuhn, and WalterWhiteley. Michael F. Thorpe Phillip M. Duxbury East Lansing, Michigan viii CONTENTS RIGIDITYTHEORY Generic and Abstract Rigidity ................................................................................................ 1 Brigitte Servatius and Herman Servatius Rigidity of MolecularStructures: Generic and Geometric Analysis ..............................................................................21 Walter Whiteley Tensegrity Structures: Why Are They Stable? ....................................................................47 R. Connelly The Role of Tensegrity in Distance Geometry .....................................................................55 Timothy F. Havel APPLICATIONS TO NETWORKS Comparison of Connectivity and Rigidity Percolation.........................................................69 Cristian F. Moukarzel and Phillip M. Duxbury Rigidity Percolation on Trees............................................................................................... 81 P.L. Leath and Chen Zeng Rigidity as an Emergent Property of Random Networks: A Statistical Mechanical View .................................................................................95 Paul M. Goldbart Granular Matter Instability: A Structural Rigidity Point of View..................................... 125 Cristian F. Moukarzel Rigidity and Memory in a Simple Glass............................................................................. 143 P. Chandra and L.B. Ioffe ix APPLICATIONS TO GLASSES Constraint Theory, Stiffness Percolation and the Rigidity Transition in Network Glasses................................................................................ 155 J.C.Phillips Topologically Disordered Networks of Rigid Polytopes: Applications to Noncrystalline Solids and ConstrainedViscousSintering..................................... 173 Prabhat K. Gupta Rigidity Constraints in Amorphization of Singly- and Multiply-Polytopic Structures................................................................................. 191 Linn W. Hobbs, C. Esther Jesurum, and Bonnie Berger Floppy Modes in Crystalline and Amorphous Silicates .....................................................217 Martin T. Dove, Kenton D. Hammonds, and Kostya Trachenko Generic Rigidity of Networks Glasses................................................................................239 M.F. Thorpe, D.J. Jacobs, N.V. Chubynsky, and A.J. Rader Rigidity Transition in Chalcogenide Glasses......................................................................279 P. Boolchand, Xingwei Feng, D. Selvanathan, and W.J. Bresser Rigidity, Fragility, Bond Models and the “Energy Landscape” for CovalentGlassformers......................................................................................297 C.A. Angell Entropic Rigidity.........................................................................................................................315 Béla Joós, Michael Plischke, D.C. Vernon, and Z. Zhou APPLICATIONS TO PROTEINS MolecularDynamics and Normal Mode Analysis of Biomolecular Rigidity.....................329 David A. Case EfficientStochastic Global Optimization for Protein Structure Prediction........................345 YingyaoZhou and RubenAbagyan Flexible and Rigid Regions in Proteins ..............................................................................357 Donald J. Jacobs, Leslie A. Kuhn, and Michael F. Thorpe Flexibly Screening for Molecules Interacting with Proteins ..............................................385 Volker Schnecke and Leslie A. Kuhn Studying MacromolecularMotions in a Database Framework: From Structure to Sequence....................................................................................401 Mark Gerstein, Ronald Jansen, Ted Johnson,JerryTsai, and Werner Krebs List of Participants..............................................................................................................421 Index...................................................................................................................................429 x
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