statistical mechanics algorithms and computations krauth 2006 11 16 Cấu trúc dữ liệu và giải thuật

com OXFORD MASTER SERIES IN STATISTICAL, COMPUTATIONAL, AND THEORETICAL PHYSICS CuuDuongThanCong.com OXFORD MASTER SERIES IN PHYSICS The Oxford Master Series is designed for final year undergraduate and beginning graduate students in physics and related disciplines It has been driven by a perceived gap in the literature today While basic undergraduate physics texts often show little or no connection with the huge explosion of research over the last two decades, more advanced and specialized texts tend to be rather daunting for students In this series, all topics and their consequences are treated at a simple level, while pointers to recent developments are provided at various stages The emphasis in on clear physical principles like symmetry, quantum mechanics, and electromagnetism which underlie the whole of physics At the same time, the subjects are related to real measurements and to the experimental techniques and devices currently used by physicists in academe and industry Books in this series are written as course books, and include ample tutorial material, examples, illustrations, revision points, and problem sets They can likewise be used as preparation for students starting a doctorate in physics and related fields, or for recent graduates starting research in one of these fields in industry CONDENSED MATTER PHYSICS M.T Dove: Structure and dynamics: an atomic view of materials J Singleton: Band theory and electronic properties of solids A.M Fox: Optical properties of solids S.J Blundell: Magnetism in condensed matter J.F Annett: Superconductivity, superfluids, and condensates R.A.L Jones: Soft condensed matter ATOMIC, OPTICAL, AND LASER PHYSICS 15 C.J Foot: Atomic physics G.A Brooker: Modern classical optics S.M Hooker, C.E Webb: Laser physics A.M Fox: Quantum optics: an introduction PARTICLE PHYSICS, ASTROPHYSICS, AND COSMOLOGY 10 D.H Perkins: Particle astrophysics 11 Ta-Pei Cheng: Relativity, gravitation and cosmology STATISTICAL, COMPUTATIONAL, AND THEORETICAL PHYSICS 12 M Maggiore: A modern introduction to quantum field theory 13 W Krauth: Statistical mechanics: algorithms and computations 14 J.P Sethna: Statistical mechanics: entropy, order parameters, and complexity CuuDuongThanCong.com Statistical Mechanics Algorithms and Computations Werner Krauth Laboratoire de Physique Statistique, Ecole Normale Sup´erieure, Paris CuuDuongThanCong.com Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Oxford University Press 2006 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Printed in Great Britain on acid-free paper by CPI Antony Rowe, Chippenham, Wilts ISBN 0–19–851535–9 (Hbk) ISBN 0–19–851536–7 (Pbk) 10 CuuDuongThanCong.com 978–0–19–851535–7 9780198515364 Fă ur Silvia, Alban und Felix CuuDuongThanCong.com This page intentionally left blank CuuDuongThanCong.com Preface This book is meant for students and researchers ready to plunge into statistical physics, or into computing, or both It has grown out of my research experience, and out of courses that I have had the good fortune to give, over the years, to beginning graduate students at the Ecole Normale Sup´erieure and the Universities of Paris VI and VII, and also to summer school students in Drakensberg, South Africa, undergraduates in Salem, Germany, theorists and experimentalists in Lausanne, Switzerland, young physicists in Shanghai, China, among others Hundreds of students from many different walks of life, with quite different backgrounds, listened to lectures and tried to understand, made comments, corrected me, and in short helped shape what has now been written up, for their benefit, and for the benefit of new readers that I hope to attract to this exciting, interdisciplinary field Many of the students sat down afterwards, by themselves or in groups, to implement short programs, or to solve other problems With programming assignments, lack of experience with computers was rarely a problem: there were always more knowledgeable students around who would help others with the first steps in computer programming Mastering technical coding problems should also only be a secondary problem for readers of this book: all programs here have been stripped to the bare minimum None exceed a few dozen lines of code We shall focus on the concepts of classical and quantum statistical physics and of computing: the meaning of sampling, random variables, ergodicity, equidistribution, pressure, temperature, quantum statistical mechanics, the path integral, enumerations, cluster algorithms, and the connections between algorithmic complexity and analytic solutions, to name but a few These concepts built the backbone of my courses, and now form the tissue of the book I hope that the simple language and the concrete settings chosen throughout the chapters take away none of the beauty, and only add to the clarity, of the difficult and profound subject of statistical physics I also hope that readers will feel challenged to implement many of the programs Writing and debugging computer code, even for the naive programs, remains a difficult task, especially in the beginning, but it is certainly a successful strategy for learning, and for approaching the deep understanding that we must reach before we can translate the lessons of the past into our own research ideas This book is accompanied by a compact disc containing more than one hundred pseudocode programs and close to 300 figures, line drawings, CuuDuongThanCong.com viii Preface and tables contained in the book Readers are free to use this material for lectures and presentations, but must ask for permission if they want to include it in their own publications For all questions, please contact me at www.lps.ens.fr/ krauth (This website will also keep a list of misprints.) Readers of the book may want to get in contact with each other, and some may feel challenged to translate the pseudocode programs into one of the popular computer languages; I will be happy to assist initiatives in this direction, and to announce them on the above website CuuDuongThanCong.com Contents Monte Carlo methods 1.1 Popular games in Monaco 1.1.1 Direct sampling 1.1.2 Markov-chain sampling 1.1.3 Historical origins 1.1.4 Detailed balance 1.1.5 The Metropolis algorithm 1.1.6 A priori probabilities, triangle algorithm 1.1.7 Perfect sampling with Markov chains 1.2 Basic sampling 1.2.1 Real random numbers 1.2.2 Random integers, permutations, and combinations 1.2.3 Finite distributions 1.2.4 Continuous distributions and sample transformation 1.2.5 Gaussians 1.2.6 Random points in/on a sphere 1.3 Statistical data analysis 1.3.1 Sum of random variables, convolution 1.3.2 Mean value and variance 1.3.3 The central limit theorem 1.3.4 Data analysis for independent variables 1.3.5 Error estimates for Markov chains 1.4 Computing 1.4.1 Ergodicity 1.4.2 Importance sampling 1.4.3 Monte Carlo quality control 1.4.4 Stable distributions 1.4.5 Minimum number of samples Exercises References 3 15 21 22 24 27 27 29 33 35 37 39 44 44 48 52 55 59 62 62 63 68 70 76 77 79 Hard disks and spheres 2.1 Newtonian deterministic mechanics 2.1.1 Pair collisions and wall collisions 2.1.2 Chaotic dynamics 2.1.3 Observables 2.1.4 Periodic boundary conditions 2.2 Boltzmann’s statistical mechanics 2.2.1 Direct disk sampling 81 83 83 86 87 90 92 95 CuuDuongThanCong.com 328 Dynamic Monte Carlo methods zk+1 zk xk+1 xk construction is to build a cone section model that we can imagine cut out in paper and glued together inside the unit sphere (see Figs 7.21 and 7.22) Hexagonally close-packed configurations are drawn on the paper strips, and centers of disks that not cut a boundary (the gray disks in Fig 7.22) are projected onto the sphere As the disks get smaller, fewer and fewer of them are on the boundaries (the white disks in the figure), and the packing fraction on the strip approaches the hexagonal close-packing density Moreover, as the strips get smaller, their total area approaches the area of the sphere It suffices to let the disks decrease in size more quickly than the strips to reach ηmax (See Exerc 7.13 for a paper-cutting competition.) Fig 7.21 Cone section on the inside of the unit sphere, formed by one of the strips in Fig 7.22 Fig 7.22 Cone section model consisting of close-packed strips of equal width, to be glued together and assembled inside the unit sphere The area of one strip in Fig 7.22 is area (one strip) = 2Ô xk+1 + xk average strip length (zk+1 − zk )2 + (xk+1 − xk )2 strip width It is better to use angles φk , where xk = cos φk and zk = sin φk (φk+1 − φk = Ô/(2n)) The strip width is sin [Ô/(4n)], and we find the following for the total area of a cone section model with 2n strips: Sn = 4Ô n−1 k=0 Ô (cos φk+1 + cos φk ) sin 4n (where φk = k Ô/(2n)), which may be rewritten and expanded in terms CuuDuongThanCong.com 7.3 Disks on the unit sphere 329 of 1/n: Sn = 4Ô n−1 cos k=0 Ô (2k + 1)Ô sin 4n 2n 1− Ô2 + ··· 32n2 sin−1 [π/(4n)] As the disks get smaller (in the large-N limit), the fraction of cut-up disks that cannot be transferred from the strips to the sphere goes to zero We may suppose the disks to have identical size on the strip and on the surface of the sphere, and then the packing density on a cone section model with only 12 strips (n = 6) already approaches 99% of the close-packing density in the plane (see Table 7.3) We now increase the number of strips, in order to reach 100% of the close-packing density This obliges us to think about the strip boundaries, and to exclude a zone of width 2r from the strip The total length of the boundaries is proportional to n, and we find free area with 2n strips = 4Ô − c − c · nr n2 This free area is maximized for n = c /r1/3 ; it varies according to free area with ∝ r−1/3 strips = 4Ô − const · r2/3 (7.8) Equation (7.8) proves that the disposable area, in the limit r → 0, goes to 4Ô, although this limit is reached extremely slowly, and at the price of introducing a large number of grain boundaries separating regions with mutually incompatible hexagonal close-packed crystals Nevertheless, in the large-N limit, these crystallites contain infinitely many disks The conclusion of our paper-cutting exercise is that the best packing of N disks on the unit sphere, in the limit N → ∞, reaches the hexagonal close-packing density of disks in the plane Furthermore, this optimal packing density grows very slowly with N From eqn (7.8), it follows that the density of the homogeneous planar system is approached as 1− const N 1/3 The best packing thus has long grain boundaries, regions where the hexagonal ordering is perturbed The number of grains increases with N , but less than proportionally, so that, in the limit N → ∞, the grains contain more and more particles Their extension, measured in multiples of the disk radius, diverges Evidently, however, we not expect the grain boundaries of the true optimal packing of N disks to form concentric circles around the z-axis In the thermodynamic limit, packing densities on a sphere and in a plane thus become equivalent In retrospect, however, we were welladvised in Chapter to study the liquid–solid phase transition with periodic boundary conditions on an abstract torus, rather than on a CuuDuongThanCong.com Table 7.3 Surface area Sn of a cone section model with 2n strips compared with the surface area of the unit sphere n Sn /(4Ô) − Ô2 /(32n2 ) 0.70711 0.92388 0.96593 0.99144 0.69157 0.92289 0.96573 0.99143 330 Dynamic Monte Carlo methods sphere: in that case, the system sizes available for simulation are too small to capture the difference between a polycrystalline material and an amorphous block of matter 7.3.3 rk rl θk θl Polydisperse disks and the glass transition Unerringly, the simulated-annealing algorithm in Section 7.3.1 reaches the globally optimal solution, sidestepping the many local minima on its way Notwithstanding this success, we must understand that simulated annealing is more a great “first try” than a prodigious scout of ground states and a true solver of general optimization problems To see this in an example, we simply consider unequal (polydisperse) disks on the surface of the unit sphere instead of the equal disks studied so far (see Fig 7.23) For concreteness, let us assume that disk k has an opening angle (7.9) θk = θ · (1 + δk ) , so that the density is equal to N Fig 7.23 Polydisperse disks (with central vectors xk and xl , and opening angles θk and θl ) on the surface of the unit sphere Disks overlap if their scalar product (xk · xl ) is greater than cos (θk + θl ) η= k=1 − cos θk The optimization problem, a generalization of eqn (7.6), now consists in maximizing θ for a fixed ratio of the opening angles Disks k and l overlap if arccos (xk · xl ) > θ · (2 + δk + δl ), and we must solve the following optimization problem: max {x1 , ,xN } |x1 |=···=|xN |=1 k