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Mathematics and Materials IAS/PARK CITY MATHEMATICS SERIES Volume 23 Mathematics and Materials Mark J. Bowick David Kinderlehrer Govind Menon Charles Radin Editors American Mathematical Society Institute for Advanced Study Society for Industrial and Applied Mathematics Mathematics and Materials IAS/PARK CITY MATHEMATICS SERIES Volume 23 Mathematics and Materials Mark J. Bowick David Kinderlehrer Govind Menon Charles Radin Editors American Mathematical Society Institute for Advanced Study Society for Industrial and Applied Mathematics Rafe Mazzeo, Series Editor Mark J. Bowick, David Kinderlehrer, Govind Menon, and Charles Radin, Volume Editors. IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program 2010 Mathematics Subject Classification. Primary 82B05, 35Q70, 82B26, 74N05, 51P05, 52C17, 52C23. Library of Congress Cataloging-in-Publication Data Names: Bowick, Mark J., editor. | Kinderlehrer, David, editor. | Menon, Govind, 1973– editor. | Radin, Charles, 1945– editor. | Institute for Advanced Study (Princeton, N.J.) | Society for Industrial and Applied Mathematics. Title: Mathematics and materials / Mark J. Bowick, David Kinderlehrer, Govind Menon, Charles Radin, editors. Description: [Providence] : American Mathematical Society, [2017] | Series: IAS/Park City math- ematics series ; volume 23 | “Institute for Advanced Study.” | “Society for Industrial and Applied Mathematics.” | “This volume contains lectures presented at the Park City summer school on Mathematics and Materials in July 2014.” – Introduction. | Includes bibliographical references. Identifiers: LCCN 2016030010 | ISBN 9781470429195 (alk. paper) Subjects: LCSH: Statistical mechanics–Congresses. | Materials science–Congresses. | AMS: Sta- tistical mechanics, structure of matter – Equilibrium statistical mechanics – Classical equilibrium statistical mechanics (general). msc | Partial differential equations – Equations of mathematical physics and other areas of application – PDEs in connection with mechanics of particles and sys- tems. msc | Statistical mechanics, structure of matter – Equilibrium statistical mechanics – Phase transitions (general). msc | Mechanics of deformable solids – Phase transformations in solids – Crystals. msc | Geometry – Geometry and physics (should also be assigned at least one other classification number from Sections 70–86) – Geometry and physics (should also be assigned at least one other classification number from Sections 70–86). msc | Convex and discrete geometry – Discrete geometry – Packing and covering in n dimensions. msc | Convex and discrete geometry – Discrete geometry – Quasicrystals, aperiodic tilings. msc Classification: LCC QC174.7 .M38 2017 | DDC 530.13–dc23 LC record available at https://lccn. loc.gov/2016030010 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Contents Preface ix Introduction xi Veit Elser Three Lectures on Statistical Mechanics 1 Lecture 1. Mechanical foundations 3 Lecture 2. Temperature and entropy 15 Lecture 3. Macroscopic order 25 Bibliography 41 Henry Cohn Packing, Coding, and Ground States 43 Preface 45 Lecture 1. Sphere packing 47 1. Introduction 47 2. Motivation 48 3. Phenomena 50 4. Constructions 52 5. Difficulty of sphere packing 54 6. Finding dense packings 55 7. Computational problems 57 Lecture 2. Symmetry and ground states 59 1. Introduction 59 2. Potential energy minimization 60 3. Families and universal optimality 61 4. Optimality of simplices 65 Lecture 3. Interlude: Spherical harmonics 69 1. Fourier series 69 2. Fourier series on a torus 71 3. Spherical harmonics 73 Lecture 4. Energy and packing bounds on spheres 77 1. Introduction 77 2. Linear programming bounds 79 3. Applying linear programming bounds 81 v vi CONTENTS 4. Spherical codes and the kissing problem 82 5. Ultraspherical polynomials 83 Lecture 5. Packing bounds in Euclidean space 89 1. Introduction 89 2. Poisson summation 91 3. Linear programming bounds 92 4. Optimization and conjectures 94 Bibliography 99 Alpha A. Lee and Daan Frenkel Entropy, Probability and Packing 103 Introduction 105 Lecture 1. Introduction to thermodynamics and statistical physics 107 1. Classical equilibrium thermodynamics 107 2. Statistical physics of entropy 113 3. From entropy to thermodynamic ensembles 119 4. Exercises 123 Lecture 2. Thermodynamics of phase transitions 125 1. Thermodynamics of phase equilibrium 125 2. Thermodynamic integration 126 3. The chemical potential and Widom particle insertion 130 4. Exercises 133 Lecture 3. Order from disorder: Entropic phase transitions 135 1. Hard-sphere freezing 136 2. Role of geometry: The isotropic-nematic transition 140 3. Depletion interaction and the entropy of the medium 145 4. Attractive forces and the liquid phase 151 5. Exercises 154 Lecture 4. Granular entropy 157 1. Computing the entropy 158 2. Is this “entropy” physical? 160 3. The Gibbs paradox 161 Bibliography 165 MichaelP.Brenner Ideas about Self Assembly 167 Introduction 169 Lecture 1. Self assembly: Introduction 171 1. What is self-assembly 171 2. Statistical mechanical preliminaries 172 CONTENTS vii Lecture 2. Self assembly with identical components 175 1. Introduction 175 2. The (homogeneous) polymer problem 176 3. Cluster statistical mechanics 179 Lecture 3. Heterogeneous self assembly 187 1. Heteropolymer problem 187 2. The yield catastrophe 189 3. Colloidal assembly 191 Lecture 4. Nucleation theory and multifarious assembly mixtures 195 1. Nucleation theory 195 2. Magic soup 196 Bibliography 199 P. Palffy-Muhoray, M. Pevnyi, E. Virga, and X. Zheng The Effects of Particle Shape in Orientationally Ordered Soft Materials 201 Introduction 203 Lecture 1. Soft condensed matter and orientational order 205 1. Soft condensed matter 205 2. Position and orientation 206 3. Orientational order parameters 210 Lecture 2. The free energy 213 1. Helmholtz free energy 214 2. Trial free energy 215 3. Configurational partition function 215 4. Pairwise interactions 215 5. Soft and hard potentials 216 6. Mean-field free energy 217 7. Density functional theory 219 Lecture 3. Particle shape and attractive interactions 227 1. Polarizability of a simple atom 228 2. Dispersion interaction 230 3. Polarizability of non-spherical particles 231 Lecture 4. Particle shape and repulsive interactions 239 1. Onsager theory 240 2. Excluded volume for ellipsoids 241 3. Phase separation 242 4. Minimum excluded volume of convex shapes 244 5. Systems of hard polyhedra 247 viii CONTENTS Summary 249 Bibliography 251 Roman Koteck´y Statistical Mechanics and Nonlinear Elasticity 255 Introduction 257 Lecture 1. Statistical mechanics 259 Statistical mechanics of interacting particles 259 Lattice models of nonlinear elasticity 261 Ising model 263 Lecture 2. Phase transitions 267 Existence of the free energy 267 Concavity of the free energy 268 Peierls argument 269 Lecture 3. Expansions 275 The high temperature expansion 275 Intermezzo (cluster expansions) 277 Proof of cluster expansion theorem 280 Lecture 4. Gradient models of random surface 283 Quadratic potential 283 Convex potentials 285 Non-convex potentials 286 Lecture 5. Nonlinear elasticity 291 Free energy 291 Macroscopic behaviour from microscopic model 293 Main ingredients of the proof of quasiconvexity of W 294 Bibliography 297 Peter Bella, Arianna Giunti, and Felix Otto Quantitative Stochastic Homogenization: Local Control of Homogenization Error through Corrector 299 1. A brief overview of stochastic homogenization, and a common vision for quenched and thermal noise 301 2. Precise setting and motivation for this work 304 3. Main results 307 4. Proofs 309 Bibliography 326 Preface The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the “Regional Geometry Institute” initiative of the National
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