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 10.1090/pcms/023

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

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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 Science Foundation. In mid-1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS/Park City Mathematics Institute encourages both research and ed- ucation in mathematics and fosters interaction between the two. The three-week summer institute offers programs for researchers and postdoctoral scholars, gradu- ate students, undergraduate students, high school teachers, undergraduate faculty, and researchers in mathematics education. One of PCMI’s main goals is to make all of the participants aware of the total spectrum of activities that occur in math- ematics education and research. We wish to involve professional mathematicians in education and to bring modern concepts in mathematics to the attention of educators. To that end, the summer institute features general sessions designed to encourage interaction among the various groups. In-year activities at the sites around the country form an integral part of the High School Teachers Program. Each summer a different topic is chosen as the focus of the Research Program and Graduate Summer School. Activities in the Undergraduate Summer School deal with this topic as well. Lecture notes from the Graduate Summer School are being published each year in this series. The first twenty-three volumes are:

• Volume 1: Geometry and Quantum Field Theory (1991) • Volume 2: Nonlinear Partial Differential Equations in Differential Geom- etry (1992) • Volume 3: Complex Algebraic Geometry (1993) • Volume 4: Gauge Theory and the Topology of Four-Manifolds (1994) • Volume 5: Hyperbolic Equations and Frequency Interactions (1995) • Volume 6: Probability Theory and Applications (1996) • Volume 7: Symplectic Geometry and Topology (1997) • Volume 8: Representation Theory of Lie Groups (1998) • Volume 9: Arithmetic Algebraic Geometry (1999) • Volume 10: Computational Complexity Theory (2000) • Volume 11: Quantum Field Theory, Supersymmetry, and Enumerative Geometry (2001) • Volume 12: Automorphic Forms and their Applications (2002) • Volume 13: Geometric Combinatorics (2004) • Volume 14: Mathematical Biology (2005) • Volume 15: Low Dimensional Topology (2006) • Volume 16: Statistical Mechanics (2007)

ix xPREFACE

• Volume 17: Analytic and Algebraic Geometry: Common Problems, Dif- ferent Methods (2008) • Volume 18: Arithmetic of L-functions (2009) • Volume 19: Mathematics in Image Processing (2010) • Volume 20: Moduli Spaces of Riemann Surfaces (2011) • Volume 21: Geometric Group Theory (2012) • Volume 22: Geometric Analysis (2013) • Volume 23: Mathematics and Materials (2014) Volumes are in preparation for subsequent years. Some material from the Undergraduate Summer School is published as part of the Student Mathematical Library series of the American Mathematical Society. We hope to publish material from other parts of the IAS/PCMI in the future. This will include material from the High School Teachers Program and publications documenting the interactive activities that are a primary focus of the PCMI. At the summer institute late afternoons are devoted to seminars of common interest to all participants. Many deal with current issues in education: others treat mathematical topics at a level which encourages broad participation. The PCMI has also spawned interactions between universities and high schools at a local level. We hope to share these activities with a wider audience in future volumes.

Rafe Mazzeo Director, PCMI March 2016 Introduction

This volume contains lectures presented at the Park City summer school on “Mathematics and Materials” in July 2014. The central theme is a description of material behavior that is rooted in statistical mechanics. While many presentations of mathematical problems in materials science begin with continuum mechanics, these lectures present an alternate view. A rich variety of material properties is shown to emerge from the interplay between geometry and statistical mechanics. The school included approximately eighty graduate students and forty research- ers from many areas of mathematics and the sciences. This interdisciplinary spirit is reflected in a diverse set of perspectives on the order-disorder transition in many geometric models of materials, including nonlinear elasticity, sphere packings, gran- ular materials, liquid crystals, and the emerging field of synthetic self-assembly. The lecturers for the school, and the topics of their lectures, were as follows:

(1) Michael Brenner, School of Engineering and Applied Sciences, Harvard University: Ideas about self-assembly. (2) Henry Cohn, Microsoft Research: Packing, coding and ground states. (3) Veit Elser, Department of Physics, Cornell University: Three lectures on statistical mechanics. (4) Daan Frenkel, Department of Chemistry, University of Cambridge: En- tropy, probability and packing. (5) Richard D. James, Department of Aerospace Engineering and Mechanics, University of Minnesota: Phase transformations, hysteresis and energy conversion –the role of geometry in the discovery of materials. (6) Robert V. Kohn, Courant Institute, New York University: Wrinkling of thin elastic sheets. (7) Roman Koteck´y, Mathematics Institute, University of Warwick: Statisti- cal mechanics and nonlinear elasticity. (8) Peter Palffy-Muhoray, Department of Chemical Physics, Kent State Uni- versity: The effects of particle shape in orientationally ordered soft mate- rials.

In addition, L. Mahadevan (Harvard University) and Felix Otto (MPI, Leipzig) were in residence for the program as Clay Senior Scholars, and gave well-received public lectures. All the lectures in this volume contain unique pedagogical introductions to a variety of topics of current interest. Several lectures touch on the interplay between discrete geometry (especially packing) and statistical mechanics. These problems have an immediate mathematical appeal and are of increasing importance in ap- plications, but are not as widely known as they should be to mathematicians with

xi xii Introduction an interest in materials science. Both Elser and Frenkel present elegant introduc- tions to statistical mechanics from the physicist’s perspective, with an emphasis on the interplay between entropy and packings. This theme is repeated in Cohn’s lectures, which reveal the role of unexpected mathematical tools in simply stated problems about symmetric ground states. Similarly, Brenner uses discrete geome- try and statistical mechanics to model exciting new experiments on synthetic self- assembly. Palffy-Muhoray uses statistical mechanics to derive several models for liquid crystals, exploring again the interplay between shape and statistical mechan- ics. Koteck´y’s lecture contains a mathematical introduction to statistical mechan- ics, with a focus on the foundations of nonlinear elasticity. The volume also includes an account of recent work on correctors in stochastic homogenization by Otto and his co-workers. Regrettably, this volume does not contain the texts of excellent lectures by James on solid-solid phase transitions, and by Kohn on the elasticity of thin sheets. We express our thanks to the PCMI steering committe, especially Richard Hain, John Polking and Ronald Stern, for their support of the program; to Cather- ine Giesbrecht and Dena Vigil for invaluable help with organization; and to Rafe Mazzeo for his assistance in bringing this volume to publication. Finally, we ex- press our thanks to the lecturers, students and researchers for their enthusiastic participation in the summer school.

Mark Bowick, David Kinderlehrer, Govind Menon, and Charles Radin PUBLISHED TITLES IN THIS SERIES

23 Mark J. Bowick, David Kinderlehrer, Govind Menon, and Charles Radin, Editors, Mathematics and Materials, 2017 22 Hubert L. Bray, Greg Galloway, Rafe Mazzeo, and Natasa Sesum, Editors, Geometric Analysis, 2016 21 Mladen Bestvina, Michah Sageev, and Karen Vogtmann, Editors, Geometric Group Theory, 2014 20 Benson Farb, Richard Hain, and Eduard Looijenga, Editors, Moduli Spaces of Riemann Surfaces, 2013 19 Hongkai Zhao, Editor, Mathematics in Image Processing, 2013 18 Cristian Popescu, Karl Rubin, and Alice Silverberg, Editors, Arithmetic of L-functions, 2011 17 Jeffery McNeal and Mircea Mustat¸˘a, Editors, Analytic and Algebraic Geometry, 2010 16 Scott Sheffield and Thomas Spencer, Editors, Statistical Mechanics, 2009 15 Tomasz S. Mrowka and Peter S. Ozsv´ath, Editors, Low Dimensional Topology, 2009 14 Mark A. Lewis, Mark A. J. Chaplain, James P. Keener, and Philip K. Maini, Editors, Mathematical Biology, 2009 13 Ezra Miller, Victor Reiner, and Bernd Sturmfels, Editors, Geometric Combinatorics, 2007 12 Peter Sarnak and Freydoon Shahidi, Editors, Automorphic Forms and Applications, 2007 11 Daniel S. Freed, David R. Morrison, and Isadore Singer, Editors, Quantum Field Theory, Supersymmetry, and Enumerative Geometry, 2006 10 Steven Rudich and Avi Wigderson, Editors, Computational Complexity Theory, 2004 9 Brian Conrad and Karl Rubin, Editors, Arithmetic Algebraic Geometry, 2001 8 Jeffrey Adams and David Vogan, Editors, Representation Theory of Lie Groups, 2000 7 Yakov Eliashberg and Lisa Traynor, Editors, Symplectic Geometry and Topology, 1999 6 EltonP.HsuandS.R.S.Varadhan,Editors, Probability Theory and Applications, 1999 5 Luis Caffarelli and Weinan E, Editors, Hyperbolic Equations and Frequency Interactions, 1999 4 Robert Friedman and John W. Morgan, Editors, Gauge Theory and the Topology of Four-Manifolds, 1998 3 J´anos Koll´ar, Editor, Complex Algebraic Geometry, 1997 2 Robert Hardt and Michael Wolf, Editors, Nonlinear partial differential equations in differential geometry, 1996 1 Daniel S. Freed and Karen K. Uhlenbeck, Editors, Geometry and Quantum Field Theory, 1995 Articles in this volume are based on lectures presented at the Park City summer school on “Mathematics and Materials” in July 2014. The central theme is a description of material behavior that is rooted in statistical mechanics. While many presentations of mathematical problems in materials science begin with continuum mechanics, this volume takes an alternate approach. All the lectures present unique pedagogical introductions to the rich variety of material behavior that emerges from the interplay of geometry and statistical mechanics. The topics include the order-disorder transi- tion in many geometric models of materials including nonlinear elasticity, sphere packings, granular materials, liquid crystals, and the emerging field of synthetic self-assembly. Several lectures touch on discrete geometry (especially packing) and statistical mechanics. The problems discussed in this book have an immediate mathematical appeal and are of increasing importance in applications, but are not as widely known as they should be to mathematicians interested in materials science. The volume will be of interest to graduate students and researchers in analysis and partial differential equations, continuum mechanics, condensed matter physics, discrete geometry, and mathematical physics.

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