ریاضیات و مواد

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 transition 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. Cover -- Title page -- Contents -- Preface -- Introduction -- Three Lectures on Statistical Mechanics (Veit Elser) -- Lecture 1. Mechanical foundations -- Laws, hypotheses, and models -- Soft billiards -- Quantum states -- Kinetic elasticity -- Statistical equilibrium -- Problems for study -- Snell's law for particles -- Tracer particle analysis of soft billiards -- Weyl's law -- Central limit analysis of polymer -- Slow modes of a polymer -- Lecture 2. Temperature and entropy -- The thermodynamic limit -- A model for the number of states -- Temperature -- The Boltzmann distribution -- Thermal averages -- Entropy -- Thermal elasticity -- Free energy -- Thermal equilibrium -- Problems for study -- Lecture 3. Macroscopic order -- Macroscopic manifestations of microscopic order -- Hard spheres: microscopic order -- Hard spheres: macroscopic order -- Order in the height model -- Random tilings -- Problems for study -- Bibliography -- Packing, Coding, and Ground States (Henry Cohn) -- Preface -- Acknowledgments -- Lecture 1. Sphere packing -- 1. Introduction -- 2. Motivation -- 3. Phenomena -- 4. Constructions -- 5. Difficulty of sphere packing -- 6. Finding dense packings -- 7. Computational problems -- Lecture 2. Symmetry and ground states -- 1. Introduction -- 2. Potential energy minimization -- 3. Families and universal optimality -- 4. Optimality of simplices -- Lecture 3. Interlude: Spherical harmonics -- 1. Fourier series -- 2. Fourier series on a torus -- 3. Spherical harmonics -- Lecture 4. Energy and packing bounds on spheres -- 1. Introduction -- 2. Linear programming bounds -- 3. Applying linear programming bounds -- 4. Spherical codes and the kissing problem -- 5. Ultraspherical polynomials -- Lecture 5. Packing bounds in Euclidean space -- 1. Introduction -- 2. Poisson summation -- 3. Linear programming bounds Three Lectures On Statistical Mechanics / Veit Elser -- Packing, Coding, And Ground States / Henry Cohn -- Entropy, Probability, And Packing / Alpha A. Lee And Daan Frenkel -- Ideas About Self Assembly / Michael P. Brenner -- The Effects Of Particle Shape In Orientationally Ordered Soft Materials / P. Palffy-muhoray, M. Pevnyi, E. Virga, X. Zheng -- Statistical Mechanics And Non-linear Elasticity / Roman Kotecký -- Quantitative Stochastic Homogeneization: Local Control Of Homogeneization Error Through Corrector / Peter Bella, Arianna Giunti, And Felix Otto. Mark J. Bowick, David Kinderlehrer, Govind Menon, Charles Radin, Editors. 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. Content: Three lectures on statistical mechanics / Veit Elser -- Packing, coding, and ground states / Henry Cohn -- Entropy, probability, and packing / Alpha A. Lee and Daan Frenkel -- Ideas about self assembly / Michael P. Brenner -- The effects of particle shape in orientationally ordered soft materials / P. Palffy-Muhoray, M. Pevnyi, E. Virga, X. Zheng -- Statistical mechanics and non-linear elasticity / Roman Kotecký -- Quantitative stochastic homogeneization: local control of homogeneization error through corrector / Peter Bella, Arianna Giunti, and Felix Otto. Examines material behaviour 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. Contributions to this volume present unique pedagogical introductions to the rich variety of material behaviour that emerges from the interplay of geometry and statistical mechanics.