Mathematics and Materials

Cover......Page 1
Title page......Page 4
Contents......Page 6
Preface......Page 10
Introduction......Page 12
Three Lectures on Statistical Mechanics (Veit Elser)......Page 14
Soft billiards......Page 16
Quantum states......Page 20
Kinetic elasticity......Page 21
Statistical equilibrium......Page 25
Tracer particle analysis of soft billiards......Page 26
Slow modes of a polymer......Page 27
A model for the number of states......Page 28
Temperature......Page 29
Thermal averages......Page 31
Entropy......Page 32
Thermal elasticity......Page 33
Thermal equilibrium......Page 35
Problems for study......Page 36
Hard spheres: microscopic order......Page 38
Hard spheres: macroscopic order......Page 42
Order in the height model......Page 44
Random tilings......Page 46
Problems for study......Page 51
Bibliography......Page 54
Packing, Coding, and Ground States (Henry Cohn)......Page 56
Acknowledgments......Page 58
1. Introduction......Page 60
2. Motivation......Page 61
3. Phenomena......Page 63
4. Constructions......Page 65
5. Difficulty of sphere packing......Page 67
6. Finding dense packings......Page 68
7. Computational problems......Page 70
1. Introduction......Page 72
2. Potential energy minimization......Page 73
3. Families and universal optimality......Page 74
4. Optimality of simplices......Page 78
1. Fourier series......Page 82
2. Fourier series on a torus......Page 84
3. Spherical harmonics......Page 86
1. Introduction......Page 90
2. Linear programming bounds......Page 92
3. Applying linear programming bounds......Page 94
4. Spherical codes and the kissing problem......Page 95
5. Ultraspherical polynomials......Page 96
1. Introduction......Page 102
2. Poisson summation......Page 104
3. Linear programming bounds......Page 105
4. Optimization and conjectures......Page 107
Bibliography......Page 112
Entropy, Probability and Packing (Alpha A Lee, Daan Frenkel)......Page 116
Introduction......Page 118
1. Classical equilibrium thermodynamics......Page 120
2. Statistical physics of entropy......Page 126
3. From entropy to thermodynamic ensembles......Page 132
4. Exercises......Page 136
1. Thermodynamics of phase equilibrium......Page 138
2. Thermodynamic integration......Page 139
3. The chemical potential and Widom particle insertion......Page 143
4. Exercises......Page 146
Lecture 3. Order from disorder: Entropic phase transitions......Page 148
1. Hard-sphere freezing......Page 149
2. Role of geometry: The isotropic-nematic transition......Page 153
3. Depletion interaction and the entropy of the medium......Page 158
4. Attractive forces and the liquid phase......Page 164
5. Exercises......Page 167
Lecture 4. Granular entropy......Page 170
1. Computing the entropy......Page 171
2. Is this “entropy” physical?......Page 173
3. The Gibbs paradox......Page 174
Bibliography......Page 178
Ideas about Self Assembly (Michael P. Brenner)......Page 180
Introduction......Page 182
1. What is self-assembly......Page 184
2. Statistical mechanical preliminaries......Page 185
1. Introduction......Page 188
2. The (homogeneous) polymer problem......Page 189
3. Cluster statistical mechanics......Page 192
1. Heteropolymer problem......Page 200
2. The yield catastrophe......Page 202
3. Colloidal assembly......Page 204
1. Nucleation theory......Page 208
2. Magic soup......Page 209
Bibliography......Page 212
The Effects of Particle Shape in Orientationally Ordered Soft Materials (P. Palffy-Muhoray, M. Pevnyi, E. G. Virga, and X. Zheng)......Page 214
Introduction......Page 216
1. Soft condensed matter......Page 218
2. Position and orientation......Page 219
3. Orientational order parameters......Page 223
Lecture 2. The free energy......Page 226
1. Helmholtz free energy......Page 227
4. Pairwise interactions......Page 228
5. Soft and hard potentials......Page 229
6. Mean-field free energy......Page 230
7. Density functional theory......Page 232
Lecture 3. Particle shape and attractive interactions......Page 240
1. Polarizability of a simple atom......Page 241
2. Dispersion interaction......Page 243
3. Polarizability of non-spherical particles......Page 244
Lecture 4. Particle shape and repulsive interactions......Page 252
1. Onsager theory......Page 253
2. Excluded volume for ellipsoids......Page 254
3. Phase separation......Page 255
4. Minimum excluded volume of convex shapes......Page 257
5. Systems of hard polyhedra......Page 260
Summary......Page 262
Bibliography......Page 264
Statistical Mechanics and Nonlinear Elasticity (Roman Kotecký)......Page 268
Introduction......Page 270
Statistical mechanics of interacting particles......Page 272
Lattice models of nonlinear elasticity......Page 274
Ising model......Page 276
Existence of the free energy......Page 280
Concavity of the free energy......Page 281
Peierls argument......Page 282
The high temperature expansion......Page 288
Intermezzo (cluster expansions)......Page 290
Proof of cluster expansion theorem......Page 293
Quadratic potential......Page 296
Convex potentials......Page 298
Non-convex potentials......Page 299
Free energy......Page 304
Macroscopic behaviour from microscopic model......Page 306
Main ingredients of the proof of quasiconvexity of ......Page 307
Bibliography......Page 310
Quantitative Stochastic Homogenization: Local Control of Homogenization Error through Corrector (Peter Bella, Arianna Giunti, Felix Otto)......Page 312
1. A brief overview of stochastic homogenization, and a common vision for quenched and thermal noise......Page 314
2. Precise setting and motivation for this work......Page 317
3. Main results......Page 320
4. Proofs......Page 322
Bibliography......Page 339
Back Cover......Page 342