A collection of the majority of papers presented at the all German workshop 'Spatial Physics and Spatial Statistics,' held at the University of Wuppertal, February 22-24, 1999. Each of these papers present and use geometric concepts to study random spatial configurations.
Author(s): Sylvie Huet, Anne Bouvier, Marie-Anne Poursat, Emmanuel Jolivet
Series: Springer Series in Statistics
Edition: 1
Publisher: Springer
Year: 2000
Language: English
Pages: 419
LNP 554:
Statistical Physics and Spatial Statistics......Page 1
Preface
......Page 5
Contents
......Page 11
1 Introduction......Page 13
2 Stationarity and Isotropy......Page 16
3 Three Stationary Stochastic Models......Page 17
4 Edge Problems......Page 22
5 Gibbs Point Processes......Page 24
6 Statistical Tests......Page 27
Acknowledgement......Page 29
References......Page 30
1 Introduction......Page 32
2 Quantitative Analysis ofIrregular Geometric Structures......Page 33
3 Stationary Models for Random Geometric Structures......Page 34
4 The Intensit and the Campbell Theorem......Page 35
5 The Palm Distribution
and the Re .ned Campbell Theorem......Page 36
6 Marked Point Processes......Page 38
7 Application to Random Tessellations ofthe Plane......Page 40
References......Page 45
1 Introduction......Page 46
2 Cosmological Models and Observations......Page 47
3.1 Two –point Statistics......Page 53
3.2 Higher Moments......Page 55
3.3 Minkowski Functionals......Page 58
3.4 The J Function......Page 68
4 Summary and Outlook......Page 74
References......Page 75
1 Introduction......Page 82
2.1 Early Stage of Dewetting:Formation of Holes......Page 83
2.2 Late Stage of Dewetting:Growth of Holes......Page 86
3 Conclusions and Outlook......Page 88
4 Acknowledgements......Page 89
References......Page 99
1 Motivation......Page 102
2 Geometric Functionals......Page 106
3 Integral Geometry......Page 108
4 Boolean Models......Page 110
References......Page 115
1 Motivation:Complex Patterns in Statistical Physics......Page 118
2 Geometric Functionals:Curvatures in Physics......Page 122
2.1 Curvature Energy of Membranes......Page 126
2.2 Capillary Condensation of Fluids in Porous Media......Page 127
2.3 Spectral Density of the Laplace Operator......Page 132
3 Morphology:Characterization ofSpatial Structures......Page 133
3.1 Minkowski Functions as Order Parameters......Page 134
3.2 Minkowski Functions as Dynamical Quantities......Page 138
3.3 Minkowski Functions versus Structure Functions......Page 141
3.4 Minkowski Functions of Fractals:Scaling......Page 145
4 Integral Geometry:Statistical Physics ofFluids......Page 148
4.1 Kinematic Formula:Cluster Expansion and Density Functional Theory......Page 149
4.2 Excluded Volumes:Percolation Thresholds......Page 151
4.3 Additivity and Completeness: Curvature Model for Complex Fluids......Page 156
5 Boolean Model:Thermal Averages......Page 158
5.1 Widom-Rowlinson Model and Density Functional Theory......Page 159
5.2 Correlated,Inhomogeneous and Anisotropic Distributions......Page 166
5.3 Second-Order Moments of Minkowski Measures......Page 178
5.4 Lattices:Models for Complex Fluids and Percolation......Page 181
References......Page 189
1 Introduction......Page 192
2 The Stereological Equation......Page 196
3 Numerical Solutions ofthe Stereological Equation......Page 198
3.2 Repeated Trapezoidal Quadrature Rule......Page 199
3.3 Application of the EM Algorithm......Page 200
4.1 The Condition Number of the Operator P......Page 202
4.2 The Optimal Discretization Parameter......Page 204
References......Page 205
Appendix......Page 207
1 Introduction......Page 210
2.1 Physical Problems......Page 211
2.2 Geometrical Problems......Page 212
3.2 Geometric Observables......Page 213
3.3 Definition of Stochastic Porous Media......Page 216
3.4 Moment Functions and Correlation Functions......Page 217
3.6 Local Porosity Distributions......Page 219
3.7 Local Percolation Probabilities......Page 223
4.1 Description of Experimental Sample......Page 225
4.2 Sedimentation,Compaction and Diagenesis Model......Page 227
4.3 Gaussian Field Reconstruction Model......Page 229
4.4 Simulated Annealing Reconstruction Model......Page 231
5.1 Conventional Observables and Correlation Functions......Page 235
5.2 Local Porosity Analysis......Page 236
5.3 Local Percolation Analysis......Page 240
6.1 Exact Results......Page 242
6.2 Mean Field Results......Page 243
References......Page 245
1 Introduction......Page 249
2 The Boolean Model and Crystallisation Kinetics......Page 253
3.1 Microstructure and Nucleation Rate......Page 255
3.2 Growth Rate of Crystallites......Page 259
3.3 Transformed Volume Fraction......Page 260
3.4 Discussion and Application......Page 261
4 Random Tessellations......Page 264
4.1 Poisson Line and Plane Tessellations......Page 265
4.2 Voronoi and Johnson-Mehl Tessellations......Page 267
References......Page 268
1 Gibbs Measures:General Principles......Page 272
2 Phase Transition and Percolation:Two Lattice Models......Page 276
2.1 The Ising Model......Page 277
2.2 The Widom –Rowlinson Lattice Gas......Page 280
3 Continuum Percolation......Page 282
4 The Continuum Ising Model......Page 285
4.1 Uniqueness and Phase Transition......Page 286
4.2 Thermodynamic Aspects......Page 289
4.3 Projection on Plus-Particles......Page 292
4.4 Simulation......Page 293
References......Page 298
2 Introduction:The Model......Page 300
3.2 One-Dimensional Case......Page 304
3.5 Degenerate Cases......Page 305
4.1 Intuitive Arguments......Page 306
4.3 Percus-Yevick Theory......Page 307
4.4 Estimation of the Freezing Transition......Page 309
4.6 Density Functional Theory......Page 310
5 Hard Spheres and Phase Transitions: Computer Simulations......Page 313
6.1 Binary Mixtures......Page 316
6.3 Hard Spheres near Hard Plates......Page 317
6.4 Hard Spherocylinders......Page 318
Acknowledgements......Page 319
References......Page 332
1 Introduction......Page 337
2 Densit Definitions......Page 338
3 Cell Complexes in the Plane......Page 340
4 Finite Packings in the Plane......Page 342
5 Finite Coverings in the Plane......Page 345
6 Large Packings of Spheres......Page 347
7 Facet Densities......Page 348
8 Wulff Shapes......Page 349
References......Page 351
1 Introduction......Page 354
2 Conventional Markov Chain Monte Carlo......Page 355
3 Coupling......Page 356
3.1 Random Walks......Page 357
3.2 The Ising Model......Page 361
3.3 Immigration-Death Process......Page 364
3.4 Forward Coupling and Exact Sampling......Page 367
4 Coupling from the Past......Page 368
5 Dominated Coupling from the Past......Page 373
6 Perfection in Space......Page 379
Acknowledgement......Page 381
References......Page 382
1 Introduction......Page 384
2 Grand Canonical Simulations for Hard-Disk Systems......Page 387
3 Algorithms......Page 388
3.1 Basic Algorithm......Page 389
3.2 Simulated Tempering......Page 390
4 Results......Page 391
4.1 Area Fraction......Page 392
4.2 Pair Correlation Function......Page 393
4.3 Alignment Function......Page 394
4.4 Hexagonality Number......Page 396
References......Page 397
2 Simulation ofGranular Media......Page 399
3.1 Dynamic Triangulations......Page 400
3.2 Distinct Element Modeling and Simulation......Page 405
3.3 The Physical Models......Page 406
3.5 Examples......Page 407
4 When the Grains Are Polygonal......Page 409
4.1 In 2D......Page 410
4.2 In 3D......Page 411
References......Page 413
Index
......Page 415
