Morphology of Experimental and Simulated Turing Patterns

5.5 Minkowski-maps of patterns in the CIMA reaction . . . . . . . . . . 725.6 Minkowski analysis of the CDIMA reaction (De Kepper et al.) . . . 755.7 Minkowski functionals of patterns in other reaction-diffusion models 786 Influence of additive noise on pattern formation in the LE model 816.1 Influence on pattern stability . . . . . . . . . . . . . . . . . . . . . . 816.2 Influence on convergence . . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Influence of noise on pattern morphology . . . . . . . . . . . . . . . 827 Statistical ensemble of superposed basic patterns 877.1 A simple model with superposition of patterns . . . . . . . . . . . . 877.2 Minkowski analysis for superposed patterns . . . . . . . . . . . . . . 907.2.1 Hexagonal state . . . . . . . . . . . . . . . . . . . . . . . . . 907.2.2 Lamellar state . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2.3 Turbulent state . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3 Interacting pattern gas . . . . . . . . . . . . . . . . . . . . . . . . . . 977.3.1 Derivation of the interaction energy . . . . . . . . . . . . . . 977.3.2 Second order approximation . . . . . . . . . . . . . . . . . . . 987.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 997.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 Summary and outlook 1039 Pattern formation in granular systems 1059.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059.2 Measurements and observed patterns . . . . . . . . . . . . . . . . . . 106A Details on experimental datasets 10910