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Random Johnson-Mehl tessellations

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Jesper Møller
Jesper Møller*
Affiliation:
University of Aarhus
*
∗Postal address: Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, DK-8000 Aarhus C, Denmark.
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Abstract

A unified exposition of random Johnson–Mehl tessellations in d-dimensional Euclidean space is presented. In particular, Johnson-Mehl tessellations generated by time-inhomogeneous Poisson processes and nucleation-exclusion models are studied. The ‘practical' cases d = 2 and d = 3 are discussed in detail. Several new results are established, including first- and second-order moments of various characteristics for both Johnson–Mehl tesselations and sectional Johnson–Mehl tessellations.

Keywords

CRYSTAL CHARACTERISTICSHARD CORE PROCESSESINHOMOGENEOUS POISSON PROCESSESMEAN VALUE RELATIONSNUCLEATION–EXCLUSION MODELPALM DISTRIBUTIONRANDOM TESSELLATIONSSECTIONAL TESSELLATIONSSPATIAL BIRTH PROCESSESSTEREOLOGYSTOCHASTIC GEOMETRYVORONOI TESSELLATIONS

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 4 , December 1992 , pp. 814 - 844
Copyright
Copyright © Applied Probability Trust 1992 

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References

Brakke, K. A. (1987) Statistics of random plane Voronoi tessellations. Department of Mathematical Sciences, Susquehanna University. Unpublished manuscript.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes . Springer-Verlag, New York.Google Scholar
Evans, U. R. (1945) The laws of expanding circles and spheres in relation to the lateral growth of surface films and the grain size of metals. Trans. Faraday Soc. 41, 365–374.Google Scholar
Frost, H. J. and Thompson, C. V. (1987) The effect of nucleation conditions on the topology and geometry of two-dimensional grain structures. Acta Metall. 35, 529–540.CrossRefGoogle Scholar
Gilbert, E. N. (1962) Random subdivisons of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
Horálek, V. (1988) A note on the time-nonhomogeneous Johnson–Mehl tessellation. Adv. Appl. Prob. 20, 684–685.Google Scholar
Horálek, V. (1990) ASTM grain-size model and related random tessellation models. Materials Characterization 25, 263–284.Google Scholar
Johnson, W. A. and Mehl, R. F. (1939) Reaction kinetics in processes of nucleation and growth. Trans. Amer. Inst. Min. Engrs. 135, 416–458.Google Scholar
Leistritz, L. and Zähle, M. (1992) Topological mean value relations for random cell complexes. Math. Nachr. 155, 57–72.Google Scholar
Lorz, U. and M⊘ller, J. (1992) Large scale simulations of Johnson-Mehl models. In preparation.Google Scholar
Mahin, K. W., Hanson, K. and Morris, J. W. (1980) Comparative analysis of the cellular and Johnson–Mehl microstructures through computer simulation. Acta Metall. 28, 443–453.CrossRefGoogle Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Mecke, J., Schneider, R. G., Stoyan, D. and Weil, R. R. (1990) Stochastische Geometrie . Birkhäuser, Basel.Google Scholar
Meijering, J. L. (1953) Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.Google Scholar
Miles, R. E. (1972) The random division of space. Suppl. Adv. Appl. Prob. , 243–266.Google Scholar
Miles, R. E. (1988) Matschinski's identity and dual random tessellations. J. Microsc. 151, 187–190.CrossRefGoogle Scholar
M⊘ller, J. (1989) Random tessellations in ℝ d . Adv. Appl. Prob. 21, 37–73.CrossRefGoogle Scholar
M⊘ller, J. (1992) Generation of Johnson-Mehl crystals and comparative analysis of models for random nucleation. In preparation.Google Scholar
M⊘ller, J., Vedel, E., Petersen, J. S. and Gundersen, H. J. G. (1989) Modelling an aggregate of space-filling cells from sectional data. Research Report No. 182, Department of Theoretical Statistics, University of Aarhus.Google Scholar
Nguyen, X. X. and Zessin, H. (1979) Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.Google Scholar
Shannon, C. E. (1949) Communication in the presence of noise. Proc. I.R.E. 37, 10–21.Google Scholar
Slivnyak, I. M. (1962) Some properties of stationary flows of homogeneous random events. Teor. Veruyat. Primen. 7, 347–352. (Translation in Theory Probab. Appl. 7, 336–341).Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and Its Applications . Akademie-Verlag, Berlin.Google Scholar
Weiss, V. and Zähle, M. (1988) Geometric measures for random curved mosaics of ℝ d . Math. Nachr. 138, 313–326.Google Scholar
Zähle, M. (1988) Random cell complexes and generalized sets. Ann. Prob. 16, 1742–1766.CrossRefGoogle Scholar