Abstract
It has been shown by a substantial body of literature that the hazard function plays an important role in the derivation of evolution equations of volume and n-facet densities of Johnson-Mehl tessellations generated by germ-grain models associated with spatially homogeneous birth-and-growth processes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Avrami, Kinetic of phase change. Part I, J. Chem. Phys., 7 (1939), 1103–112.
G. Barles, H.M. Soner, P.E. Souganidis, Front propagation and phase-field theory, SIAM J. Contr.Optim., 31 (1993), 439–469.
P. Brémaud, Point Processes and Queues, Martingale Dynamics, Springer-Verlag, New York, 1981.
M. Burger, Growth fronts of .rst-order Hamilton-Jacobi equations. SFB Report 02–8, J. Kepler University, Linz (2002).
M. Burger, V. Capasso, L. Pizzocchero, Mesoscale averaging of nucleation and growth models. SIAM J. Multiscale Modeling and Simulation, (2006). In press.
V. Capasso, Ed., Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing, Springer-Verlag, Heidelberg, 2003.
V. Capasso, A. Micheletti, Local spherical contact distribution function and local mean densities for inhomogeneous random sets, Stochastics and Stoch. Rep., 71 (2000), 51–67.
V. Capasso, A. Micheletti, Stochastic geometry of spatially structured birthand-growth processes. Application to crystallization processes. In Topics in Spatial Stochastic Processes (E. Merzbach, Ed.). Lecture Notes in Mathematics, Vol. 1802 - CIME Subseries, Springer-Verlag, Heidelberg, 2002, 1–39.
V. Capasso, A. Micheletti, Stochastic geometry and related statistical problems in Biomedicine In “Complex Systems in Biomedicine” (A. Quarteroni et al, Eds.) Springer, Milano, 2006.
V. Capasso, C. Salani, Stochastic-birth-and-growth processes modelling crystallization of polymers with spatially heterogeneous parameters, Nonlinear Analysis: Real World Application, 1 (2000), 485–498.
V. Capasso, E. Villa, On the evolution equations of mean geometric densities for a class of space and time inhomogeneous stochastic birth-and-growth processes In “Stochastic Geometry” (W. Weil, Editor) Lecture Notes in Mathematics - CIME subseries - Springer, Heidelberg, 2005.
V. Capasso, E. Villa, Continuous and absolutely continuous random sets. Stoch. Anal. Appl., 24 (2006), 381–397.
V. Capasso, E. Villa, On the geometric densities of random closed sets, 2005. RICAM Report 13/2006, Linz, Austria.
U. Hahn, A. Micheletti, R. Pohlink, D. Stoyan, H. Wendrock, Stereological Analysis and Modeling of Gradient Structures, J. of Microscopy, 195 (1999), 113–124.
W.A. Johnson, R.F. Mehl, Reaction kinetics in processes of nucleation and growth, Trans. A.I.M.M.E., 135 (1939), 416–458.
A.N. Kolmogorov, On the statistical theory of the crystallization of metals, Bull. Acad. Sci. USSR, Math. Ser.,1 (1937), 355–359.
G. Last, A. Brandt, Marked Point Processes on the Real Line. The Dynamic Approach, Springer, New York, 1995.
G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1975.
J. Møller, Random Johnson-Mehl tessellations, Adv. Appl. Prob., 24 (1992), 814–844.
G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi, F. Bussolino, Modeling the early stages of vascular network assembly, EMBO J., 22 (2003), 1771–1779.
D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Application, John Wiley & Sons, New York, 1995.
T. Ubukata, Computer modelling of microscopic features of molluscan shells. In Morphogenesis and Pattern Formation in Biological Systems (T. Sekimura etal. eds.), Springer-Verlag, Tokyo, 2003, 355–368.
V.S. Vladimirov, Generalized Functions in Mathematical Physics, Mir Publishers, Moscow, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
Burger, M., Capasso, V., Micheletti, A. (2007). An Extension of the Kolmogorov-Avrami Formula to Inhomogeneous Birth-and-Growth Processes. In: Aletti, G., Micheletti, A., Morale, D., Burger, M. (eds) Math Everywhere. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44446-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-44446-6_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44445-9
Online ISBN: 978-3-540-44446-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)