Skip to main content

An Extension of the Kolmogorov-Avrami Formula to Inhomogeneous Birth-and-Growth Processes

  • Chapter
Math Everywhere

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. A. Avrami, Kinetic of phase change. Part I, J. Chem. Phys., 7 (1939), 1103–112.

    Article  Google Scholar 

  2. G. Barles, H.M. Soner, P.E. Souganidis, Front propagation and phase-field theory, SIAM J. Contr.Optim., 31 (1993), 439–469.

    Article  MATH  Google Scholar 

  3. P. Brémaud, Point Processes and Queues, Martingale Dynamics, Springer-Verlag, New York, 1981.

    MATH  Google Scholar 

  4. M. Burger, Growth fronts of .rst-order Hamilton-Jacobi equations. SFB Report 02–8, J. Kepler University, Linz (2002).

    Google Scholar 

  5. M. Burger, V. Capasso, L. Pizzocchero, Mesoscale averaging of nucleation and growth models. SIAM J. Multiscale Modeling and Simulation, (2006). In press.

    Google Scholar 

  6. V. Capasso, Ed., Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing, Springer-Verlag, Heidelberg, 2003.

    MATH  Google Scholar 

  7. 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.

    Article  MATH  Google Scholar 

  8. 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.

    Google Scholar 

  9. 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.

    Google Scholar 

  10. 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.

    Article  MATH  Google Scholar 

  11. 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.

    Google Scholar 

  12. V. Capasso, E. Villa, Continuous and absolutely continuous random sets. Stoch. Anal. Appl., 24 (2006), 381–397.

    Article  MATH  Google Scholar 

  13. V. Capasso, E. Villa, On the geometric densities of random closed sets, 2005. RICAM Report 13/2006, Linz, Austria.

    Google Scholar 

  14. U. Hahn, A. Micheletti, R. Pohlink, D. Stoyan, H. Wendrock, Stereological Analysis and Modeling of Gradient Structures, J. of Microscopy, 195 (1999), 113–124.

    Article  Google Scholar 

  15. W.A. Johnson, R.F. Mehl, Reaction kinetics in processes of nucleation and growth, Trans. A.I.M.M.E., 135 (1939), 416–458.

    Google Scholar 

  16. A.N. Kolmogorov, On the statistical theory of the crystallization of metals, Bull. Acad. Sci. USSR, Math. Ser.,1 (1937), 355–359.

    Google Scholar 

  17. G. Last, A. Brandt, Marked Point Processes on the Real Line. The Dynamic Approach, Springer, New York, 1995.

    MATH  Google Scholar 

  18. G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1975.

    MATH  Google Scholar 

  19. J. Møller, Random Johnson-Mehl tessellations, Adv. Appl. Prob., 24 (1992), 814–844.

    Article  MATH  Google Scholar 

  20. 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.

    Article  Google Scholar 

  21. D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Application, John Wiley & Sons, New York, 1995.

    Google Scholar 

  22. 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.

    Google Scholar 

  23. V.S. Vladimirov, Generalized Functions in Mathematical Physics, Mir Publishers, Moscow, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints 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

Publish with us

Policies and ethics