An extension of the Kolmogorov-Avrami formula to inhomogeneous birth-and-growth processes

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 n-facet densities of Johnson-Mehl tessellations generated by germ-grain models associated with spatially homogeneous birth-and-growth processes. In this paper we analyze a more general class of space-time inhomogeneous birth-and-growth processes emphasizing the role that the hazard function plays in this generalization. A special result is the extension of the well known formula that Kolmogorov and Avrami had found in connection with problems of crystal growth. Recent literature shows the relevance of a theory for the hazard function in other important areas of application such as tumor growth and angiogenesis, crystallization of sea shells, etc. In this paper a detailed analysis of the hazard function in terms of relevant volume and surface densities is carried out ; its relationship with the local spherical contact distribution function is also given.