As mentioned in the Introduction, we shall consider the case in which all the nucleation takes place at t = 0 (namely, site-saturated case), and the case in which the nucleation takes place in time (namely, time-dependent case).
Site-saturated nucleation processes and space-time dependent nucleation processes can be modeled by point processes and marked point processes, respectively.
We intentionally use the same notation N for the site-saturated nucleation process and for the time-dependent one because the site saturated process may be seen as a particular case of the time-dependent one by assuming [T.
In this paper we shall assume Poissonian nucleation, that is N will be a Poisson process (in [R.
It is defined as the space-time region in which at least one nucleation event has to take place in order to cover the point x at time t; namely, it is the subset of [R.
moreover, under Poissonian assumption on the nucleation process it holds
Since, in general, nucleation and growth are random in time and space, then the transformed region [[THETA].
Birth-and-growth and nucleation and growth will be used as synonyms in this paper.
i] the spatial location of the j-th nucleus of a site-saturated nucleation process, and by [[THETA].
Then, a space-time nucleation process can be modeled by a marked point process, identifying [T.
In other words, if N is a site-saturated nucleation process, then [LAMBDA](A) represents the mean number of nuclei born in A [subset] [R.
under assumptions of homogeneous nucleation and growth), then [V.