In the preceding sections I raised several questions on the application of population synthesis models, ranging from their use in CMD diagrams for semi-resolved systems to fitting metrics and the still unexplored arena of covariance coefficients. In fact, most of the applications presented here have not been fully developed and have been presented in conference papers. In this respect I refer to work by Buzzoni (2005), which has been a source of inspiration to most of the applications described here and some additional applications not discussed. However, let me finish with two comments about additional unexplored areas in which stochasticity could play a role.
The first is chemical evolution. At the start of this paper I noted that any modelling that makes use of the stellar birth rate (IMF and SFH) is intrinsically probabilistic. Hence, the present discussion also applies to chemical evolution models. Here, there is no stellar luminosity function, but a stellar yield function, which is the amount of material that a star has ejected (instantaneously or accumulatively). Obviously, such a stellar yield function refers mainly to dead stars, stars with ℓ = 0 in the stellar luminosity function, and is hence correlated with live stars. The application of such ideas is suggestive, but not simple. Chemical evolution affects the metallicity and depends on the SFH; hence, metallicity and SFH cannot be separable functions in the stellar birth rate. Even worse, the actual metallicity of a system depends on its previous metallicity evolution, which is itself described as a probability distribution. Determination of the evolution of a system would require a change in the ordinary differential equations defining chemical evolution to stochastic differential equations. Interested readers can find very preliminary approaches to the subject in work by Cerviño et al. (2000a, 2001), Cerviño and Mollá (2002), and more detailed studies by Shore and Ferrini (1995), Carigi and Hernandez (2008).
The second question is the inclusion of highly variable phases such as thermal pulses in synthesis models. The present idea is actually a re-elaboration of the so-called fuel consumption theorem proposed by Renzini and Buzzoni (1983) and Buzzoni (1989) applied over isochrone synthesis. A requirement of isochrone synthesis is that evolutionary tracks must be smooth enough to allow interpolations and compute isochrones. In particular, variation is not allowed for a star with a given mass and a given age. However, we can include variability as far as we can model it by a probability distribution function. In fact, we can define such a probability as proportional to the period of variation and include it directly in the stellar luminosity function. I refer readers interested in this subject to the paper by Gilfanov et al. (2004), whose ideas provide a formal development of the probabilistic modelling of stellar populations.