  A varying number of particles is the most relevant characteristic of systems
of interest in nature and technology; ranging from the exchange of energy and
matter with the surrounding environment to the change of particle numbers
through internal dynamics such as reactions. The physico-mathematical modeling
of these systems is extremely challenging, with the major difficulty being the
time dependence of the number of degrees of freedom and the additional
constraint that the increment or reduction of the number and species of
particles must not violate basic physical laws. Theoretical models, in such a
case, represent the key tool for the design of computational strategies for
numerical studies that deliver trustful results. In this manuscript, we discuss
complementary physico-mathematical approaches of varying numbers of particles
inspired by different specific numerical goals and provide a generalized
equation that embeds all of them.
