The integro-differential Master Equation (ME) provides a fairly generic framework for describing
the statistical dynamics of complex systems. In this work we address the mathematical analysis of
two different limits of the ME.
1) It well known that diffusive dynamics does arise as long-wavelength limit of the ME, and is
usually described in terms of the Fokker-Planck equation. Fick’s form is another expression for the
diffusion equation. In homogeneous environments the two expressions are trivially identical,
whereas in inhomogeneous environments they are no longer equivalent. Use of the improper form
within a given context leads to appearance of artefacts when modelling experimental data, an
instance being given by the modelling of mass and energy in laboratory plasmas [1,2]. We carry out
a study in terms of microscopic dynamics of the conditions that lead to either the one or the other
form of the diffusive equation.
2) The ME can be interpreted within a Bayesian viewpoint as the mathematical expression of the
rule for a statistical distribution to evolve from a prior one by the gradual assimilation of novel
information (e.g., new data from measurements, etc …); it is, under this regard, an instance of an
evolutionary process and—as such—its dynamics can be modelled using the mathematics of
evolutionary theory, i.e., the replicator equation . We study the asymptotic limit of this equation
pointing out how several common statistical distributions, often encountered in disparate fields of
science, do arise as its limiting solutions.
 B. Ph. Van Milligen, B.A. Carreras, R. Sànchez, Plasma Phys. Control. Fusion 47 (2005) B743
 F. Sattin, A. Bonato. L. Salasnich, arxiv:1512.07428 (submitted to Springer)
 P.D. Taylor, L.B. Jonker, Math. Biosciences 40 (1978) 145