of particle dynamics. The temporal evolution for this cumbersome problem, even for a modest number of particles, requires a statistical description which introduces the concept of probability distribution function (PDF) for the phase space of the system. Irreversibility, equilibrium, and more important non-equilibrium properties, surge from this probability conception of systems (with a large number of degrees of freedom) and its deterministic evolution. Briefly, the methodology reduces (under some assumptions) to a kinetic description which displays an irreversible behavior to equilibrium observed in macroscopic systems. The assumptions for this approach are: i) Macroscopically, a system is described by a finite set of observables; ii) The robust instability of the microscopic motions which is at the basis of the sensibility to initial conditions and the ergodic assumption. iii) A [*Stosszahlansatz*]{} that introduces explicitly a broken before-after symmetry of the probability distribution evolution.

About 20 years ago, Nicolis [*et al.*]{} [@nicolis; @et] introduced a systematic coarse-graining approach to the treatment of the macroscopical variables. As a consequence, this coarse-graining breaks naturally the past-future symmetry in time, leading to an irreversible master equation, for a reduced probability distribution function of the system. In the present paper, we apply this systematic approach to a 