We introduce a high
dimensional symplectic map, modeling a large system, to analyze the
interplay between single-particle chaotic dynamics and particles
interactions in thermodynamic systems. We study the initial growth of the
Boltzmann entropy, S_B, as a function of the coarse graining resolution
(the late stage of the evolution is trivial, as the system is subjected
to no external drivings). We show that a characteristic
scale emerges, and that the behavior of S_B vs time, at variance with the
Gibbs entropy, does not depend on the resolution, as far as it is finer
than this scale. The interaction among particles is crucial to achieve
this result, while the rate of entropy growth, in its early stage,
depends essentially on the single-particle chaotic dynamics. It is
possible to interpret the basic features of the dynamics in terms of a
suitable Markov approximation.