  We introduce a high dimensional symplectic map, modeling a large system
consisting of weakly interacting chaotic subsystems, as a toy model to analyze
the interplay between single-particle chaotic dynamics and particles
interactions in thermodynamic systems. We study the growth with time of the
Boltzmann entropy, S_B, in this system as a function of the coarse graining
resolution. We show that a characteristic scale emerges, and that the behavior
of S_B vs t, at variance with the Gibbs entropy, does not depend on the coarse
graining 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 depends essentially on the single-particle chaotic dynamics (for t not
too small). It is possible to interpret the basic features of the dynamics in
terms of a suitable Markov approximation.
