Initial growth of Boltzmann entropy and chaos in a large assembly of weakly interacting systems

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, SB, 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 SB vs t, 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.