Do you know of an elementary proof for the second law of thermodynamics, for example, from the Newton laws or perhaps some particular model in which it is equivalent/reduces to it?
My naive concept of entropy was it was some concave function of all velocities of a system, for example, $min(\{v\})$. Is it correct at all?
You cannot derive the second law of thermodynamics from Newton's laws. Boltzmann's H-theorem was intended to do that, but it's not an actual theorem: the proof is flawed. However, although it's not a theorem, it works in reality. Go figure.
Reversible Newtonian mechanics is not enough; it can describe both physical behaviour (systems in non-equilibrium state spontaneously relaxing to equilibrium state) and reversed behaviour we don't observe (systems spontaneously leaving equilibrium state and going into the non-equilibrium state).
To select the "correct" behaviour of mechanical systems consistent with 2nd law, we need additional assumption, such as assumption on the initial conditions that lead to the "correct" behaviour. Most "random" initial conditions will lead to physical behaviour. But initial conditions may be fine-tuned to lead to unphysical behaviour, where the system evolves towards states of lower entropy.
