Liouville's theorem alone does not suffice to obtain the Second Law. You might want to look up the objections to Boltzmann's derivation of H-theorem made by Zermelo (wait long enough and the system will return to a state arbitrarily close to the original state, due to Poincare's recurrence theorem) and Loschmidt (reverse the speeds of all particles and the entropy will decrease to its original value). Boltzmann killed himself in a bout of depression because he could not find a satisfactory answer to these objections. More than a century later, we still don't have satisfactory answers.
Liouville's theorem alone does not suffice to obtain the Second Law. You might want to look up the objections to Boltzmann's derivation of H-theorem made by Zermelo (wait long enough and the system will return to a state arbitrarily close to the original state, due to Poincare's recurrence theorem) and Loschmidt (reverse the speeds of all particles and the entropy will decrease to its original value). Boltzmann killed himself in a bout of depression because he could not find a satisfactory answer to these objections. More than a century later, we still don't have satisfactory answers.