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.
There’s nothing magical about reversing particle speeds. For entropy to decrease to the original value you would have to know and be able to change the speeds with perfect precision, which is of course meaningless in physics. If you get it even the tiniest bit off you might expect _some_ entropy decrease for a while but inevitably the system will go “off track” (in classical chaos the time it’s going to take is only logarithmic in your precision) and onto a different increasing-entropy trajectory.
Jaynes’ 1957 paper has a nice formal explanation of entropy vs. velocity reversal.
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.
There’s nothing magical about reversing particle speeds. For entropy to decrease to the original value you would have to know and be able to change the speeds with perfect precision, which is of course meaningless in physics. If you get it even the tiniest bit off you might expect _some_ entropy decrease for a while but inevitably the system will go “off track” (in classical chaos the time it’s going to take is only logarithmic in your precision) and onto a different increasing-entropy trajectory.
Jaynes’ 1957 paper has a nice formal explanation of entropy vs. velocity reversal.
(the paper: https://journals.aps.org/pr/abstract/10.1103/PhysRev.106.620)