[cyphar]

Entropy in thermodynamics is a statistical effect which acts like a "force" because of the immense number of particles and sub-states in play. A perfect simulation of gas particles bouncing in a two-chamber system will result in the "pressure" equalising because that is overwhelmingly the most likely state to end up in.

To be honest, I hadn't heard of Louisville's Theorem before but it doesn't seem to imply what you're saying -- in fact it is used to prove the fluctuation theorem which quantifies the probability of entropy spontaneously decreasing (as thermodynamic entropy is a statistical effect).

[kgwgk]

Liouville's Theorem does indeed seem to imply that entropy doesn't change:

https://physics.stackexchange.com/questions/202522/how-is-li...