This example is a bit far-fetched, but hopefully you can get enough intuition from it to see that uncertainty can only grow over time for any situation.
I think you’re glossing over the best part of the story here! If you know that the system is in a phase-space region of volume V now, then in an hour or a year you will know (in principle) that the system is in a (different) phase-space region of the same volume V. (Cf. “Liouville’s theorem”.) So in a straightforward sense, your uncertainty doesn’t go up as the system evolves in time.
However, your information about the system gradually converts from “actionable” to “not actionable”, as you will eventually have information about complicated correlations involving huge numbers of particles, e.g. “if the 17th decimal digit of the velocity of particle 79 is 4, then the 19th decimal place of the velocity of particle 857 is 6”. The phase-space volume V will be this beautiful, horrible mess of infinitesimally-fine tendrils arbitrarily close to every point in phase space. You can’t do anything with this information that you have about the system, so you might as well forget that information altogether. And that’s how entropy goes up.
I think you’re glossing over the best part of the story here! If you know that the system is in a phase-space region of volume V now, then in an hour or a year you will know (in principle) that the system is in a (different) phase-space region of the same volume V. (Cf. “Liouville’s theorem”.) So in a straightforward sense, your uncertainty doesn’t go up as the system evolves in time.
However, your information about the system gradually converts from “actionable” to “not actionable”, as you will eventually have information about complicated correlations involving huge numbers of particles, e.g. “if the 17th decimal digit of the velocity of particle 79 is 4, then the 19th decimal place of the velocity of particle 857 is 6”. The phase-space volume V will be this beautiful, horrible mess of infinitesimally-fine tendrils arbitrarily close to every point in phase space. You can’t do anything with this information that you have about the system, so you might as well forget that information altogether. And that’s how entropy goes up.
Thanks for the comment, this is indeed an important component! I’ve added a couple of sentences pointing in this direction.