As a final nail in the coffin, if you wanted there to be a physical "arrow of time" you'd need not just T-symmetry breaking, but CPT breaking. Worse, CPT-symmetry breaking also breaks lorentz symmetry, meaning you broke General Relativity, and you broke it HARD.
In short, physics is perfectly okay with t -> -t, absolutely nothing interesting happens to the laws themselves, so long as CP is also reversed. Admittedly this does imply a somewhat more difficult problem than "merely" reversing the velocity of every particle in the universe, but it is technically doable (t -> -t is thought to result in C -> -C, leaving only the parity to be fixed). Well, doable in a very estoric sense, in that IF we could left-right swap everything, reverse the momentum of everything, and start calling ourselves "antimatter" instead of "matter" you'd get the universe to run backwards. That that is so INCREDIBLY unlikely is what causes the thermodynamic arrow of time. Note also that for certain initial conditions this MAY be impossible on a universal scale (i.e. if the universe as a whole has a momentum associated with it, then we can't reverse all the velocities without violating momentum conservation). A particularly interesting problem would be the expansion of the universe, which may result in some energy conservation violation (well, the energy remains conserved if you consider the space-time metric as having a certain energy, but it starts getting interesting), which would imply that the universe would also need to be contracting, so that that energy could be restored.
However, the point is that the physics laws CAN be reversed, under a very strict set of rules. Your system by contrast, can be reversed under the much less stringent "reverse all changes". This would infact be a side effect of the reversal I described above. More to the point however, is that according to your definition, I should be able to locally reverse time by simply undoing all changes inside a certain volume. For example, I could throw a ball into a wall, and make a dent. According to your description, if I could reverse the velocities of everything inside a certain volume around our ball/wall system, then time would reverse, as the ball would leap back up, hit the wall which would undent itself and push the ball back along it's trajectory, where I would catch it.
Now you may state "aha! But you've simply added one more change! Now the changes are the ball whipping up, the wall UNdenting and so on!". To this I respond "Choose an initial configuration and stick with it". You can't declare that the initial condition is "me with the ball, wall undented", that it changes to "ball on ground, wall dented", and then declare that the initial conditions are now "ball on ground, wall dented" which changes to "me with ball, wall undented". You have to choose an initial condition and remain with it. In particular, you're instinct is most likely to be to choose an initial condition with the lowest entropy.
Now, in the examples you gave, ignoring the pencil and eraser one (that wouldn't be reversing the velocities of the particles) you mentioned that entropy increases. This is because you chose your system incorrectly. If I roll a ball up a hill, no it will not roll back down in such a way that I get all the energy I put into it back. However if instead of just reversing the ball's velocity, I also reverse the velocity of all the particles in the hill, then yes, yes I will get all my momentum back. At this point you'll want to start wondering what the fuck happened to friction. The answer is that by reversing all those particles, any collisions that initially resulted in ball -> ground energy transfer (which is what friction IS), now result in ground -> ball energy transfer.
However, there are a LOT more ways for things to move that result in ball-> ground energy transfer than the reverse. Based purely on statistics, it is likely that entropy increases. But the point is that this isn't a physical law, it's a statistical one. Yes, in something with 10^80 particles kicking around the likelyhood of them all reversing direction is so wildly unlikely we consider it impossible, but it technically isn't. And that's what we mean when we say there's no physical "arrow of time". There is no physical law that states the universe couldn't run backwards completely fine (Well, there is, but as long as we also flip CP as above, it works).
So your system doesn't have an "arrow of time" anymore than the real system does.
You want an experiment you can reproduce, I can't give you one. It has been done in certain isolated systems under very strict conditions and for a certain value of "done" (involing the directions of angular momentum spin vectors in a magnetic field) but realistically for any system that you can see there's going to be trillions of particles involved, and more possible states than can be computed, so this isn't something you can do in your backyard. You can make an approximation in things with little friction (i.e. throwing a ball straight up in the air and watching it's return is ALMOST time symmetric), but it's not quite there. This is however, still science. You can confirm that there is a very small amount of energy removed in the above experiment. You can further confirm that small random movements of air can give energy to something place a ping-pong ball on a flat clean table in a large room, mark it's location, return much, much later and the ball will have moved very slightly. The fact that the second experiment takes so much more time to get even a fraction of the energy is irrelevant and only a consequence of it being very unlikely for the air to have any reasonable pressure difference across the surface of the ping pong ball. But it is POSSIBLE. Just very, very, very, very unlikely.
tl;dr: in practice yes entropy always increases. But it's not inevitable that it does, it's simply overwhelmingly likely.
Wolydarg wrote:That was like a roller coaster of mathematical reasoning. Problems! Solutions! More problems!
****************Signature Dehosted, New Signature Under Construction************************