Isn't this like the Maxwell's demon? Deciding what to do on micro level can create macro level changes that break physics, but it's not actually possible, so there's no paradox.
A good way to ask "what does our model mean, exactly" is to imagine the perfect processes with no unnecessary energy losses. Real mirrors might not be infinitely thin mathematical abstractions, but if there isn't an actual, physically defined fundamental limit to how thin a mirror can be, then it would be weird if we could violate fundamental laws of the universe, but for want of such a mirror.
Maxwell's Demon is neat because we can whittle things down and eventually get to needing to account for the physical cost of the bits in the thing's 'brain.' An interesting example of the fact that information is actually a physical quantity.
How does the demon attain the knowledge of what is "fast" and "slow" without continuous observation (and thus interaction) with the particles. Velocity is just function of position over time, so the demon needs at least 2 samples to make the most basic approximation. Where is the entropy for doing that coming from? How does the interference of the measuring apparatus factor into the whole process - what if the sole act of measurement changes the state of the particle from "fast" to "slow" or vice versa? Do we need to measure twice? But what if the second measurement causes the transition it was meant to detect?
In order to derive those heat equations from the second law, we now know it requires an assumption: that you (or any entropy-containing component not modeled in the system) cannot have specific knowledge of the microstate of the system, only its macroscopic properties. For a long time it seemed absurd that such an assumption would necessary at all, for these seemingly universal laws, and there was no clear way to thermodynamically model the knowledge of the actor inside this system such that you wouldn't need such an assumption.
Not so long. The second law dates from 1850 and the requirement of such an assumption is what Maxwell’s demon illustrated less than forty years later.
> and there was no clear way to thermodynamically model the knowledge of the actor inside this system such that you wouldn't need such an assumption.
There is still no way to model an actor inside a thermodynamical system in equilibrium - by definition.
The initially apparent problem is that the demon appears to be generating a temperature gradient "for free" by just swinging a gate open (for fast molecules) or closed (for slow ones) (because there's no fundamental physical cost for gate-swinging).
It is resolved by taking into account the fact that the demon must use at least a bit of memory, while acquiring the 'status' of the molecule (let through or don't). So, we can at least say that the demon, no matter how slow and lazy (efficient) it wants to be, must pay the information-theory based cost of erasing that bit.
I agree.
> The initially apparent problem is […]
Why would that be a problem? If it’s because the second law of thermodynamics says that it cannot happen spontaneously in a thermodynamical system in equilibrium why would that be applicable when we’re not considering just a system in thermodynamical equilibrium?
As Maxwell wrote: “This is only one of the instances in which conclusions which we have drawn from our experience of bodies consisting of an immense number of molecules may be found not to be applicable to the more delicate observations and experiments which we may suppose made by one who can perceive and handle the individual molecules which we deal with only in large masses.”
This quote seems more compelling in cases where the statistics are known, but and we just need to have enough samples, such that the expected values, etc, show themselves.
It's the very example that he was discussing, which he had described just before: "Now let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A."
Another interesting quote of his: "Available energy is energy which we can direct into any desired channel. Dissipated energy is energy we cannot lay hold of and direct at pleasure, such as the energy of the confused agitation of molecules which we call heat. Now, confusion, like the correlative term order, is not a property of material things in themselves, but only in relation to the mind which perceives them. A memorandum-book does not, provided it is neatly written, appear confused to an illiterate person, or to the owner who understands thoroughly, but to any other person able to read it appears to be inextricably confused. Similarly the notion of dissipated energy could not occur to a being who could not turn any of the energies of nature to his own account, or to one who could trace the motion of every molecule and seize it at the right moment. It is only to a being in the intermediate stage, who can lay hold of some forms of energy while others elude his grasp, that energy appears to be passing inevitably from the available to the dissipated state."
The interaction between the demon and the particles is the key. The final version, at least as far as I'm concerned, comes from Landauer and Bennett [https://en.wikipedia.org/wiki/Maxwell%27s_demon#Criticism_an...] -- the demon must accumulate information (which it can only do finitely) or erase it. Erasing information has a real physical cost, resolving the issue.
When the demon was first invented, information theory hadn't been developed yet. So the mystery to Maxwell was that it looked like it was violating energy conservation, but that's just because there's a sneaky place we can store entropy temporarily or finitely.
So, I think I was imprecise (or... wrong). A demon that does what we really want (generates free energy) is impossible. But it is impossible for weird reasons that Maxwell wouldn't have been aware of, and I don't think he explicitly explored them.
That's just my gut though, I'm not a professional physicist.