Another difference from separated hot & cold reservoirs is that the time horizon for being able to make use of the information is on the order of nanoseconds before the information is useless. Even without quantum messiness and prescribing perfect billiard-ball molecules, just a few stray thermal photons from outside and a few collisions will scramble the speeds and angles hopelessly.
As far as temperature goes it is really undefined, since the energy in the water is not thermal from the point of view of the extremely well informed observer. It has essentially zero entropy, like the kinetic energy of a car or that of a static magnetic field. If you go ahead and try to define it using statistical mechanics anyway, you get a division by zero error: temperature is the marginal ratio of energy to entropy, and the entropy is an unchanging zero regardless of energy.
I think that last bit only applies if we suppose that you are equipped not only with a complete specification of the state of the molecules but with a constantly instantly updating such specification. Otherwise, if you put more energy in then the entropy will increase too and you can say T = dE/dS just fine even though the initial entropy is zero. (But you make a good point about the energy being not-thermal from our near-omniscient viewpoint.)
If you just have a snapshot state (even with an ideal model of internal interactions from that state) then any thermal contact with the outside will almost instantly raise entropy to near maximum regardless of whether energy is added or removed or on balance unchanged. I don’t think it makes sense to talk about temperature there either, since the entropy is not a function of energy and does not co-vary with it in any smooth way.
Another difference from separated hot & cold reservoirs is that the time horizon for being able to make use of the information is on the order of nanoseconds before the information is useless. Even without quantum messiness and prescribing perfect billiard-ball molecules, just a few stray thermal photons from outside and a few collisions will scramble the speeds and angles hopelessly.
As far as temperature goes it is really undefined, since the energy in the water is not thermal from the point of view of the extremely well informed observer. It has essentially zero entropy, like the kinetic energy of a car or that of a static magnetic field. If you go ahead and try to define it using statistical mechanics anyway, you get a division by zero error: temperature is the marginal ratio of energy to entropy, and the entropy is an unchanging zero regardless of energy.
I think that last bit only applies if we suppose that you are equipped not only with a complete specification of the state of the molecules but with a constantly instantly updating such specification. Otherwise, if you put more energy in then the entropy will increase too and you can say T = dE/dS just fine even though the initial entropy is zero. (But you make a good point about the energy being not-thermal from our near-omniscient viewpoint.)
If you just have a snapshot state (even with an ideal model of internal interactions from that state) then any thermal contact with the outside will almost instantly raise entropy to near maximum regardless of whether energy is added or removed or on balance unchanged. I don’t think it makes sense to talk about temperature there either, since the entropy is not a function of energy and does not co-vary with it in any smooth way.