Hi! This is a fun question. Certainly, thermal equilibrium involves something called the equipartition theorem, which says that all of the energy is divided equ

Hi! This is a fun question. Certainly, thermal equilibrium involves something called the equipartition theorem, which says that all of the energy is divided equally across all of the degrees of freedom so that each has an average small amount of energy which we call the temperature. (It's hard to define temperature in any other circumstance! But let's not get into ensembles.) The big caution with the gradient idea is that you can get uncomfortably close, if you are not paying attention, to saying that refrigerators are impossible. An energy flow from degree of freedom A to degree of freedom B manages to “slurp up” some of this thermal energy from a system's degrees of freedom and move them to another system, one gradient undoes a flow across a different gradient. And like, this happens all the time in everyone's kitchen. Information erasure looks like this: a bit needs to look like a degree of freedom with some energy stored above the baseline average. It could be energy neutral in that you might have two degrees of freedom, {ifZero, ifOne}, so that a NOT gate doesn't take any energy, but just swaps the energy in the two degrees of freedom. But the energy stored in one or the other needs to be roughly one temperature*, or so. And the reason why is that “equipartition” specifies an average, but there is also a standard deviation: these are the energy fluctuations in thermal equilibrium. So, as a result of Ludwig Boltzmann we know that this is an exponentially-distributed random variable and therefore the standard deviation is equal to the mean. So if you want your bits to not be thermally excited, they need to hold a few temperatures’ worth of energy. Information erasure says that the only circuits on these bits which preserve the energy levels inside them, are reversible circuits like NOT. Irreversible dynamics can look like reversible dynamics plus erasure: in the simplest erasure, we drain both ifZero and ifOne energies into the environment and then we refill ifZero from some energy reservoir. And this says basically that there is nothing much better than that for erasing a bit. So, just like the ratchet had a spring that was kept at low energy in order to facilitate a transfer of energy from cold to hot, and could only work as long as you kept that spring well-refrigerated, Maxwell's demon has a similar “coldness buffer” in terms of these bits that are present in a definite prepared state ahead of time. The information processing step takes it out of that zero state, and if it wants to go back there it has to erase, but that requires dumping some energy out into an environment, and getting energy from some reservoir to compensate. And then Maxwell's demon becomes just a normal refrigerator like the other ones that we know about.