During the summer, I wrote a paper that I dumped in an arxiv category called cond-mat.stat-mech, and then managed to entirely forget about it. So somewhat belatedly, here is a summary.Pretty much the only recollection I have of my stat mech lectures is that every single one of them was inevitably accompanied by the always same divided box with two sides labeled A and B. Let me draw this for you:
Maxwell’s demon in its original version sits in this box. The demon’s story is a thought experiment meant to highlight the following paradox with the 2nd law of thermodynamics.
Imagine the above box is filled with a gas, and the gas is at a low temperature on side A and at a higher temperature on side B. The second law of thermodynamics says that if you open a window in the dividing wall, the temperatures will come to an average equilibrium value, and in this process entropy is maximized. Temperature is basically average kinetic energy, so the average speed of the gas atoms approaches the same value everywhere, just because this is the most likely thing to happen
The system can only do work on the way to equilibrium, but no longer once it’s arrived there. Once you’ve reached this state of maximum entropy, nothing happens any more, except for fluctuations. Unless you have a Maxwell demon...
Maxwell’s demon sits at the dividing wall between A and B when both sides are at the same temperature. He opens the window every time a fast atom comes from the left or a slow atom comes from the right, otherwise he keeps it closed. This has the effect of sorting fast and slow atoms so that, after some while, more fast atoms are on the right side than on the left side. This means the temperatures are not in equilibrium anymore and entropy has decreased. The demon thus has violated the second law of thermodynamics!
Well, of course he hasn’t, but it took a century for physicists to pin down the exact reason why. In brief it’s that the demon must be able to obtain, store, and use information. And he can only do that if he either starts at a low entropy that then increases, or brings along an infinite reservoir of low entropy. The total entropy never decreases, and the second law is well and fine.
It has only been during recent years that some versions of Maxwell’s demon have been experimentally realized in the laboratory. These demons use essentially information to drive a system out of equilibrium, which can then, in principle, do work.
It occurred to me that this must mean it should be possible to replace transfer of energy from a sender to a receiver by transfer of information, and this information transfer could take place with a much smaller energy than what the receiver gets out of the information. In essence this would mean one can down-convert energy during transmission.
The reason this is possible is that the relevant energy here is not the total energy – a system in thermal equilibrium has lots of energy. The relevant energy that we want at the receiving end is free energy – energy that can be used to do work. The signal does not need to contain the energy itself, it only needs to contain the information that allows one to drive the system out of equilibrium.
In my paper, I have constructed a concrete example for how this could work. The full process must include remote measuring, extraction of information from the measurement, sending of the signal, and finally making use of the signal to actually extract energy. The devil, or in this case the demon, is in the details. It took me some while to come up with a system simple enough so one could in the end compute the energy conversion and also show that the whole thing, remote demon included, obeys the Carnot limit on the efficiency of heat engines.
In the classical example of Maxwell’s demon, the necessary information is the velocity of the particles approaching the dividing wall, but I chose a simpler system with discrete energy levels, just because the probability distributions are then better to deal with. The energy extraction that my demon works with is a variant of stimulated emission that is also used in lasers.
The atoms in a laser are being “pumped” into an out-of equilibrium state, which has the property that as you inject light (ie, energy) with the right frequency, you get out more light of the same frequency than you sent in. This does not work if the system is in equilibrium though, it is then always more likely that the injected signal is absorbed rather than that it stimulates a net emission.
However, a system in equilibrium always has fluctuations. The atoms have some probability to be in an excited state, a state in which they could be stimulated to emit light. If you just knew which atoms were in the excited state, then you could target them specifically, and end up with twice the energy that you sent in.
So that’s what my remote demon does: It measures out of equilibrium fluctuations in some atomic system and targets these to extract energy. The main point is that the energy sent to the system can be much smaller than the extracted energy. It is, in essence, a wireless battery recharger. Except that the energies in question are, in my example, so tiny that it’s practically entirely useless.
I’ve never worked on anything in statistical mechanics before. Apparently I don’t even have a blog label to tag it! This was a fun project and I learned a lot. I even made a drawing to accompany it.