Using Information to Extract Energy

There was some excitement last week about a Maxwell’s-Demon-type experiment conducted by Shoichi Toyabe and collaborators in Japan. (Costly Nature Physics article here; free arxiv version here.) It’s a great result, worth making a fuss about. But some commentators spun it as “converting information into energy.” That’s not quite right — it’s more like “using information to extract energy from a heat bath.”

Say you have a box of gas with a certain temperature at maximum entropy — thermodynamic equilibrium. That is, the gas is smoothly spread throughout the box. (We can safely ignore gravity.) There’s certainly energy in there, but it’s not very useful. Indeed, one way of thinking about entropy is as a measure of how useless a certain amount of energy is. If we have a low-entropy configuration, we can extract useful work from the energy inside, such as pushing a piston. If we have a high-entropy configuration, the energy is useless; there’s nothing we can do to consistently extract it.

Here’s an example from my book. Consider two pistons with the same number of gas particles inside, with the same total energy. But the top container is in a low-entropy state with all the gas on one side of the piston; the bottom container is in a high-entropy state with the gas equally spread out.

You see the difference — from the top configuration we can extract useful work by simply allowing the piston to expand. In the process, the total energy of the gas goes down (it cools off). But in the bottom piston, nothing’s going to happen. There’s just as much energy inside there, but we can’t get it out because it’s in a high-entropy state.

In 1929, Leó Szilárd used a similar setup to establish an amazing result: the connection between energy and information. The connection is not that “information carries energy”; if I tell you some information about gas particles in a box, that doesn’t change their total energy. But it does help you extract that energy. Effectively, learning more information lowers the entropy of the gas. That’s a loosey-goosey statement, because there is more than one way to define “entropy”; but one reasonable definition is that the entropy is a measure of the information you don’t have about a system. (In the piston above, we know more about the gas in the low-entropy setup, since we have a better idea of where it is localized.)

This being physics, there are equations. Szilárd showed that, if you have one bit of information about the system, you can use that to extract an amount of energy given by

Here, k is Boltzmann’s constant, and T is the temperature of the system. That temperature is crucial; that’s where the energy actually comes from, not from the “information” (and the temperature will go down when you extract the energy).

The idea is called Szilárd’s Engine, and is beautiful in its simplicity. As a physicist, one always looks for the spherical cow. In this case, it’s a box of gas with just one particle, moving in one dimension. (In the real experiment, they used knowledge of a particle’s position to make it hop up a staircase.)

The equivalent of “maximum entropy” here is “we have no idea where the particle is.” There is energy in the box, equal to kT/2, but we can’t get it out.

But now imagine that someone gives us one bit of information: they tell us which side of the box the particle is on. Now we can get some energy out! All we have to do is wait until the particle is on the left-hand side of the box, and quickly slip in our piston coming out the right:

The particle will now bump into the piston, pushing it to the right, allowing us to useful work, like lifting a very tiny bucket or something. In the process some of the particle’s energy is transferred to the piston, so we’ve extracted some energy from the box. Note that we could not have done this if we hadn’t been given the information — without knowing where the particle is, our piston would have lost energy on average just as often as it gained energy, so we couldn’t have done any useful work.

This clever thought experiment launched an entire field of research, connecting information to energy. “Information is physical,” the motto now goes. It’s fun to see (a version of) Szilárd’s engine actually be created in the lab.