Divide a box of gas in two and put a small door in the partition. Suppose something sits by that door able to tell the molecules apart one at a time. Maxwell set this up in 1867, and it later came to be called a demon.
When a fast molecule comes from the left, open the door and let it through to the right. When a slow one comes from the right, open it and let it through to the left. Otherwise keep it shut. The door itself is light, so opening it takes no work.
The figure is that at work. The fast ones gather on the right, the slow ones on the left. They are only sorted and let through; nothing pushes them.
Keep it up and the right grows hot while the left grows cold. The two distributions in the figure are those, drawing apart from where they began together.
No work has been done, and yet a difference of temperature is there. With such a difference a heat engine can be run, which means work has been drawn out of nothing.
The second law appears to be broken. But somewhere along the argument something has gone uncounted. The place to look is not the gas.
What went uncounted was the thing sitting by the door. To tell fast from slow, it must measure, and the result of a measurement is left somewhere.
The figure is that record. Each molecule told apart adds one entry to it. Fast or slow being two possibilities, each entry is one bit.
The room for records is finite. To keep going indefinitely, something has to be erased at some point. It is at that erasing that the bill is waiting.
To erase one record is to take a state that was one of two and make it one of one. The left of the figure is before erasing, the right after, with the possibilities reduced to a single one.
The count of states is halved, so the entropy on the record side falls by k\log 2. The entropy of the whole cannot fall, so that much leaves as heat: at temperature T, exactly kT\log 2. This is Landauer's principle.
The bill comes not at measuring but at erasing. This was Bennett's point: measurement alone can be arranged so that it can be undone. What cannot be undone is the erasing.
The figure is the entropy of the whole. The gas side falls by however much is sorted; the record side rises with every measurement.
Added together, they never fall. At the very best they balance exactly, and nothing does better than that. The work the demon can draw out never exceeds the heat paid to erase its records. The second law was not broken after all.
What happened here is that information went onto the same account as heat. One bit is k\log 2 of entropy. Entropy was a matter of counting from the very beginning. All that widened was what gets counted: from the arrangements of a gas to the arrangements of a record.