[raducu]

The work will never be zero, and I have a hunch it could well be greater than your overall temperature gain.

If the opening works on non-charged particles, it has to have mass and opening and closing it is not possible without work.

The work might be tiny, but your temperature gain is probably event tinier.

If the particle is charged, you don't need a physical gate, but you do work on the particle by exerting a force on it.

You also somehow must detect the particle, and that also takes energy.

[FabHK]

> If the opening works on non-charged particles, it has to have mass and opening and closing it is not possible without work.

You could open the gate by pulling it up, which would require work, but you could recover the entire work by letting it sink down again.

It has been shown that the solution was not the physical work of opening/closing the gate, but the work required in information processing, in particular deleting information.