Much of the noise comes from the thermal bouncing of the atoms in a metal. They occasionally bounce into an electron which causes the electron to emit electromagnetic radiation. While at high temperatures this could be light or an x-ray, at room temperatures it is normally a radio wave. Some of the noise may come from Zero-Point Energy as mentioned in some of the previous articles as virtual particles would bounce into an electron.
So it should be possible to collect these radio waves and perform some form of useful work with them. They could be collected in the manner of the crystal radio where you have a multi-turn coil and a diode which taps direct current off of the coil. The direct current could then be stored in a battery.
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