What is to stop us running an AC nano-motor with Johnson-Nyquist noise? You can say, that the deformations in the motor will cancel out the current. But what if we keep two thermally insulated coils on a transformer. The hot coil is the source of the noise, the cold coil is the one fed into the AC motor. Then the power received by the motor will be greater than its own noise and it can spin?
The energy required to generate the noise will be greater than what can be extracted from it.