Imagine a parallel plate capacitor, where one plate is mounted on a spring and is allowed to move. Now connect a resistor across the terminals, so you have thermal noise driving the capacitor.
The reason this case is very interesting is that the force of attraction between the plates is proportional to the square of the voltage across the plates. The squaring process removes negative values and so the noise is rectified.
However, the question is how to mathematically model this system? It is more difficult than it looks.
Here is our miserable failed attempt at trying to model it:
https://www.researchgate.net/publication/10730976
What did we miss?
Article Thermodynamic energy exchange in a moving plate capacitor
I guess one can describe this effect in terms of the temperature of the spring and temperature of the resistor. The thermal noise coming from the resistor will indeed push the spring, but however at the same time the thermal fluctuation comming from the spring will result in current pushed through the resitor. So we are dealing with thermal equazalization between the resistor and the spring.
I don't think this solves the overall modelling problem, because in principle one could heat the resistor to be at a much higher temperature than the spring. Then the fluctuations of the spring are negligible and the problem of correctly describing the system still remains.
The dissipation in the spring might be essential here. If there is no dissipation there, the applied force even being rectified (always in one direction) will result in endless oscillations of the spring. The phase of such oscillation will be undefined, so it is not no-entropy energy.
If we consider spring without dissipation we have to either assign some temperature to it in the beggining and corresponding kT energy ( spring plus plate as oscillatory degree of freedom). If we don't assign that energy then we effectivelly declare the spring and plate to be and T=0, what happens then is the heat transfer (thermal equalization) from the resistor to spring plus plate degree of freedom.
If however there is a dissipation in the spring, since it has its own temperature we are talking about equzalization between the resistor and the spring
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