University of Arkansas researchers collecting the displacement current using a nearby small-area metal electrode and present an Ito-Langevin model for the motion coupled to a circuit containing diodes. Numerical simulations show that the system reaches thermal equilibrium and the average rates of heat and work provided by stochastic thermodynamics tend quickly to zero. Power is dissipated by the load resistor, and its time average is exactly equal to the power supplied by the thermal bath. The exact power formula is similar to Nyquist's noise power formula, except that the rate of change of diode resistance significantly boosts the output power, and the movement of the graphene shifts the power spectrum to lower frequencies. They have calculated the equilibrium average of the power by asymptotic and numerical methods. Experiments have proven the theory.
NOTE: These are big claims and the power levels must be very, very small. They will need millions of miniaturized circuits powering a capacitor to replace the battery for a tiny low power chip.