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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.It make sense, but the problem is more subtle. It's better compare the clock with a brick.
In the example of the clock there is a temperature difference that can be used to extract "useful" energy. You can use it to move the clock, or make a sound or light or something.
(I'm saying "useful" as in the a quote of the main author of the paper "What we did was reroute the current in the circuit and transform it into something useful.” .)
If you put a brick in a oven and keep it at the same temperature for some time, until the temperature, humidity and other properties have stabilized, you reach thermal equilibrium. It doesn't mean that the heat energy in the brick is perfectly constant, it exchanges heat with the oven. The heat energy has small random variations.
But unlike in the clock scenario, you can't use this additional accumulated energy for something "useful". To check if it is in a high or a low, you need a variant od the Maxwell's Demon. You can measure the energy exchange and analyze the theoretical and experimental properties anyway.
(Also, if you put the clock in the oven at a constant temperature for a long enough time, it will stop working.)
Stealing the name from a sibling thread, to make the grapheme device produce "useful" energy, you probably need a Maxwell's Diode.