The temperature can be defined even for very small systems using Statistical Mechanics, and also the distribution of speeds and energy in the Brownian motion of the surrounding molecules and movement of the membrane (and phonons?) depend on the temperature.
I'm not sure was is your point. I'll try with another answer.
I don't remember something like the clock at the molecular level, but I think it is "theoretically possible" (or to be more accurate, "not theoretically impossible").
[Weird example warning]
In the mitochondria the ATP synthase https://en.wikipedia.org/wiki/ATP_synthase use the H+ difference of the inner and outer part to produce ATP. The main problem is how to create this difference without sugar or pyruvate, using only a change of temperature.
Perhaps it is possible to put a weak acid outside of a mitochondria, and select the weak acid that changes the dissociation constant a lot with the temperature.
So when temperature is high it is mostly dissociated and the acidity is high and the mitochondria produces ATP. But after some time there is no difference in the concentration of H+ inside and outside of the mitochondria and the process stop.
Then reduce the temperature so the acidity outside is low, and the H+ inside the mitochondria can escape by other pores. (Perhaps we need to make some additional pores for this? The pores must be small and not very polar.) After some time, the concentration of H+ inside the mitochondria is low again. And we can repeat the cycle.
[/Weird example warning]
This will be painfully slow and painfully inefficient. My biochemistry level is too low to be sure this is possible, but at lest I think it is not theoretically impossible.
It is a small system, so I think it is a relevant comparison in spite it is very different of a graphene membrane in a vacuum chamber.
