Around here wood burning stoves are very popular, and in the places where they are sold you usually find some variant of a device which uses a Stirling engine to power a fan. A Stirling engine uses heat differential as its power source, so if you sit it on a surface that is appreciably warmer than the surrounding air, (Such as the top of a stove) it begins to run. They are very appealing to a certain type of person (Me) for their self-sufficient elegance.
These engines set me to thinking that if you had an engine of sufficient efficiency, a thermo-electric cell akin to the solar power photo-electric cells but which generated electricity from heat rather than light, you could have air conditioning which was self powered, moving the air in the room whilst robbing it of some of its heat. This could, if it could be made cheap enough, not only bring a little comfort to millions of people who haven’t got, or can’t afford to use, an electricity supply. It could also have the significant beneficial effect of reducing the use of traditional air conditioning which uses enormous amounts of power and really only transfers heat from the inside of a building / vehicle to the outside rather than converting the heat into another form of energy.
So far so Sci-Fi, but reading this :- http://www.wired.co.uk/news/archive/2012-03/09/230-percent-efficient-leds makes me think it may be nearer then we would at first expect. Of course an LED is not a motor, (And it needs an external power supply to engender the effect) but there are other ways than motors and fans to move air (The Dyson Air Multiplier for one) so maybe it can be made to happen.
I can’t, off the top of my head, think of any other solution to a problem that would use the problem itself as its source of power so elegantly. A virtuous circle indeed and pure poetry.