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How can my dragon convert heat to usable energy?

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In my world, dragons are big, and require lots of energy just to support their massive bodies and huge brains - and of course, to fly, using enormous oversized wings. Rather than causing mass extinctions when they have breakfast, I want my dragons to gain part or all of their energy needs by other means.

I would like to keep these dragons as classic as possible, so I had the idea that they should make lairs inside of volcanoes, and have the ability to hibernate in these lairs. They would sustain their organism on the geothermal heat that's plentiful in their environment. Efficiency is not a huge concern, as the dragons can have a fairly arbitrary amount of heat in the lair, and equally arbitrary energy needs when hibernating.

But how do they turn heat into energy that their organism can use? As far as I can tell, even bacteria near hydrothermal vents don't use the heat for energy, but rather "eat" sulfur compounds ejected by the vent. I do not particularly care if these dragons are warm-blooded, cold-blooded, reptilian, mammalian, whatever, so long as they are organic beings and don't require magic or particularly intense hand-waving.

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Try thermosynthesis.

Thermosynthesis is a hypothetical mechanism, usually applied to the RNA world theory for the rise of life on Earth. In a nutshell, it states that an organism could function as a heat engine, where thermal energy is gained from a heat reservoir - in this case, a volcano.

Thermosynthesis allows you to synthesize molecules via anabolic pathways; it's not too far a conceptual leap to imagine that this organism could store chemical energy through thermosynthesis in the form of sugars, and from there, use a mechanism similar to respiration to unlock the energy from those sugars. More specific details can be found in an answer by Dubukay.

Would it be evolutionary feasible? Well, that's another question entirely. From an evolutionary perspective, it would be unlikely that thermosynthesis would be selected as an animal's method of energy generation, because a primarily thermosynthetic organism would need to stay close to the source of heat. Thus, it doesn't make sense for the creature to be mobile. So you'd probably see something less complicated - like, say, a fungus - evolve first.

Perhaps an animal of sorts would arise to use thermosynthesis, but only on a world where most other forms of energy generation are impractical or impossible. It has been suggested that thermosynthesis could be handy on a world that's cold, where the creature would take advantage of the temperature difference between the atmosphere and frozen ground. This is sort of the reverse scenario; the ground is hot and the air is comparatively cold.

I think thermosynthesis is the closest you can get to what you're looking for. The odds are against such a creature, but it's not impossible at all.

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