Ah, yes, parameters... Well, seeing as this is a concept that somebody posted in this thread that I happened to like, I'm not really too fuzzy about exactly what the parameters are, to be honest. I will do my very best to describe them as far as I see the character, so far:
I don't really like the "doing something to the sun" approach, that just seems powerful in a boring kind of way. Basically no ramp-up, just "take heat from sun, win". Same thing with "move heat into an area faster than its particles move out", at least not without some technological kind of help (doomsday machine?).
I like the "taking heat from stuff, like breathing"-idea. I'm not sure if the heat can build up infinitely, or if some dissapates after a while (giving the character a need to have a really big source of heat to "feed" off of before he wants to do anything really fancy, or having a continuous source of heat).
I dislike the "turn people's brains off" idea on a purely game and story-level - too much win for too little effect/fun/possibility to evade. So, maybe he has to see or somehow percieve the things he takes heat from? Touch? Maybe he can do the brain-thing if he's close enough (i.e. hugging distance)?
I like the idea of him walking into burning buildings as a way of building up heat. I don't know whether he should be able to do the same with a volcano, though - maybe he can only absorb so much heat per time, and a volcano is simply too much? Again, technology could possibly help with that, like an insulating suit that helps him absorb the heat somehow? (If he could, why hasn't he destroyed the earth yet?)
In general, I think I would like some combination of parameters that doesn't allow him to instantly win without needing some kind of Master Plan and/or Doomsday Device, but still being very powerful if using his powers creatively. Just below instant world-shattering, maybe?
I'll try to answer your points, Frank:
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How much heat?
Not sure, see above.
How fast can the heat be transferred?
Not sure. Touching the object in question probably makes it go faster, though.
How obvious is it that this is happening?
Depending on how much he's currently taking, i.e. ranging from "not at all" to "I should have brought my winter jacket".
How far away can the heat be moved from its source?
Probably not that far, unless stored inside him. I'm thinking maybe 10 meters? His range is probably not as good as the speed and amount of heat he can transfer.
What maximum and minimum temperatures limit the transfer?
Not really sure what this means. If it means how cold something can get before he can't take any heat from it again... probably some way above 0 K, at the least. Maybe -100°C, or somewhere along those lines? Maybe -40°C is more "realistic"?
How big or small of an area can the heat be transferred from?
Probably all of the immediate area around him, i.e. same range as above. Not sure if the heat transfer is uniform or faster closer to him or not.
How big or small of an area can the heat be transferred to?
Not sure. Heating himself/his hand up and touching somebody to damage them is probably a lot easier than targeting somebody else's hand and heating that up (without touching it, that is). Actually, I think touching stuff makes whatever he wants to do be easier for him, in general.
How rigorously do you have to define and understand and perceive the two targets?
See brain discussion above.
Can you move things against their entropy gradient at all? Or do you just act as a conductor from one place to another?
I'm not a physicist, so I'm not really sure what moving things against their entropy gradient means...
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All of you guys seem to be quite a lot more versed in heat-transfer mechanics than I am, so I don't want to set any parameters I make in stone - I'd rather have well-thought out parameters given to me by what idea of the character I'm trying to convey. Granted, re-reading what I just wrote doesn't really make it all that clear...
