If I have, say a gallon or two of warm water, is there a way to transfer that heat into an ounce of boiling water? (With heat syncs or something?)
You would need a heat exchanger, like what you would find in an air conditioner. A heat exchanger uses the phase changing properties of a refrigerant to move heat from cold place to a hot place. Essentially, it expands the refrigerant into a gas that is colder than your large water volume, so that the large water volume can dump heat into it. Then the refrigerant is compressed into a hot liquid so it can transfer the heat to your smaller water volume. The range of temperatures you can achieve is limited by operational temperature of your refrigerant, and the heat exchanger does require energy, so its probably not very feasible.
No, not without some complicated set up that would make you question why you just didnt heat up the tiny bit of water on its own as opposed to building a complicated setup. What are you trying to do?
Laws of Thermodynamics still apply. You can't heat something from a colder source without adding energy. Even a refridgerator needs energy. Quite a bit.
The answer is no. It is impossible to raise the temperature of any amount of any substance, by adding any amount of a second substance which is at a lower temperature, no matter how much equipment you use. With perfect heat transfer and perfect mixing the highest temperature you can ever achieve is the temperature of the coolest substance in the mix, unless you add supplemental heat.
If there's no complex set ups and containers for the water, you cannot add heat to a boiling water anymore. The boiling water is already in its specific heat level. You can no longer add more heat because it will evaporate. But for the sake of mind game, it could be done using a very complicated set up that will seem to break the law of thermo-dynamics. Search the net about SOLAR BROILER (?) Maybe it can give you an idea.
