The measure of a cold magician is not really how low they can bring the temperature. As you say there's a hard limit. They call it "absolute" zero for a reason. The real measure of a cold magician is "how much heat can they absorb?". If they can absorb more calories of heat than Fire-guy can generate then they win a magical arm wrestling competition regardless of how cold they can make things. In the end all they need is "cold enough to keep themselves from being burned and cold enough over there to freeze their opponent". Or you know keep your razor sharp icicles from melting as they skewer the target.

Unless you want laser wizards or some other form of mages that emit huge amounts of radiation, there's going to have to be something "magical" going on to get any kind of usable fireball. A fire mage whose ability is that they can heat the palms of their hands to arbitrary temperatures will still have to deal with the limits of thermal conductivity, and even after that they're just going to heat the air to a plasma that'll dissipate before it gets anywhere -- they're going to burn themselves a lot worse than any target more than a few feet away.

Given that, it's actually quite a lot easier to solve this issue at the magic level than the physics level. In order to transmit this magic power, suppose there's an as-yet-undiscovered boson, which I'll call the handwaveon. Handwaveons interact with matter by binding to it, transferring energy until their own energy drops to zero. They come in two types:

• "up" handwaveons (let's call them flameons), which increase the energy of the matter they're bound to while reducing their own energy by a like amount


• "down" handwaveons (let's call them freezeons), which reduce the energy of the matter they're bound and their own by equal amounts

When a cluster of handwaveons passes through air (or any other medium), those on the surface interact with the matter as described above, ablating the cluster until it dissipates (or collides with enough matter to bind all the remaining handwaveons).

In the case of flameons, this superheats the air, creating a plasma that releases photons all across the spectrum -- something an unsophisticated observer would describe as a fireball. In the case of freezeons, this supercools the air, freezing the water in it (and the air itself), creating an effect one would be tempted to describe as an iceball. However, these aren't fire or ice, and their respective heating and cooling abilities aren't limited like those of matter.

At this point, mages are just people with some (learned or innate) ability to gather (or create) handwaveons and handle them without binding to them (maybe we'll need another boson -- a handholdon? -- to carry a force analogous to magnetism, which repels handwaveons of a particular flavor). Whatever allows them to do this could either be exclusive to a particular flavor of handwaveon (e.g. opposite handholdons annihilate), or it could just be that they need to be handled in different ways so the odds of someone being able to handle both are really low.

But bosons are way too deep into physics for a fantasy story!

Low-tech societies (and the average people from high-tech societies) don't go around talking about broad-spectrum electromagnetic radiation emitted from the naked fusion reaction at the center of the solar system, they talk about sunlight. It would make perfect sense for a society without knowledge of fundamental particles to interpret handwaveons through the lens of more common phenomena. People may talk of "magefire" or "wizard's ice", and describe them as hotter/colder than anything else in the world. The characters (and even the readers) don't need to fully understand the physics as long as they can figure out the general rules for how it works.

One handwave you could use would be to give ice magic the ability to magically increase the specific heat of an object while it is chilled. The result would be that the iced object would act like a thermal reservoir. A hot object with a reasonable normal specific heat could naturally get as hot as expected, glowing hot if needed, but would have a similar heat capacity to a cold object which had a supernaturally increased specific heat.

It would still have to chill the environment slowly, because it's temperature would still not be able to go below 0K, but it would be able to chill much more massive objects because it would be a huge cold reservoir, due to its artificially high specific heat.

Then it's up to you and your magic system to explain why fire mages do not have a similar capability, but that's another question.

The fire mage is adding heat energy. The ice mage is substracting heat energy. Once the ice mage has subtracted all the heat energy he cant go lower... But he could try to do alternative things.

1: the ice mage adds potential negative energy. A fire mage (or just the outside world heating it up) requires more energy to get it from zero Kelvin to 1 degree higher than it would normally take.

2: after sucking all the heat energy out he'll start sucking potential energy. First the potential energy of say a fire mage who heats things up, then potential chemical energy, like the energy used to activate muscles or send nerve impulses...

3: why would the cap of absolute zero be less dangerous than heating something to thousands of degrees? We know that the colder things get, the weirder things get as well. Superfluids anyone? Or other effects? https://www.google.com/amp/s/phys.org/news/2015-06-absolute-molecules-exotic-states.amp

Heat is the movement of atoms. When you heat something up, you make the molecules and atoms vibrate more quickly. When you cool something down, you make them vibrate less. So "cold" magic is really the magic of stillness.

When you use a spell to cool something, what you are doing is applying a force to each molecule equal and opposite to it's current movement - it vibrates up, you push it down; it goes right, you go left; etc.

Heat is the opposite. Your fire mage heats things by rapidly shaking them, increasing their thermal energy. So when they fight an ice mage, what is actually happening, is one of them is trying to hold things still while the other is trying to move them.

You've got, I think, a bit of a conceptual error as well. Absolute 0 is NOT just -270 degrees. What it actually is, is no movement. You can make something move less than not at all; you can't make something colder than no heat. The colder you get, the harder it is to make things colder. Raising the temperature from 10 to 20 degrees takes the same efforts as 100 to 120; you're just pumping energy into it. But lowering the temperature to 0 is a whole different thing, because you are trying to hold every molecule perfectly still.

So fire and ice are not just opposites - they are actually different magical disciplines, requiring entirely different skill sets.

This also lets you use cold for more conceptual things, like 'freezing' light to make it dark (with a burst of light when you let it go), or stilling the air to cancel out sound waves.

If all you want is an unlimited power-scale in both directions, the solution is simple:

Use a logarithmic scale for temperature.

So a good icemage can cool down the temperature by a factor of 3

(So from 20°C = 290K to 290/3 = 95K = -170°C)

while a good fire-mage can increase the heat by a factor of 3

(So from 20°C = 290K to 290*3=870K = +600°C)

This scale is unlimited in both directions, so a great mage could change the temperature by a factor of 10 and a godly creature maybe by a factor of 100. Just like in reality the point of absolute zero is practically unreachable (like an absolute vacuum in macroscopic size, or light-speed particles with a mass) - the closer you get to zero, the harder it becomes to get just a little closer. It is like with pureness of gold, you can reach 99,99% pureness, but to reach 99,999999% pureness are worlds of difference.

So the more powerful your ice-mage becomes the closer he can get to absolute zero, which is a mystically frightening state. Like other answers have stated, it means no movement of particles whatsoever. 0K is a state which no matter has ever reached in the real world and which breaks all kind of physics formulas. Approaching it is like approaching a black hole, so while an impossible godlike factor-10.000 firemage can reach 3 Million °C (almost as hot as the suns corona) - a factor-10.000 icemage can reach 0,029 K, which is a hundred times lower than the temperature of outer space. Some gases start to show crazy properties like super-fluidity, super-conductivity and more at these temperatures, as you create in principle a macroscopic coherent quantum wave.

With a little bit of handwaving you can create all kinds of crazy effects the closer you get to absolute zero.

TL;DR: Don't underestimate just how impossible badass absolute zero is.

You could view ice/fire magic as a means of transferring energy, and cap the rate at which energy can be transferred. Changing the temperature of 1kg of material X by 10 degrees will result in the same magnitude of energy change, whether that change is positive or negative. An ice mage could lower the temperature of an object by 100 degrees, but if a fire mage wants to raise the temperature by 1000 degrees, he needs to be 10 times as powerful.

One issue is that this nerfs the fire mage side of things, as an ice mage could hit absolute zero, while an equally-powered fire mage couldn't even start a campfire in the same amount of time. Maybe fire mages are generally more powerful, or maybe there's some conversion factor that heating an object requires only half the magical energy as cooling it, or perhaps it just takes fire mages longer to cast their inferno spells. I don't see any way for ice mages to hit temperatures below absolute zero without ignoring physical principles.

Is there any way to make ice magic work, bringing it to negative degrees in the range of thousands, while still having it follow the laws of physics, even if loosely?

Well, no. According to the kinetic model, at any temperature corresponds a certain velocity of the molecules in any substance. At the temperature of 0 K the molecules are still. Once they are still they cannot get more still. Thus it's meaningless to talk about negative absolute temperatures.

After all, a being that could produce fire of several thousand degrees easily wouldn't really be bothered by a mere -270 degrees.

That's not exactly true. As said before, at 0 K molecules are perfectly still. According to Heisenberg indetermination principle, one cannot known both position and velocity of a molecule with arbitrarily high accuracy.

If you skew a bit the interpretation of this principle, since you know the velocity with perfect accuracy (you know they are still), you know nothing about their position, that is the thing at 0 K ceases to have a given position and could be everywhere.

In my World, magic is driven by energy from an other dimension. Mages can manipulate the flow of this energy between our dimension and this energy-dimension. With that, Fire-Mages can transfer energy in this world and use this to heat things up, making fire etc. Ice magic could work the complete opposite way.
While heat obviosly can be very destructive, cold can be so, too. Removing all the energy of something can be quite powerful. If something's burning, heating it up even more won't make that much of a different. But to remove ALL the energy out of a certain area would enable mages that are "infinitly powerful" to even freeze the sun, or whatever they please. Don't underestimate what 0°K means, just because it doens't consume things like fire does. Also Ice-Magic would be able to cancel out Fire-Magic completly in this concept.

Throwing an icicle at someone is indeed not really that frightening. Even if it's at nearly 0K, most of the damage it causes will be due to its kinetic energy, not due to frostbite. If that's your superpower, then you might label it rather as telekinesis than cold.

But what about the ability to shoot a "magic" cold ray which directly removes thermal energy from whatever it hits? Not through conductive heat transfer (the cold you feel when you hold an ice cube in your hand) but by actually changing the temperature of the target directly.

This could be a terrifying ability because low temperatures affect the speed of many physical processes. If you partially freeze your enemy, their powers will become weaker, even if they survive. Superman might barely feel warm if you heat him up to 300°C, but if you chill him to -273°C, he will lose most of his power. If you get him down to -273.15°C, then any physical processes in his body will come to a halt and he will be frozen solid.

Balance in the Force

When it comes down to it, cold magic and heat magic are just adding and subtracting energy. This energy comes from somewhere. What if this reservoir was depleted by the heat mages, and restored by the cold mages? If you have more and more powerful heat magic being performed, this reservoir becomes depleted, and it becomes harder and harder to do heat magic (draw energy out), but easier and easier to perform powerful cold magic (pull energy in).

As a twist, you could have combination cold/heat mages, who would operate on their own personal reservoir, drawing it in with cold magic and expending it with heat magic.

You could make an "Ice magician" someone that can lower temperatures and a "Fire Magician" someone that can raise temperatures.
(If you want realism we need to remember about the law of conservation of energy. Therefore, lowering a temperature somewhere will raise it elsewhere, and vice versa.)
The grade and speed of which would probably be determined by the raw power, ability, skill or level of training your Magician has reached or possess.

Just because we don't have a defined absolute limit on heat (minus taking all energy in existence and concentrating it into one point) doesn't make the lack of heat an inferior ability. On the contrary, you get to freeze molecules on an atomic level. Even cosmic beings of unimaginable power must consist of something, even if it's just a thought, thy are made of energy. Energy can be brought to a complete standstill at certain temperatures.

Adding energy, or "making hotter" excites molecules. Enough energy causes them to break apart, combusting in many cases. Lets face it, you wont feel a difference if you touch 1000 degrees Celsius or 10,000 degrees Celsius as long as its on the same point. You can, however, increase the range the heat radiates to.

Its possible to take this further, and define your "magic" as transfer of energy. In other words, a "Fire Magician" in essence draws energy from the area around an object into that object. The result is a hot object, but an area, or other objects in that area, devoid of their energy, leaving them cold, with the opposite being true for the other side of the coin.

Remember, you cannot have ice without water (but you can also solidify other liquid substances at low enough temperatures). You can use moisture from the air, but that will leave the air dry. This can be incorporated as a side effect and ability.
With fire its more complicated. You need fuel and a source of ignition. Many gasses and gas mixtures will combust without a source of ignition if heated enough. You will need an oxidizer (oxygen works well) as well as a second substance that likes to oxidize (like metal) to create a ball of flame. This could be anything from VERY fine sand to biological substances, coal, certain gasses and the like. For one of these Magicians, the ability to compress and depress air, create magnetic fields, "molecularize" items by snipping them up into small components, as well as air flow manipulation can assist in creating the form of fire they wish. Without all these components your "Fire Magician" could just as well walk around with an aerosol can, Molotov or paraffin-dipped charcoal and a lighter.

If you want scientific ice/fire magic, you could try to respect the laws of thermodynamics except for one thing: entropy would be the thing you could mess with.

Ice/fire mages could then be entropy modifiers: one adds entropy, the other removes it. To make it more scientifically sound, they could both transport it to/from the environment. In this sense, every fire mage is an ice mage (wherever something heats up, the environment (or something else) must cool down, conserving energy.

That could stall a fire/ice mage duel though, but could bring more creativity to these showdowns.

Absolute zero and planck temprature are irrelevant.

The final temperatures they can reach is not important, the important metric is how fast can they move heat (kinetic energy).

They are not creating hot or cold matter, they are adding or subtracting heat from matter, so it doesn't really matter how much heat that mass can hold. In essence your mages are either directly altering the kinetic energy of a mass or they are directly controlling heat transfer, either way the capacity of the material is irrelevant, since it only effects normal heat transfer which they are not using. What matters is how fast you can add or subtract that heat. The upper and lower limits on heat only matter if they are using normal thermal conductivity and temprature gradients, they are not. Your mages are directly adding or subtracting kinetic energy from the matter.

If an ice mage can subtract 5X Joules per second and an fire mage can add 5X Joules per second they cancel each other out.

We use lasers to cool down molecules in a similar fashion to how a telephone pole stops a speeding car. The result is a change in the form of energy, direction, etc. If your ice magic were to create some type of hawking radiation, put the heat energy elsewhere through entanglement, make noise or lights (non IR), etc., then there's no reason it couldn't work magitificallly. After a certain limit a Bose Einstein condensate approach could even make the process work better. E.g. Larger scale, faster, etc., as long as there is somewhere to put the heat. Say, the sun?

Ice Magic and Fire Magic are actually the same thing.

In both cases heat is transferred between a tangible, nearby object and a "reservoir" somewhere.

The parameters that define a magic user's power are then:

• how large, and how small, a volume he can affect. Transferring the same amount of heat to a smaller volume leads to a higher end temperature. Being able to transfer heat from a large volume allows freezing faster, without waiting for thermal conduction.


• how much heat they can transfer per unit of time.


• whether they are better at transferring heat to the reservoir or from the reservoir.


• the speed with which they can concentrate on a different volume.

Now, a fire mage transfers heat to a volume of air in such a way as to create a flame jet with a temperature of, potentially, thousands of degrees. Due to the low specific heat of air (around 1 kJ per cubic meter per K), this translates to the range of tens of millions of joules.

An ice mage capable of siphoning that much heat back to the reservoir would create a volume of air where the temperature is not particularly low, but it stays low. When the plasma jet from the fire mage crosses that volume of space, it is cooled and its heat returned to the reservoir.

It now becomes a matter of which mage can transfer more heat; whether the fire mage is capable of launching more attacks than the ice mage can cope with, or conversely whether the latter can create a heat-sucking, omnidirectional protective "dome"; and who tires first.

As an offensive capability, an ice mage can freeze the air around a person so that oxygen becomes a liquid. The victim can't breathe, and if the could, the air would kill him nonetheless.

A subtler method allowed by precise thermal control would be to deep-freeze air at some distance from the victim forming an air-tight "dome", and reheating proportionally the volume immediately adjacent so that nothing is noticed by the victim (except for a different quality of the sounds from the outside, perhaps). The carbon dioxide in the air would then precipitate as carbonic snow outside the "dome", which within a few minutes would start inducing hypocapnia and loss of consciousness, unless the victim was aware of the events and started breathing voluntarily.

Trade temperature change with area affected. Make it so a fire or cold mage has to split his effort between how hot or cold he makes something and the amount of area or mass they want to affect.

Make it harder to go to extreme lower temperatures than going to extreme high temperatures. That just means that cold mages are going to be more focused on area effect attacks. Or they store some of their attack energy into the cold to make it more persistent or to make it spread faster than actual conduction.

Ice can be used to trap or slow someone.

Also, heat magic is adding energy while cold magic subtracts energy. A well placed and angled shield may be able to deflect the majority of the energy of a fire bolt. where a bolt of cold may not "splash" off and suck energy directly from the shield (and the person with the shield).

So you end up with two different styles:

1. Fire magic which has higher straight damage output but is less
   flexible.


2. Ice/Cold magic which has less max damage but greater penetration,
   area and flexibility.

Consider making the Ice Mage able to work magic over a larger area than the Fire Mage. Perhaps fire magic only works up close, while Ice Magic might have a range of a hundred miles, even if at that range the ice isn't deadly. What if Ice Mage causes a killing frost for a single day over many many square miles of his enemy's grain fields, in June.

The power of Ice:
https://www.tk421.net/lotr/film/fotr/19.html

I'll take "following the laws of physics" to mean "maintain conservation of energy"

The Easy Way: Heat energy can be converted into magic energy

If heat can be readily converted to magic energy, this is incredibly trivial and actually quite terrifying in its versatility and power. You'd be able to freeze anything and everything by drawing magic energy from the heat of your target. Even if magic energy is unstable and must quickly convert to another type, you can simply exhaust heat anywhere else. Ice mages become incredibly overpowered from this method and would easily be able to completely neutralize and overwhelm fire mages as long as they had enough space to properly exhaust the absorbed energy.

The only way to balance this is to limit the absorption rate of the heat-magic conversion.

The Hard Way: Transfer the heat to something else

There's a reason refrigerators are a lot harder to make than heaters. Heat is a stubborn form of energy that doesn't like to convert easily into other types, while just about everything produces heat as a waste product. Basically, you need to somehow facilitate the transfer of heat energy to another substance in the environment, preferably using materials that are commonly available.

Possibilities:

• Refrigeration using wind and water. Unfortunately this process is far too slow to be viable, though if it could be somehow sped up might be viable.


• Only a handful of chemical reactions are endothermic and the chemicals are generally not readily accessible in most environments, so this isn't really a viable option.


• Laser Cooling. Essentially you have to emit radiation at the target 180 degrees out of phase with its natural heat vibration. This won't be practical at the scale ice mages will want to work at because it generally only works for individual atoms and molecules and requires continual knowledge of the particle's momentum.


• Radiation. Radiation is naturally the slowest type of heat transfer, but if this could be accelerated by, say 1000x, you could cool objects by forcing them to radiate heat faster. Unfortunately, if not kept under control, this would radiate deadly amounts of gamma radiation, cause sunburns from UV radiation, and lead to cancer from UV and X-Rays.


• Liquid Nitrogen. Nitrogen is plentiful in the atmosphere and can be compressed easily through magical means and kept in a pressure-resistant container. When released from pressure, a large amount of heat is absorbed from the atmosphere. The issue? Liquid Nitrogen takes a while to cool down to the ambient temperature before it can be used to cool things. This can be viable, but requires ice mages to prepare in advance and limits how much they can do to how much liquid nitrogen they can carry.


• Use magic to transfer heat from a solid to a liquid. Water in particular is a good choice due to its abundance and high heat capacity in liquid state (ammonia would be better but is harder to come by. Hydrogen gas would be an even better choice were it not for the fact that it reacts with oxygen rather explosively- obviously counterproductive). Throw water at your target and absorb the heat into the water, producing steam and a popsicle.
  
  
  
  
  • A similar idea could be used to produce icicles by boiling water using the energy from the rest of the water.
  
  
  • Naturally, this method is limited by the availability of water.
  
  
  • Air isn't really a viable option here due to its low density. In order to get a sufficient cooling effect, you basically need wind so fast that you're more likely to kill your target via suffocation. You might as well just be a wind mage at that point.